Computer Aided Design in Control and Engineering Systems. Advanced Tools for Modern Technology

410 Pages·1986·39.295 MB·English

by  P. M. Larsen

IFAC Proceedings Series
1985
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IFAC Related Titles
BROADBENT & MASUBUCHI: Multilingual Glossary of Automatic Control Technology
EYKHOFF: Trends and Progress in System Identification
ISERMANN: System Identification Tutorials (Automatica Special Issue)
COMPUTER AIDED DESIGN IN
CONTROL AND ENGINEERING
SYSTEMS
ADVANCED TOOLS FOR MODERN
TECHNOLOGY
Proceedings of the 3rd IFAC I IF IP Symposium
Lyngby, Denmark, 31 July—2 August 1985
Edited by
P. MARTIN LARSEN
Technical University of Denmark
and
N. E. HANSEN
NEH Engineering, Denmark
Published for the
INTERNATIONAL FEDERATION OF AUTOMATIC CONTROL
by
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Copyright © 1986 IFAC
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First edition 1986
British Library Cataloguing in Publication Data
Computer aided design in control and engineering systems :
advanced tools for modern technology : proceedings of the 3rd
IFAC/IFIP symposium, Lyngby, Denmark, July 31 - August 2,
1985. — (IFAC proceedings series; 1986, no. 5)
1. Automatic control — Data processing
I. Larsen, P. Martin II. Hansen, N. E. III. International
Federation of Automatic Control IV. International Federation for
Information Processing V. Series
629.8'312 TJ213
ISBN 0-08-032557-2
In order to make this volume available as economically and as rapidly as possible the author's typescript lias been
reproduced in its original form. This method unfortunately has its typographical limitations but it is hoped that
they in no way distract the reader.
Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter
COMPUTER AIDED DESIGN IN CONTROL AND
ENGINEERING SYSTEMS
Sponsored by:
International Federation of Automatic Control (IFAC)
IFAC Committee on Systems Engineering (SECOM)
Co-sponsored by:
IFAC Committee on Application (APCOM)
IFAC Committee on Computers (COMPUT)
IFAC Committee on Education (EDCOM)
IFAC Committee on Economic & Management Systems (ENSCOM)
IFAC Committee on Social Effects (SOC:EFF)
IFAC Committee on Theory (THEORY)
International Federation for Information Processing (IFIP)
Institute of Electrical and Electronics Engineers (IEEE)
Control System Society
Denmark Section
Organized by:
The Danish Automation Society
National Organizing Committee
Ole Feddersen
Niels Erik Hansen
Mogens Kümmel (Chairman)
Peer Martin Larsen
J0rgen Pontoppidan
Christoffer Svendsen
Holger B. Thomasen (Secretary)
International Programme Committee
H. Akashi, Japan P. Martin Larsen (Chairman), Denmark
S. P. Bingulac, Yugoslavia R. Lauber, Germany
A. van Cauwenberghe, Belgium G. G. Leininger, USA
Chen Zhen-Yu, China L. Ljung, Sweden
J. B. Cruz, USA M. Mansour, Switzerland
M. Cuenod, Switzerland D. Q. Mayne, UK
A. R. Curran, USA L. F. Pau, Switzerland
C. Foulard, France M. Paul, Austria
L. Fransson (IFIP), Sweden W. Schaufelberger, Switzerland
R. W. Gellie, Australia H. Austin Spang, III Junior, USA
G. Guardabassi, Italy B. Tamm, France
N. E. Hansen, Denmark A. Titli, France
R. Isermann, Germany M. Wozny, USA
Z. Kozar, Czechoslovakia K. J. Aström, Sweden
PREFACE
The object of the 3rd International IFAC Symposium on
Computer aided Design in Control and Engineering Systems
was to present and review the state of the art and the
trends in development of Computer Aided Design of Control
Systems (CADCS), techniques, procedures and concepts.
The two preceeding symposia were held in Zürich, Switzer-
land, 1979, and in West Lafayette, USA, 1982.
The 3rd symposium was held at the Technical University in
Lyngby, Denmark, from July 31 to August 2, 1985. 264 par-
ticipants from 33 countries attended the symposium, where
totally 11 papers were presented in 4 plenary sessions, 3
special sessions, and 15 technical sessions. Addiditiona1 -
ly the program included a panel discussion on standardi-
zation of software.
These Proceedings contain the text of 73 papers selected
for publication of the Symposium Editors and a summary of
the panel discussion. The papers are ordered alphabetical-
ly with respect to the first listed author within 15 sec-
tions corresponding to the headings of the symposium ses-
sions. An author index is included for^the convenience of
the reader.
The next symposium on the same subject is planned to take
place in 1988 in Beijing, PRC.
P. Mar t i n Lar sen
N. E. Hansen
VI
Copyright © IFAC Computer Aided Design in Control PLENARY PAPERS
and Engineering Systems, Lyngby, Denmark, 1985
EXPERT SYSTEMS TECHNIQUES IN A
COMPUTER-BASED CONTROL SYSTEM
ANALYSIS AND DESIGN ENVIRONMENT
J. D. Birdwell*, J. R. B. Cockett*, R. Heller*, R. W. Rochelle*,
A. J. Laub**, M. Athans*** and L. Hatfield****
^Departments of Electrical Engineering and Computer Science, The University of
Tennessee, Knoxville, TN 37996-2100, USA
**Department of Electrical and Computer Engineering, The University of
California, Santa Barbara, CA 93106, USA
***Laboratory of Information and Decision Systems, Massachusetts institute of
Technology, 35-406, Cambridge, MA 02139, USA
****Lawrence Livermore National Laboratory, PO Box 5504, L-156, Livermore,
CA 94550, USA
Abstract. We explore the fusion of expert control system analysis and design tools
into a prototype computer environment. The use of expert systems technology allows the
transfer of recent developments in control system design to users who may not be
experts in either theoretical developments or computer technology, in a vehicle which
provides considerable design flexibility. The technology does not provide as general a
framework as, for example, a command language based environment; however, in specific
applications, such as system modeling and linear control systems, it is useful and
appropriate. Criteria for such applications is that they have a mature theoretical
basis, and that the user interaction influence the sequence of the design process.
Keywords. Expert systems; computer-aided design; artificial intelligence; linear
systems; computer-aided system design; natural languages.
engineering to which the environment was to be
applicable would be sharply limited; the area
INTRODUCTION also had to be mature and well-defined to fit
within the expert system framework. Linear
The interaction of two areas is examined: the multivariable control system design was chosen as
development of knowledge-based systems, and the the application area. In particular, the
analysis and design of control systems. Our environment would emphasize the very recent
viewpoint is the result of work performed on the results using the robust properties of the linear
CASCADE (Computer-Aided Systems and Control quadratic regulator or the Kaiman filter, and loop
Analysis and Design Environment) project, transfer recovery techniques. This methodology
sponsored by the U. S. Department of Energy. allows synthesis of multivariable control design
The project attempts to generate new concepts in which satisfy frequency domain constraints on
computer environments for the systems community by robustness to unmodeled plant uncertainty, command
bringing together an interdisciplinary team of following error limits or disturbance rejection,
researchers from computer science, numerical and closed loop bandwidth.
analysis, and systems science. The emphasis of
this project is the examination of ideas relevant To address this problem, it was proper to use an
to the development of computer-based aides for the expert system which would sit over numerical and
systems community, rather than implementation of a graphical routines and provide a knowledgeable
specific design environment. The first phase of interface to the application of these routines. A
this project studied the issues which affect the system written over Prolog (Clocksin (1981)) is
design of a computer-based environment; this being used as the knowledge interpreter; this
phase is summarized in Birdwell (1983a, 1983b, paper describes how it is intended to work and the
1984). In the past year, the majority of this progress to date.
effort has focused on the feasibility of using
expert system technology to construct the user There is a clear distinction in the design of a
interface portion of a computer environment system which is concerned with highly numerical
relevant to system engineers. Some of this effort problems between the numerical calculations that
was reported in Birdwell (1985). the computer carries out and what the human wants
to see. The human is concerned with general
An expert system-based environment for aiding the descriptions (is this greater than that; have I
analysis and design of control systems has two met my design specifications?), while often the
primary features: The environment must have solid major concern in performing the numerical
numerical routines for the necessary calculations; calculations is to maintain the precision of the
it must also have knowledge of the design methods computation. Humans can quickly digest graphical
available, and be able to judge when and which to and conceptual data; however, the precise
apply to the particular problem. It was clear numerical values which gave rise to these are
early in the project that the area of system often difficult to digest. Thus, particularly for
1
2 J. D. Birdwell et al
numerically-based casks, there is a serious and graphics software. With the exception of the
requirement to provide a mechanism for translating command interpreter, these functions will be
the numerical calculations into human terms. available in any expert system-based environment;
here, the emphasis is on a clean, well-defined
This distinction gives rise to an important partition. A diagram of this generic environment
observation: If a human cannot handle large and its inceraction with the user and the computer
numbers of high precision numbers, there is no resources is shown in Fig. 1.
need for the machinery at the interface to
particularly good at that task. The machinery at USER
the interface needs to be good at the sorts of
things humans are good at; for example, providing EXPERT SYSTEM
illustrative sketches and discussing concepts.
This sort of task is matched by the abilities of
COMMAND INTERFACE
workstations and the current wave of symbolic
machines.
DRIVER PROGRAMS TO PERFORM SPECIFIC TASKS
This distinction implies the software for handling
the interface could be on a different type of BASIC SUBROUTINE LIBRARY FOR NUMERICAL METHODS, GRAPHICS
machine than that for handling the numerical
calculations. There can be a physical division of
labor: work-station versus work-horse. The types Fig. 1. Generic environment.
of task which the workstation needs to be capable
at include: There are valid reasons for separating the expert
system from the remainder of the environment.
1. symbolic manipulations, First, the command interpreter and
2. graphical displays, and numeric/graphics software could be an existing
3. interactive computing, software package. The expert system then assumes
the primary functions of data organization and
while the task that the work-horse needs to be repository for a knowledge base which specifies
good at is numerical calculations. how the software package is applied to specific
problem classes. If the expert system generates
In this environment, the role of the workstation commands to the package, over a defined interface,
is as a facilitator where the needs of a human can then little or no change is required in the
be handled adequately, rather than a place for existing code.
major computation. The expert system task
provides interaction with the user and arranges Second, many artificial intelligence/expert system
that the necessary programs be run and the results languages are not designed to efficiently
be appropriately presented to the user. Most manipulate numerical information in the quantity
important, the expert system must guide the user or accuracy required in a systems application.
through the steps of a design. If possible, this CProlog on the VAX is an example of this, where
should be done in a way which does not aggravate the floating point implementation is single
the experienced user, while allowing the precision. This requires the numeric, and
inexperienced user to have every step explained possibly graphics, processing to be implemented in
and thus learn from the process. another language; the designer is then forced to
provide an interface between the expert system
Finally, the expert system must not be so environment and the computation environment.
complicated to use that it requires a hefty There are exceptions to this generalization,
manual; it must be able to explain itself. A specifically, the Lisp Machine; however, one is
user should be able to do something almost still faced with the large amount of existing code
immediately, so that there is no lengthy learning written in other languages which is well-tested,
cycle, and so the program can be self contained. and applicable to systems analysis and design.
The CASCADE prototype expert system was developed The command interpreter is required as an
in DECIDE, which is an interface implemented in interface between the implementation language of
Prolog. It allows a problem solving method to be the expert system (Prolog in our application) and
programmed in such a way that the computer can numeric/graphics software (Fortran 77 and C). The
lead a user through a problem, while being alternative is to provide a direct linkage between
tolerant of user typing errors and the implementation language, which is usually an
misunderstandings. At the same time, it is interpreter, and the remaining software. The
capable of explaining its actions and any terms it feasibility of this is governed not onlv by the
uses. constraints of the machine environment (for
example, VMS or UNIX), but also by the details of
The DECIDE system is based on a decision the expert system implementation language. We
expression methodology, which allows a highly used CProlog, and avoided almost all modification
modular design of a knowledge base (Cockett of the prolog system. The single exception was
(1983a, 1983b)). The knowledge is clustered into the addition of a "system" function to CProlog,
self-contained subtopics, each of which can be which allowed the expert system to spawn tasks to
independently designed and implemented. The perform specific computations.
clusters are direct analogs of subroutines.
The expert system/command interpreter interface
can be implemented in a variety of ways. We
GENERIC ENVIRONMENT experimented with two approaches, and some
variants. The interface must provide two
functions: commands to perform a specific task,
The designs we investigated in the CASCADE project
and locations of any data which is required or
can be generalized into a structure for a generic
will be generated. Thus, one interface is for the
computer environment for systems analysis and
expert system to run a specific program, with data
design. This environment is primarily a partition
in named files. The program is expected to place
of the software into three distinct parts: The
results in a file which is mutually agreeable.
expert system; a command interpreter, which
This is the first interface we implemented, and is
serves as an interface between the expert system
found in other packages. There are some reasons
and numeric or graphics software, and the numeric
that this is not an ideal implementation;
Expert Systems Techniques 3
notably, it is not efficient. For example, on a Perhaps the major lesson from that experience is
VAX using the VMS operating system, this requires the importance of face-to-face discussions between
spawning a process in which to run the program, the knowledge engineer and the expert. As with
and VMS has a significant overhead associated with any computing language there will be different
task creation and termination. We experimented ways of approaching problems. The task of the
with a multi-tasking interface between the knowledge engineer is to implement in the chosen
subsystems, with the expert system passing language the subtleties of the knowledge. The
commands to the command interpreter through VMS
eventual form may not be quite what the expert
mailboxes (the VMS analog of UNIX pipes). This is
imagined nor indeed what the knowledge engineer
rather difficult to set up under VMS, due in part
expected.
to the large number of file or device attributes,
and rather easy under UNIX.
DETAILS
We did learn a valuable rule in designing this
interface; regardless of the manner in which the
In the following sections, we present an overview
expert system/command interpreter interface is
of some of the implementation details of the
done, the expert system should be in complete
CASCADE prototype software.
control of the location of information. This is
required if the expert system provides a mechanism
for backtracking, or going back to a previous
Purpose and Area of Application
point in the design and allowing the user to
choose a different path of decisions. When
The prototype CASCADE provides a methodology for
backtracking is performed, the expert system
the design of multivariable linear control
should avoid recomputation of information which
systems, using linear quadratic regulator (LQR) or
has been previously obtained. The expert system
Kalman-Bucy filter (KBF) designs to meet frequency
associates the location of computational results
domain measures of performance, and loop transfer
with the information those results depend upon.
recovery (LTR) techniques to preserve the
If the command interpreter is forcing an arbitrary
performance characteristics in the full design.
(to the expert system) naming convention for
An overview of this design method is available in
results, then it becomes very hard for the expert
Doyle (1981). The primary criticism of the linear
system to maintain these associations. For
quadratic gaussian (LQG) methodology is that the
example, if the name for a particular result is
cost functional which is being optimized and the
derived from that of one input parameter, then if
noise characteristics which determine the filter
the expert system recomputes the results using a
design are unrealistic. This criticism is not
different set of input data which contains the
valid for LQR/LTR or KBF/LTR designs. The
same parameter value, the expert system must
selection of a controller is not based on an
provide an ad hoc resolution of the duplicate
optimization problem; rather, the LQR or KBF
output names. This could be done either by
algorithm is used as a vehicle to construct a
renaming an existing file, or destroying existing
design which satisfies frequency-based performance
information and recomputing the information later
constraints. These constraints are intuitively
if required. Either method is more difficult that
satisfying, and are direct extensions of standard
forcing the command interface to rely on the
measures of performance used for
expert system for all names.
single-input/single-output (SISO) compensator
design (gain margin, closed loop bandwidth,
disturbance rejection, command following error,
DEVELOPMENT TEAM robustness to plant perturbations).
The most difficult part of expert system
development is the integration of expertise in the
Theoretical Details
application area (systems analysis and design)
with expertise in knowledge base design. This
We assume the plant for which we wish to construct
expertise can not generally reside in an
a controller is a linear time-invariant (LTI)
individual due to the depth of knowledge required
multi-input/multi-output system represented by the
in both areas. The structure of the decision
unknown transfer function G(s). The design
process must be designed by the systems analysis
problem is to find a controller of the structure
and design expert; however, that structure is
given in Fig. 2 which satisfies a set of design
influenced by the characteristics of the inference
constraints. We will discuss the characteristics
engine.
of the design problem in three parts: The plant
and plant model, the controller structure, and the
Our solution was the formulation of a
performance criteria.
"meta-language" which was used for communications
between the implementor of the inference engine
and the systems engineer. The knowledge base was ^ SERIGRNOARL
written in this meta-language, and its form was COMMAND V) PLANT
modified by the knowledge engineer. INPUT
CONTROLLER
This was not completely satisfactory for a number SIGNAL
of reasons. First, the inference engine DECIDE
was so new that there was no mature knowledge ESTIMATE
REGULATOR
engineer. This eventually forced the designer of
the engine to devote most of his efforts to
knowledge base development. Second, the Fig. 2. The controller structure.
understanding that the expert system was based on
a type of decision tree methodology let the The plant and plant model. We assume the designer
experts to produce a rather lengthy (and difficult has a model, GO, of the plant, which can be
to digest) manuscript. It was not until the described in state-space form as a LTI system:
knowledge engineer and the experts got together
that it was discovered that there were fundamental x(t) Ax(t) + Bu(t) (1)
symmetries (of which the expert was well aware) in J£(t) Cx(t) (2)
the design technique. These could be heavily
exploited to significantly reduce the knowledge where x is the state vector and has dimension n, u
base, and the knowledge engineering effort. is the input vector and has dimension m, and y^ is
4 J. D. Birdwell et al.
the output vector and has dimension p. This is a .Since the system is MIMO, the location of the
general representation of a linear time-invariant break at which L(s) is calculated is significant.
(LTI) multi-input/multi-output (MIMO) system which The design must be based on performance at either
has finite bandwidth. the input or the output of the plant. The choice
of the point of analysis is made by the expert
The design methodology assumes the accuracy with system, as described in the next section. The
which the designer can represent the plant is high return ratio at the plant input, LI(s), is:
for low frequencies, and decreases monotonically
with frequency. The error in the model's LI(s) = K(Z{s)~l + BK + FÇr^CZisjB. (5)
representation of the plant is modeled by a
transfer function D(s): where
G(s) = G0(s)U + D(s)) or (I + D(s))G0(s) (3) Z(s) = (si - A)"1 (6)
The methodology assumes the designer knows some The return ratio at the plant output, LO(s), is:
information about the uncertainties of his model,
at either the input or the output of the plant, in LO(s) = ÇZ(s)BK(Z(s)_1 + BK + FC)"^ (7)
the form of a frequency-dependent bound.
II D(jw) II < h(w) for w > 0 (4) The three performance measures are calculated from
either LO or LI. The robustness performance
measure is the minimum singular value of the
The controller structure. The feedback controller inverse return difference transfer function.
can be decomposed into two parts, a filter, which Either the input node or output node return ratio
is based on a nominal plant transfer function, and is used, depending on whether the plant
a regulator gain. This structure is sometimes uncertainty bound is known at the plant input or
referred to as a model-based controller (MBC). plant output, respectively. The command following
There are two free parameters of the controller error performance measure is the minimum singular
design, as shown in Fig. 3: The regulator gain value of the return difference transfer function.
matrix G, and the filter gain matrix F. The The input node return ratio must be used. The
design objective is to select these matrices so disturbance rejection performance measure is the
several frequency-domain performance criteria are minimum singular value of the return difference
satisfied; these will be developed in the next transfer function. The output node return ratio
section. must be used. All MOP's are evaluated on the
imaginary axis; details are in Laub (1979, 1981),
CONTROLLER PLANT and Klema (1980).
Construction of performance constraints. For each
design, two performance measures are used. We
will assume the design is based on command
following error and robustness measures in the
following discussion; thus, the design is
evaluated at the input node of the plant. The
development using the disturbance rejection
performance measure is similar.
In CASCADE, constraints are divided into two
classes: low and high frequency constraints. Low
frequency constraints are specified as minimum
values for the minimum singular value of the
return difference for all frequencies below a
Fig. 3. Closed-loop structure. specified frequency. These constraints correspond
to a minimum allowable loop gain for the
Performance criteria. Three measures of closed-loop system. High frequency constraints
performance are used to evaluate a MBC design: are specified as minimum values for the minimum
Robustness to unstructured perturbations in the singular value of the inverse return difference
plant dynamics, command following performance, and for all frequencies above a specified frequency.
rejection of disturbance signals. Robustness is Each candidate design is evaluated against these
defined as the maximum bound h(w), such that for constraints, and the expert system advises the
all perturbations in the plant dynamics D(s) which user to make appropriate adjustments in design
satisfy Eqn. (4), the closed-loop system is parameters if the design does not meet certain
stable; it is developed in detail in Lehtomaki constraints.
(1981) and Doyle (1981).
The design proceeds in two stages. The first
Command following error is defined as the signal stage creates either a LQR or a KBF design,
at node R in Fig. 3. Disturbance rejection is depending on the node selected for analysis. This
measured as the minimum attenuation along the path design is developed to satisfy the performance
from d(s) to y(s). These concepts are developed constraints, and may require some iteration. The
fully in Safonov (1981). second stage (LTR) uses the remaining component of
the controller (filter or regulator) to force the
For this design methodology, the engineer must return ratio of the closed-loop system to
decide which performance measure is more approximate the return ratio of the initial design
important: command following error or disturbance (LQR or KBF).
rejection. Either performance measure is then
traded off against the robustness measure in the LQR/KBF design properties. The expert system uses
design. the performance constraints to calculate a maximum
desirable crossover frequency for the return ratio
These measures of performance are determined by of the closed-loop design. It then performs
the characteristics of the system return ratio, either the regulator or the filter design based on
L(s), the return difference, I + L(s), or the information provided by the user about what type
inverse return difference, _I + L(s)**(-1), as in of performance measure is more important, whether
Safonov (1981). or not there is a preference for either design,
Expert Systems Techniques 5
whether sufficient information exists to construct At this point, there is sufficient information to
either design, and the number of plant inputs and develop an initial LQR design. This design is
outputs. The regulator requires the design evaluated by the expert system using the algorithm
evaluation to be at the plant input node; the in Laub (1981), and calculating the gain margin by
filter requires it to be at the plant output node. which the design satisfies the performance
The regulator and filter design methods are constraints. The minimum gain margin and the
symmetric; thus, the regulator-based design is corresponding frequency for each measure of
developed here. A detailed development is given performance is calculated and provided to the user
in Lehtomaki (1981). for evaluation.
The regulator constructed in the first stage is If a constraint is violated, the corresponding
evaluated using its return ratio. The return gain margin is negative. If all the constraints
ratio of the regulator is equivalent to the return are satisfied, the expert system recommends
ratio of a closed-loop system consisting of the proceeding to the next stage of the design, loop
plant with full state observation and the transfer recovery. If a constraint is violated,
regulator implementing full state feedback. The the expert system recommends an adjusted value for
regulator's return ratio is the return ratio of the parameter t and allows the user to iterate the
the closed-loop system in Fig. 3, calculated at design. Under certain conditions, the expert
node R, and given by system recommends the system be balanced at high
frequencies if it had been balanced at low
LR(s) = KZ(S)B (8) frequencies.
The first step in constructing the regulator is to The singular value and gain margin information is
"balance" the singular values of the return ratio presented to the user both numerically and
of the LQR design at either low or high graphically. The user can select views of the
frequencies. This balancing operation causes the minimum and maximum singular values of the return
singular values in the selected frequency range to ratio, return difference, and inverse return
be approximately equal (equal in the limiting case difference versus frequency. The constraints
as frequency tends toward either zero or which are appropriate for each graph are displayed
infinity). The singular values are balanced by with the singular value curves. A sample plot of
constructing a state weighting matrix with the the return ratio singular values is shown in
following structure for the LQR cost functional: Fig. 4.
£ = ETP (9) min and «ax singular values for G(jw) FILE:new.lqr
T 90
where for high frequency balancing P is the least
squares solution to the equation PB = I and for
low frequency balancing it is the least squares
solution to the equation BP = A.
The choice of high or low frequency balancing is
decided by the expert system, based on input from
the user and an estimate of the width of the
window of frequencies within which the crossover
frequencies of the closed-loop system return ratio
must lie. The expert system prefers to use low
frequency balancing, since by equalizing the
singular values at low frequencies, the command
following error performance measure is maximized.
If, however, the window for the crossover Fig. 4. Singular value plot of return ratio.
frequencies is narrow, then it may be easier to
satisfy all the performance criteria by using high Recovery of the LQR/KBF design properties. The
frequency balancing, since the singular values objective of stage one of the design process is to
near the crossover frequency are grouped more satisfy the performance criteria using only a
closely. In CASCADE, the threshold for using high regulator or filter design. These properties
frequency balancing is a window which spans less exist at either node R or node F in the system.
than a factor of four in frequency. This is a The objective of stage two of the design process
heuristic test. is to construct the remainder of the controller
loop such that the properties of the system
Given the matrix from the balancing procedure, if evaluated at node I (0) approximate those at node
the control weighting matrix parameter required by R (F). If this approximation is sufficiently
the LQR design is set to R = tl, then the maximum close, then the final design will satisfy the
crossover frequency of a singular value of the performance criteria.
return ratio of the regulator system, LR(s), is
approximately The loop transfer recovery (LTR) procedure only
works if the plant has no non-minimum phase zeros
wcmax = σ (PB)//t (10) "near" the bandwidth of the initial design. Note
max — that non-minimum phase zeros represent a
An important property of this LQR design is that fundamental limitation on system performance, and
near the crossover frequencies of the singular are not a limitation of this design methodology.
values a plot of either the maximum or minimum Neglecting this limitation sacrifices the
singular value versus frequency has a slope of robustness properties of the controller.
approximately -20dB/decade. This property allows
a rough approximation of the minimum crossover The regulator return ratio can be recovered at the
frequency required to satisfy the low frequency plant output node, 0, given that the plant is
constraints and the maximum crossover frequency minimum phase over a suitable frequency range and
which will allow the high frequency constraints to that the plant has at least as many outputs as
be satisfied. The expert system uses this to set inputs, by constructing a Kaiman filter using
a desired value for the maximum crossover specific noise covariance matrices parametized by
frequency, and solves for the parameter t. a positive number, q, As q goes to infinity, the
return ratio at the plant output node approaches
the return ratio of the regulator design