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Conventions
The following conventions are used in this manual:
[ ]
Square brackets enclose optional items—for example, [response].
»
The » symbol leads you through nested menu items and dialog box options
to a final action. The sequence File»Page Setup»Options directs you to
pull down the File menu, select the Page Setup item, and select Options
from the last dialog box.
This icon denotes a note, which alerts you to important information.
bold
Bold text denotes items that you must select or click in the software, such
as menu items and dialog box options. Bold text also denotes parameter
names.
italic
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to a key concept. Italic text also denotes text that is a placeholder for a word
or value that you must supply.
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Chapter 1
Document Organization...................................................................................1-1
Bibliographic References ................................................................................1-2
Related Publications........................................................................................1-2
Chapter 2
wcbode( ).........................................................................................................2-9
pfscale( )..........................................................................................................2-16
optscale( ) ........................................................................................................2-17
Reducibility....................................................................................................................2-17
Worst-Case Performance Degradation (wcgain) ...........................................................2-18
Conversion to a Stability Margin Problem......................................................2-18
wcgain( )..........................................................................................................2-19
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Contents
Chapter 3
System Evaluation
Singular Value Bode Plots............................................................................................. 3-1
L Infinity Norm (linfnorm)............................................................................................ 3-3
linfnorm( )....................................................................................................... 3-4
Singular Value Bode Plots of Subsystems .................................................................... 3-7
clsys( )............................................................................................................. 3-10
Chapter 4
Controller Synthesis
Weight Selection............................................................................................. 4-5
Linear-Quadratic-Gaussian Control Synthesis.............................................................. 4-14
fsregu( )........................................................................................................... 4-14
fsesti( )............................................................................................................. 4-16
Frequency-Shaped Control Design Commands.............................................. 4-17
Loop Transfer Recovery (lqgltr) ................................................................................... 4-22
lqgltr( ) ............................................................................................................ 4-23
Appendix A
Appendix B
Technical Support and Professional Services
Index
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1
Introduction
The Xmath Robust Control Module (RCM) provides a collection of
analysis and synthesis tools that assist in the design of robust control
systems.
This chapter starts with an outline of the manual and some use notes. It
continues with an overview of the Xmath Robust Control Module (RCM)
functions.
Using This Manual
This manual provides complete documentation for all the RCM functions
examples.
Document Organization
This manual includes the following chapters:
•
Chapter 1, Introduction, describes the Robust Control Module (RCM)
and shows the RCM function structure.
tools and introduces the concepts of uncertainty, robustness, and
performance degradation in the framework of closed-loop systems.
interested in robustness analysis or performance degradation, which
are explained in the Stability Margin (smargin) section and the
Worst-Case Performance Degradation (wcbode) section. The
Advanced Topics section provides additional information but this
•
•
Chapter 3, System Evaluation, describes system analysis functions that
create singular value Bode plots, performance plots, and calculate the
L∞ norm of a linear system. This chapter should be of interest to all
users.
Chapter 4, Controller Synthesis, discusses synthesis tools in two
categories, H∞ and H2. This manual does not attempt to explain all
of the theory of H∞, LQG/LTR, and frequency shaped LQG design
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Chapter 1
Introduction
techniques. The general problem setup is explained together with
known limitations; the rest is left to the references.
Bibliographic References
Throughout this document, bibliographic references are cited with
bracketed entries. For example, a reference to [DoS81] corresponds
to a document published by Doyle and Stein in 1981. For a table of
bibliographic references, refer to Appendix A, Bibliography.
Commonly-Used Nomenclature
This manual uses the following general nomenclature:
•
•
Matrix variables are generally denoted with capital letters; vectors are
represented in lowercase.
G(s) is used to denote a transfer function of a system where s is the
Laplace variable. G(q) is used when both continuous and discrete
systems are allowed.
•
•
H(s) is used to denote the frequency response, over some range of
frequencies of a system where s is the Laplace variable. H(q) is used to
indicate that the system can be continuous or discrete.
A single apostrophe following a matrix variable, for example, x',
denotes the transpose of that variable. An asterisk following a matrix
variable (for example, A*) indicates the complex conjugate, or
Hermitian, transpose of that variable.
Related Publications
For a complete list of MATRIXx publications, refer to Chapter 2,
MATRIXx Publications, Help, and Online Support, of the MATRIXx
Getting Started Guide. The following documents are particularly useful
for topics covered in this manual:
•
•
•
•
•
•
MATRIXx Getting Started Guide
Xmath User Guide
Xmath Control Design Module
Xmath Interactive Control Design Module
Xmath Interactive System Identification Module, Part 1
Xmath Interactive System Identification Module, Part 2
Xmath Model Reduction Module
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Chapter 1
Introduction
•
•
•
Xmath Optimization Module
Xmath Robust Control Module
Xmath Xμ Module
MATRIXx Help
Robust Control Module function reference information is available in the
MATRIXx Help. The MATRIXx Help includes all Robust Control functions.
Each topic explains a function’s inputs, outputs, and keywords in detail.
Refer to Chapter 2, MATRIXx Publications, Help, and Online Support, of
Help feature.
Overview
RCM functionality is structured as shown in Figure 1-1.
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Chapter 1
Introduction
Analysis Functions
smargin
wcbode
wcgain
ssv
pfscale
optscale
osscale
lqgltr
Synthesis Functions
fslqgcomp
fsesti
fsregu
hinfcontr
singriccati
clsys
Utility Functions
linfnorm
perfplots
Figure 1-1. RCM Function Structure
Many RCM functions are based on state-of-the-art algorithms implemented
in cooperation with researchers at Stanford University. The robustness
analysis functions are based on structured singular value calculations.
The synthesis tools expand on existing LQG (H2) techniques (LQG/LTR
and frequency shaping) while adding new H∞ design functions.
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2
Robustness Analysis
This chapter describes RCM tools used for analyzing the robustness
of a closed-loop system. The chapter assumes that a controller has been
designed for a nominal plant and that the closed-loop performance of
this nominal system is acceptable. The goal of robustness analysis is to
determine whether the performance will remain acceptable if the plant
differs from the nominal plant.
Modeling Uncertain Systems
This section describes the method RCM uses to model an uncertain system.
The closed-loop system is modeled as a known or nominal closed-loop
system with input w and output z, together with k unknown or uncertain
transfer functions δ1(jω), …, δk(jω), as shown in Figure 2-1.
Uncertain Transfer Function
Known Nominal
System
q1
r1
δ1
w
z
q2
r2
δ2
Figure 2-1. Model of an Uncertain System
The following transfer functions are assumed to be stable:
δi( jω) ≤ li( jω)
(2-1)
where the li are given non-negative functions of frequency. This type of
uncertainty model is known as structured nonparametric uncertainties.
To describe this model, you also must describe the nominal closed-loop
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Chapter 2
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system, including how the uncertain transfer functions are connected to the
system and the magnitude bound functions li(w).
To do this, extract the uncertain transfer functions and collect them into a
k-input, k-output transfer matrix Δ, where:
Δ( jω) = diagonal(δ1( jω),...,δk( jω))
(2-2)
The resulting closed-loop system can be viewed as a feedback connection
of the nominal closed-loop system with transfer matrix H(jω) and the
uncertain transfer matrix Δ(jω). You describe your nominal closed-loop
system as a linear system with
w
r
z
input
and output
.
q
Note The signals r and q are not really inputs and outputs of the nominal system; r and q
show how the uncertain transfer functions connect to your nominal system. The signals r
and q each have k components.
You will partition H into the four submatrices,
Hzw Hzr
H =
Hqw Hqr
so that Hzw is the nominal transfer matrix from w to z, Hzr is the nominal
transfer matrix from r to z, Hqw is the nominal transfer matrix from w to q,
and Hqr is the nominal transfer matrix from r to q.
The magnitude bound functions li(jω) from Equation 2-1 are described
with the PDM delb:
ω1 l1(ω1) … lk(ω1)
DELB =
,
:
:
:
ωm l1(ωm) … lk(ωm)
Thus, a complete description of your system requires the system SysH
to represent Hjw and the response delbto represent the bounds.
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Chapter 2
Robustness Analysis
Stability Margin (smargin)
Assume that the nominal closed-loop system is stable. That belief raises a
question: Does the system remain stable for all possible uncertain transfer
functions that satisfy the magnitude bounds (Equation 2-1)? If so, the
system is said to be robustly stable. If the magnitude bounds are small
enough, the uncertainties will not destabilize the system; your system will
be robustly stable.
Roughly speaking, the stability margin of your system is defined as the
factor by which you can increase all the magnitude bounds li and still
number is larger than one (0 dB), then you know that there are no uncertain
transfer functions that satisfy the magnitude bound and destabilize your
system. Moreover, the number tells you how much more uncertainty your
system could tolerate than the given bounds li(ω). If the margin is less than
one, then there are uncertain transfer functions that satisfy the magnitude
bound (Equation 2-1) and result in an unstable system. In this case, the
margin tells you how much you must reduce the magnitude bounds before
you have robust stability.
More precisely, the stability margin at frequency ω is defined as the
smallest α such that the system can have a pole at jω, with the uncertain
transfer functions satisfying |δi(jω)| ≤ αli(ω):
margin(w) = min{ α| systems can have a pole at jω with magnitude bounds αli(jω) }
The stability margin also can be expressed as:
margin(w) = min{ α| det I – HqrjωΔ ≠ 0 such that |Δii| ≤ αli(α) }
Note The stability margin only depends on Hqr.
The margin often is expressed in dB. If the margin is greater than zero for
all frequencies, then your system is robustly stable. If the margin is less
than zero for some frequencies, then your system is not robustly stable.
In particular, there are uncertain transfer functions that satisfy the
magnitude bound (Equation 2-1) and cause the system to have a pole at
those frequencies where the margin is negative. This does not mean that any
δi values that satisfy the magnitude bound will destabilize the system: it
and destabilize the system.
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Chapter 2
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smargin( )
marg = smargin(SysH, delb {scaling, graph})
The smargin( )function plots an approximation to the stability margin
of the system as a function of frequency. For a full discussion of
smargin( )syntax, refer to the MATRIXx Help. The approximation is
exact if the number of uncertain transfer functions is less than four and
scaling="OPT"(optimum scaling).
In other cases, the approximation is generally considered to be extremely
margin that is less than or equal to the actual margin.
The smargin( )function counts the columns in delbto calculate the
number of uncertainties k. It then assumes that the last k inputs of SysHare
signal r in Figure 2-2, and the last k outputs are signal q. To create a Nominal
System, refer to the Creating a Nominal System section.
w
r
z
Known Closed-Loop System
H(s)
q
size k
size k
Figure 2-2. Nominal Closed-Loop System
Creating a Nominal System
To better understand how to create H(s) in Figure 2-3, you will examine
a SISO tracking system with three uncertainties. δ1 is a multiplicative
actuator uncertainty, while δ2 and δ3 are multiplicative sensor uncertainties.
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reference
+
–
error
1
x1
x2
+
+
1
s
1
s
8
–
–
+
reference
2
1
K1 = 4
K2 = 8
Figure 2-3. SISO Tracking System with Three Uncertainties
The H system will have the reference input as input1 and the error output
as output1 (w and z, respectively, in Figure 2-2). Removing the δ values will
create inputs 2 through 4 and outputs 2 through 4 (r and q, respectively, in
Figure 2-2).
1. The A, B, C, D matrices of the state-space system representing H are
as follows:
A=[-4,-8;1,0];
B=[8,1,-4,-8;zeros(1,4)];
C=[0,-1;-4,-8;1,0;0,1];
D=[1,0,0,0;8,0,-4,-8;zeros(2,4)];
"r1","r2","r3"],outputNames=["error",
"q1","q2","q3"],stateNames=["x1","x2"]});
2. Specify the uncertainty bounds.
The sensor uncertainty δ3 is known to be bounded by l3(w), according
to Equation 2-1. Because the position x2 sensor model is known to be
accurate to 10% up to one radian per second, and very inaccurate at
high frequencies, the l3 shown in Figure 2-4 is selected.
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Chapter 2
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10
0
–20
0.1
30
Frequency, Radian/Second
100
1
Figure 2-4. Bound for Sensor Uncertainty
Note A value of l3 at one radian per second of –20 dB indicates that modeling
uncertainties of up to 10% (–20 dB = 0.1) are allowed.
The actuator and sensor uncertainties δ1 and δ2 are bounded by –20 dB
at all frequencies. You will use these values to interpolate to obtain l3.
First, create the bound for δ3 in Hz.
L3 = pdm([-20,-20,10,10],[0.1,1,30,100]/2/pi);
3. Now interpolate to obtain 30 points:
L3 = interpolate(L3,logspace(0.01,10,30),{xlog});
4. Create L1 and L2 (bounds for δ1 and δ2 ):
L1=-20*ones(L3); L2 = L1;
delb = [L1,L2,L3];
5. Calculate the stability margin:
marg=smargin(H,delb);
smargin --> Scaling algorithm is type: PF
smargin --> Margin computation 10% complete
smargin --> Margin computation 50% complete
smargin --> Margin computation 90% complete
The output indicates that Perron-Frobenius scaling (the default) is
used. Refer to the Approximation with Scaled Singular Values section.
The stability margin plot is shown in Figure 2-5. The minimum margin
(uncertainty bounds) could be increased (relaxed) simultaneously
by 8 dB, and the system would still remain robustly stable.
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Chapter 2
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Figure 2-5. Stability Margin
Now examine the effect on the stability margin of discretizing H(s) at
100 Hz.
dt = 0.01;
Hd = discretize(H,dt);
margD = smargin(Hd,delb);
smargin --> Margin computation 10% complete
smargin --> Margin computation 50% complete
smargin --> Margin computation 90% complete
100 Hz is a high discretization frequency for H, so the stability margin
is unchanged in the discrete-time case. The new plot is not much
different from Figure 2-6. Again, minimum margin is about 8 dB
at about 1/2 Hz.
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Chapter 2
Robustness Analysis
Worst-Case Performance Degradation (wcbode)
Even if a system is robustly stable, the uncertain transfer functions still can
have a great effect on performance. Consider the transfer function from the
qth input, wq, to the pth output, zp. With δ1 = ... = ...δk = 0, you have the
nominal system, and this transfer function is the p,q entry of Hzw. This is
called the nominal transfer function.
When the δ values are not zero, the transfer function from wq to zp is the p,q
entry of Hpert given by the formula:
Hpert = Hzw + HzrΔ(I – HqrΔ)–1Hqw
This is referred to as the perturbed transfer function. The perturbed transfer
function depends on the particular δ1, …, δk.
If the magnitude bounds are small enough, then you expect the perturbed
transfer function Hpert to be close to the nominal transfer function. Roughly
speaking, small perturbations should not significantly alter the closed-loop
transfer function from wq to zp.
The worst-case gain is defined as the largest magnitude of the perturbed
transfer function, considering all δ values that satisfy the magnitude bound.
More precisely:
wcgain(ω) = max{ Hpert,pq
Δ = diagonal(δ1,...,δk), δi ≤ li(ω)} (2-3)
wcgain(ω) is always larger than the nominal gain, |Hzw,pq(jω)|. This is not
because the uncertain transfer functions only can increase the magnitude of
the transfer function from wq to zq. In fact, it is possible that for a lucky
choice of the δ values, the perturbed transfer function actually can be
smaller than the nominal transfer function over all frequencies. But in the
worst-case gain, you consider only the worst possible δ values, and these
always increase the perturbed gain over the nominal gain.
Intuitively, if the stability margin is large, then the uncertain transfer
functions should not greatly effect the gain from wq to zp, so that wcgain(ω)
should be not much larger than the nominal gain |Hzw,pq(jω)|. If the stability
margin is small, however, wcgain(ω) could be much larger than the nominal
gain. An extreme case occurs if the stability margin is negative (in dB) at
the worst-case gain curve so that it never exceeds (the maximum nominal
gain) * 100, or +20 dB. Of course, instability is an extreme form of
performance degradation.
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Chapter 2
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wcbode( )
[WCMAG, NOMMAG] = wcbode (SysH, delb, {input, output,
graph})
The wcbode( )function computes and plots the worst-case gain of a
closed-loop transfer function.
This function is useful for checking a system that already has been verified
to be robustly stable using smargin( ). For example, a system can have a
minimum stability margin of 4 dB, so it is robustly stable. If the worst-case
gain from a function input to the output it commands has a 20 dB peak, then
even though the system is robustly stable, the design is unacceptable. On
the other hand, if you verify that the perturbed closed-loop transfer function
increases only 2 dB over the nominal, then the design is probably
acceptable.
The wcbode( )function computes and plots an approximation to
wcgain(ω), the largest possible magnitude of a perturbed closed-loop
transfer function that can be caused by uncertain transfer functions that
satisfy the magnitude bound. The wcbode( )function is conservative:
it does not under-report the maximum of the perturbed transfer function.
A large value of wcbode( )indicates instability: wcgain(ω) = ∞. In this
case, wcbode( )returns a maximum value of ten times the maximum of
the nominal transfer function over all frequencies. Consequently, the
window is clipped at 20 dB above the maximum of the nominal transfer
nominal transfer function for reference.
Using wcbode( ) to Analyze Performance Degradation
The wcbode( )function can be used to analyze performance degradation
for the system you have been using (Figure 2-3). The transfer function,
which should be small, is from reference to error (input 1 to output 1).
Figure 2-6 shows the results of the following function call:
[NOMMAG,WCMAG]=wcbode(H,delb,{input=1,output=1});
The performance degradation due to the uncertainties is small but not
negligible.
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Chapter 2
Robustness Analysis
Figure 2-6. Performance Degradation of the SISO Tracking System
Advanced Topics
This section describes the theoretical background on robustness analysis
and performance degradation.
Stability Margin
This section discusses advanced aspects of computing the stability margin
and the related scaling algorithms.
Stability Margin and Structured Singular Values (μ)
The stability margin was first defined by Safonov in [Saf82]. If you let
M = Hqrdiagonal(l1(w), ...,lk(w))
then you can express the margin at frequency d as
margin(ω) = max
{α det(I – MΔ) ≠ 0
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Chapter 2
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for all diagonal Δ such that
1
( Δii ≤ α)} = -------------
μ(M)
where μ(.) is the structured singular value, introduced by Doyle in
[Doy82]. Thus, the margin is the inverse of the structured singular value of
Hqr diagonally scaled by the magnitude bounds.
There is no numerically efficient algorithm that is guaranteed to compute
μ(M), and hence the stability margin. However, it is possible to compute
various good approximations to μ(M). One of these approximations is often
exact.
A popular but conservative method uses singular values:
1
----------------------
margin(ω) ≥
(2-4)
σ
(M)
max
Plotting the right side of Equation 2-4 gives a lower bound on the
actual stability margin. To get this plot, specify smargin( )with
scaling="SVD". This approximation can be very conservative, meaning
that the left side can be much larger than the right side. This fact spurred
the study of structured singular values and the other approximations
discussed in the following sections.
Use of Scaling Example
For this example, you will use the system in Figure 2-3. This time
smargin( )will be invoked with scaling="SVD", so smargin( )
will calculate Equation 2-4.
margSVD = smargin(H,delb,{scaling="SVD"});
smargin --> Scaling algorithm is type: SVD
smargin --> Margin computation 10% complete
smargin --> Margin computation 50% complete
smargin --> Margin computation 90% complete
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You can compare this margin to that of the example in the Creating a
Nominal System section; the following inputs produce Figure 2-7.
plot ([marg,margSVD],{xlog}
legend=["PF_SCALE","SVD"],
ylab="Stability Margin,dB",
xlab="Frequency, Hz."})
Figure 2-7. pfscale( ) versus svd Stability Margins
Note The singular value approach gives results that are too conservative, suggesting that
the uncertainties can destabilize the system. Conversely, you know from the scaled singular
value calculations that the system is robustly stable.
Approximation with Scaled Singular Values
In [Saf82] and [Doy82], the inequality
min
σmax(DMD–1) ≥ μ(M)
(2-5)
diagonal
is noted. This optimization problem can be shown to be
unimodal—for D > 0, an assumption that can be made without loss
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of generality—so, roughly speaking, it can be solved. [SD83,SD84]
discusses this optimization problem.
Notice that:
σmax(M) = σ(DMD–1)
for
D = 1
so you have the following from Equation 2-5:
σmax(M) ≥ μ(M)
This inequality is thought to be nearly an equality, so that the left side is a
generally held belief, but no example is known where the left side is more
than 15% larger than the right side. Equality can be shown to hold provided
k ≤ 3—for example, if there are three or fewer uncertain transfer functions
[Doy82].
Note The approximation equation of μ(M) (Equation 2-5) is an upper bound. This means
that the stability margin calculated using this approximation is conservative, that is, less
than the actual stability margin. This optimization problem itself can be difficult. Osborne
[Osb60] and Safonov [Saf82] provide two methods for finding good suboptimal scalings
for Equation 2-5.
Both Osborne’s and Safonov’s Perron-Frobenius scalings usually have
been found to be close to the optimum for the optimization problem
equation. The resulting approximations,
–1
ˆ
uOS(M) = σmax(DOSMDOS
)
)
–1
ˆ
uPF(M) = σmax(DPFMDPF
are thought to be good engineering approximations to μ. optscale( )
provides an iterative optimization function based on the ellipsoid
algorithm.
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Comparing Scaling Algorithms
Using the system from the first example (Figure 2-3), you can compare
the results of using the three scaling algorithms:
MARG_PF=smargin(H,delb,{scaling="PF",!graph});
MARG_OPT=smargin(H,delb,{scaling="OPT",!graph});
plot ([MARG_PF,MARG_OS,MARG_OPT],{xlog,
legend=["PF","OS","OPT"],xlab="Frequency, Hz.",
ylab="Stability margin, dB"})
Figure 2-8 plots the margins produced by the three scaling algorithms.
Notice that in this problem the three scalings yield identical stability
margins.
Figure 2-8. Results of Scaling Algorithm Options
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ssv( )
[v,vD] = SSV(M, {scaling})
The ssv( )function computes an approximation (and guaranteed upper
bound) to the Scaled Singular Value of a complex square matrix M, where
M can be a reducible matrix. The scaled singular value v(M) is defined by:
v(M) =
inf
σ(DMD–1)
n × n
D ∈ C
, det(D) ≠ 0, diagonal
Scaling can be accomplished with one of three algorithms:
•
Perron-Frobenius—If {scaling="PF"}Safonov’s
Perron-Frobenius method [Saf82] is used. This method finds the scaled
singular value for non-negative real matrices M. In general, it is
suboptimal if M is complex. This algorithm is the default because
empirical tests show that is the fastest of the three.
•
Osborne—If {scaling="OS"}, Osborne’s Method [Osb60] is used.
This method solves the problem of finding DO such that
inf
diagonal
DOMD–O1
DMD–1
F
D
where D is diagonal and positive, and ⋅ is the Frobenius norm.
F
Thus, the Osborne method minimizes the Frobenius norm, and is
therefore suboptimal.
•
Optimal—If {scaling="OPT"}, Boyd’s ellipsoid algorithm
[BYB89] is used. This algorithm computes the scaled singular value
to a guaranteed accuracy. It is, however, the most computationally
expensive of the three algorithms.
ssv( ) Examples
Consider the complex matrix M:
M = [–1, jay, 0; 0, 2*jay, 1+jay;1, 0, 1];
ssv( )can return the optimally scaled singular value of M using Osborne,
Perron-Frobenius, or Boyd methods:
VOS=ssv(M,{scaling="OS"})
VOS (a scalar) = 2.56723
VPF=ssv(M,{scaling="PF"})
VPF (a scalar) = 2.45133
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VOPT=ssv(M,{scaling="OPT"})
VOPT (a scalar) = 2.43952
VSVD = max(svd(M))
VSVD (a scalar) = 2.65886
osscale( )
[v, vD] = osscale(M)
The osscale( )function scales a matrix using the Osborne Algorithm.
A diagonal scaling DOS is found that minimizes the Frobenius norm of
DOSMDO–1S , which is the square root of the sum of the squares of its
singular values. If M is reducible, osscale( )may encounter a divide
by zero. To avoid this, use ssv( )with the Osborne scaling option:
[v,vD]=ssv(M,{scaling="OS"})
pfscale( )
[v, vD] = pfscale(M)
The pfscale( )function scales a matrix using the Perron-Frobenius
Algorithm. This scaling is optimal for matrices with all positive entries.
The matrix M must be irreducible for this function. If M is reducible,
use ssv( )with the Perron-Frobenius scaling option instead:
[v,vD]=ssv(M,{scaling="PF"})
The optimum diagonal scaling is found for M using the Perron-Frobenius
theory of non-negative matrices. This scaling is given by
pi
DPi F
=
---
qi
where p and q are right and left eigenvectors of | associated with its largest
eigenvalue:
Mp = λmaxp,
MTq = λmaxq,
p ≠ 0 ≠ q
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optscale( )
[v, vOPTD] = optscale (M, {tol})
The optscale( )function optimally scales a matrix. An iterative
optimization (ellipsoid) algorithm which calculates upper and lower
bounds on the left side of Equation 2-5 is used. If these bounds are within
a relative accuracy you have specified (tol), optscale( )stops.
optscale( )also will stop and issue a warning if the maximum number
of iterations is reached:
200 × rows(M)
optscale( )will find a μ(M) no larger than pfscale( ).
Reducibility
In some cases, the uncertain transfer functions can be divided into groups
that do not interact. This is illustrated in Figure 2-9.
δ1
δ2
δ3
δ4
As you can see, δ3 does not affect the stability margin at all because there
is no feedback through it. The system in Figure 2-9 can be reduced to the
two separate systems shown in Figure 2-10. The stability margin of
Figure 2-9 is the minimum of the stability margins of the systems in
Figure 2-10.
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δ1
δ4
δ2
Figure 2-10. Reduction to Separate Systems
In terms of the approximations to the margin discussed above, this
reducibility will manifest itself as a problem such as divide-by-zero or
nontermination. It really means that the minimum of the optimization
problem is not achieved by any finite scaling.
A matrix M can be split into its reducible components using the following
technique (refer to[BeP79]):
1. Form the matrix X = (αI + M) – 1 for any α larger than the spectral
radius of M, for example 2 M .
2. Form Y = X + XT where Y has a positive i,j entry if and only if δi and δj
are in the same reduced system; otherwise, the entries will be zero.
ssv( )checks for reducibility before invoking a scaling algorithm. The
margins of each of the reduced systems then can be calculated separately,
and the minimum taken.
Conversion to a Stability Margin Problem
In [DWS82], it is shown that a simple relation holds between the
worst-case gain defined in Equation 2-3, and the stability margin. For γ > 0,
wcgain (jw) £ g if and only if m(Hred(jw) diag(g-1,
l1(w),...,lk(w)) £ 1
where Hred is H with the rows and columns corresponding to all inputs and
outputs deleted except the ones of interest (the qth input and the pth output).
This can be interpreted as adding a fictitious uncertain transfer function
from wq to zp with magnitude bound γ–1 at the given frequency. This
reference [BoB91].
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Using this relation and any of the previously discussed approximations for
μ(.), you can compute an approximation to wcgain( ). Because the
approximations to μ(.) are upper bounds, the resulting approximations to
wcgain( )also are upper bounds. For speed purposes, wcgain( )uses
Perron-Frobenius scaling to calculate the approximation of μ.
wcgain( )
gamma = wcgain(H, gammin, gammax, gam0)
The wcgain( )function estimates the largest possible magnitude from
a given input of the system to a given output, when the other inputs are
connected to the other outputs through uncertain transfer functions
bounded by one.
For a discussion of wcgain( )syntax, refer to the Xmath Help. This is a
low-level function that calculates the worst-case gain at a single frequency,
where the magnitude bounds are normalized to one as follows:
l1 = … = lk = 1
Because it is a lower level function, there is no syntax checking. This
function is called by the wcbode( )function.
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3
System Evaluation
This chapter describes system analysis functions that create singular value
Bode plots, performance plots, and calculate the L∞ norm of a linear
system.
Singular Value Bode Plots
The singular value Bode plot is a MIMO generalization of the bode( )
magnitude plot. It is calculated as
σi(H(jw)),
i = 1…k
where
and
k = min((ninputs,noutputs))
σ1(H) ≥ σ2(H) ≥ … ≥ σk(H) ≥ 0
In these equations, σi(H(jw)) can be thought of as the maximum gain of the
system at frequency ω, and σk(H(jw)) can be thought of as the minimum
gain of the system at frequency ω.
If σi(H(jw)) » σk(H(jw)), then at the frequency ω, the system gain can be
large for some input directions and small for other input directions.
The singular value plot allows you to generalize to the MIMO case notions
such as “the command-to-tracking error transfer function is small” or “the
loop gain is large.” For example:
•
If the system represents the command-to-tracking error for a
closed-loop system, then you would hope that σ1, and hence all
σ values, are small over the bandwidth of the system. This means
that the command-to-tracking error is small in all directions at these
frequencies.
•
The singular value plot of a certain transfer matrix gives a lower bound
on the stability margin of the system.
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Chapter 3
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Refer to [BoB91] in Appendix A, Bibliography.
Creating a Singular Value Plot
Example 3-1
1. Let a system H be a 2-input/2-output system:
tf=makepoly([1,2],"s")/...
polynomial([0,-2.334,-12],"s")
tf (a transfer function) =
s + 2
--------------------
(s + 2.334)(s + 12)s
System is continuous
H = [tf, 2*tf; tf*tf, tf+3];
[outputs,inputs]=size(H)
outputs (a scalar) = 2
inputs (a scalar) = 2
2. Now plot the singular values of the system between 0.01 and 100 Hz
using svplot( ):
svplot(H,{Fmin=0.01,Fmax=100})
The result is shown in Figure 3-1. For a discussion of svplot( )syntax,
refer to the Xmath Help.
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Figure 3-1. Singular Value Plot
L Infinity Norm (linfnorm)
The L∞ norm of a stable transfer matrix H is defined as:
H
= sup σ(H( jω))
∞
ω ∈ ℜ
where σ is the maximum singular value and H(jω) is the transfer matrix
under consideration.
The L∞ norm of a stable transfer matrix is the maximum of the maximum
singular values over frequency. For example, the highest point of its
singular values plot. Observe that the L∞ norm can be calculated even if H is
not stable.
A simple interpretation of the L∞ norm of a stable system can be given as:
RMS(y)
-------------------
= max
H
∞
RMS(u)
where u and y are the input and output of H, respectively. This means that
is the root mean square (RMS) gain of the system: it is the largest
H
∞
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factor by which the RMS value of a signal flowing through H can be
increased.
By comparison, the H2 norm is defined as:
∞
k
1
σi(H( jω))2dw
------
H
=
2
2π ∫ ∑
–∞ i = 1
This norm can be interpreted as the RMS value of the output when the input
is unit intensity white noise. It can be computed in Xmath using the rms( )
function.
For discrete-time systems with a stable H,
max
H
=
σ(H(e jω))
∞
ω ∈ (–π, π)
where σ is the maximum singular value and H(ejω) is the transfer matrix
under consideration.
linfnorm( )
[sigma, vOMEGA] = linfnorm( Sys, {tol,maxiter} )
The linfnorm( )function computes the L∞ norm of a dynamic system
using a quadratically convergent algorithm. The linfnorm( )function
relies on eigenvalue calculations of a Hamiltonian matrix with twice as
many states as Sysand, consequently, may be unreliable for large systems.
A singular value plot created with svplot( )can be used as an alternative
in these cases. Refer to the Singular Value Bode Plots section.
•
The keyword tolcontrols the required relative accuracy. The default
is 0.01. maxiteris the maximum number of iterations. The default
is 15.
•
If the maximum norm is found at ω = ∞, linfnorm( )returns:
vOMEGA = Infinity
sigma = gain at infinity.
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•
•
If Ahas an imaginary eigenvalue at jω0, linfnorm( )returns:
vOMEGA = ω0
SIGMA = Infinity
where ω0 is one of the imaginary eigenvalues of A.
Even if H is unstable, linfnorm( )returns its maximum singular
value on the jω axis.
For discrete-time systems linfnorm( )converts a discrete-time L∞
norm computation problem to a continuous-time problem using a Cayley
transformation. For example, it maps the unit circle conformally onto the
complex right half plane using a linear fractional transformation. The
linfnorm( )function then calls itself to solve the continuous-time
problem, and finally converts the solution back to discrete-time.
Example 3-2
Example of linfnorm( )
Sys=system([-0.2,-1;1,0],[1,0]',[0,1],0);
[sigma,omega]=linfnorm(Sys)
sigma (a scalar) = 5.07322
omega (a scalar) = 0.157081
The linfnorm( )function will return the L∞ norm (sigma) of the transfer
matrix H(jω) described by Sys, and omegais the vector of frequencies
where it is achieved. linfnorm( )computation can be checked by
plotting the singular values of H(jω) as a function of ω (Figure 3-2).
sv=svplot(Sys,{fmin=.01, fmax=1.0});
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Figure 3-2. Singular Values of H(jω) as a Function of ω
Note svis returned in dBs. Check that sigmais within 0.01 (the default value of tol) of
10**(max(sv,{channels})/20).
[sigma,10^(max(sv,{channels})/20)]
ans (a row vector) = 5.07322 4.98731
The linfnorm( )function also can be used on discrete-time systems.
Consider a state-space system with a sample rate of 10 Hz:
SysD=system([0.5,0.5;0.8,0.5],[0.8,0.5]',
[0,1],0,{dt=0.1})
[sigma,omega]=linfnorm(SysD)
sigma (a scalar) = 5.99267
omega (a scalar) = 0
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Singular Value Bode Plots of Subsystems
To evaluate the performance achieved by a given controller rapidly, it is
useful to check four basic maximum singular value plots—for example, the
transfer matrices from process and sensor noises to the error and actuator
signals.
perfplots( )
SV = perfplots ( Sys, nd, ne, { keywords } )
The perfplots( )function plots the maximum singular value of the
four transfer matrices of the system in the following figure.
e
u
d
n
Sys
In most applications, the perfplots( )function is applied to a system of
the form shown in Figure 3-3, where Pis the plant and Kis a proposed
controller.
process
noise
error
d
e
u
y
P
sensor
noise
n
actuator
u
K
Sys
Figure 3-3. Typical System with Plant and Controller
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The four transfer matrices are labeled e/d, e/n, u/d, and u/n in the final plot.
The plots in the top row, consisting of e/d and e/n, show the regulation or
tracking achieved by the controller. If both these quantities are small, then
the disturbance d and the sensor noise n will not make the error signal e
large.
The bottom row of plots, consisting of u/d and u/n, show the actuator effort
used by the controller. If these are both small, then the actuator effort u,
which results from the disturbance d and the sensor noise n will be small.
A classic trade-off in controller design boils down to a choice between
making the top row of a perfplot( )small (good regulation/tracking)
and making the bottom row small (low actuator effort). For example, by
varying the design parameter ρ in the lqgltr( )regulator design process,
the magnitude of the top two transfer matrices can be traded off against the
magnitude of the bottom two. Increasing ρ makes the top two magnitudes
smaller but makes the bottom two larger.
The columns of a perfplot( )have a dual interpretation. The plots in the
left column, e/d and u/d, show how sensitive the system is to the process
noise or disturbance d. The plots in the right column, e/n and u/n, show how
sensitive the system is to the sensor noise n. Again, there is a trade-off
between making the magnitudes of the transfer matrices on the left small
(good disturbance rejection) and making the magnitudes of the transfer
matrices on the right small (low sensitivity to sensor noise). In the
lqgltr( )estimator design, the parameter ρ controls the relative
magnitude of the left and right plots. Increasing ρ makes the left two
magnitudes smaller but makes the right two larger. Refer to Example 3-3.
Example 3-3
Example of perfplots( )
Consider the simple closed-loop system shown in Figure 3-4.
noise
disturbance
+
e
1
s
–
+
+
n
K
Figure 3-4. Closed-Loop System
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The system matrix can be calculated using the afeedback( )function for
different values of K. Consider two cases: K = 1 and K = 5.
P = 1/makepoly([1,0],"s")
P (a transfer function) =
1
--
s
System is continuous
K1= 1/makepoly(1,"s")
K1 (a transfer function) =
1
-
1
System is continuous
K5= 5/makepoly(1,"s");
Sys1 = afeedback(P,K1);
Sys5 = afeedback(P,K5);
The effect of the value of K on closed-loop performance can be investigated
using perfplots( ).
sv1 = perfplots(Sys1,1,1);
Overlap plots:
sv5 = perfplots(Sys5,1,1,{!graph});
for i = 1:4
plot(sv5(1,i),
{graphnumber=i,line_style=2,keep})?
endfor
In Figure 3-5, you can see that over the bandwidth of 0.1 Hz, the controller
K = 5 has better regulation (e/d is smaller for K = 5 than for K = 1, with e/n
about the same for both cases) but uses slightly more actuator effort. Above
the bandwidth of 0.1 Hz, the e/n and u/n show that the K = 5 controller is
more sensitive to sensor noise. In classic terms, the K = 5 controller has a
higher bandwidth.
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Figure 3-5. Perfplots( ) for K = 1 and K = 5
clsys( )
SysCL = clsys( Sys, SysC )
The clsys( )function computes the state-space realization SysCL, of the
closed-loop system from wto zas shown in Figure 3-6.
z
y
w
u
Sys
SysC
Figure 3-6. Closed Loop System from w to z
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WhereSysC=system(Ac,Bc,Cc,Dc),Sys=system(A,B,C,D),andnz
is the dimension of zand nwis the dimension of w:
nw
nw
Bw
nz
nz
Dzw Dzu
Dyw Dyu
B is
C is
D is
Cz,Cy
Bu
Given the above, SysCLis calculated as shown in Figure 3-7.
A+Bu(I – DcDyu)–1DcCy
BcCy + BcDyu(I – DcDyu)–1DcCy Ac + BcDyu(I – DcDyu)–1Cc
Bu(I – DcDyu)–1Cc
ACL
=
Bw + Bu(I – DcDyu)–1DcDyw
BcDyw + BcDyu(I – DcDyu)–1DcDyw
BCL
=
Cz + Dzu(I – DcDyu)–1DcCy Dzu(I – DcDyu)–1Cc
CCL
=
DCL = Dzw + Dzu(I – DcDyu)–1DcDyw
Figure 3-7. Calculation of the Closed Loop System (SysCL)
be invertible). A well-posed closed-loop system assures that if two given
systems, Sysand SysC, are proper (only proper transfer functions can be
represented in state space), then the resulting closed-loop system, SysCL,
also is proper and therefore realizable in state space.
Figure 3-8 is an example of an ill-posed feedback system, where the
closed-loop transfer function is s+1, which cannot be represented as
a state-space system.
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1
s
+ 1
+
+
Figure 3-8. Ill-Posed Feedback System
Example 3-4
Example of Closed-Loop System
a = 1;
b = [1,0,1];
c = b';
d = [0,0,0;0,0,1;0,1,0];
Sys = SYSTEM(a,b,c,d);
SysC = SYSTEM(-40,2.7,-40,0);
SysCL = clsys(Sys,SysC)
SysCL (a state space system) =
A
1
-40
-40
2.7
B
1
0
0
2.7
C
1
0
0
-40
D
0
0
0
0
X0
0
0
System is continuous
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4
Controller Synthesis
This chapter discusses synthesis tools in two categories, H∞ and H2. This
chapter does not explain all of the theory of H∞, LQG/LTR, and frequency
shaped LQG design techniques. The general problem setup is explained
together with known limitations.
H-Infinity Control Synthesis
Problem Definition
The H∞ control synthesis function hinfcontr( )finds a stabilizing
multivariable controller K for the plant P, as shown in Figure 4-1.
In the closed-loop system with plant P and controller K, all frequencies ω,
σmax(Hew(jω)) ≤ γ
(4-1)
where Hew is the closed-loop transfer matrix from w to e and γ is some
specified limit. Equation 4-1 can be expressed in terms of the H∞ norm as:
Hew ∞ ≤ γ
Hew
e
w
u
y
P
K
Figure 4-1. Closed-Loop System with Plant P and Controller K
The function hinfcontr( )is based on the 2-Riccati state space
solutions presented in [GD88,DGKF89]. You can examine these references
for theoretical descriptions.
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Chapter 4
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K that is arbitrarily close to solving:
min Hew ≤ γ = γopt
(4-2)
∞
K
describes how the optimum can be found manually by decreasing γ until
an error condition occurs, or conversely by increasing γ until the error
condition is fixed.
summarized in the Restrictions on the Extended Plant section are
those imposed in [GD88,DGKF89].
Extended Transfer Matrix
Referring to Figure 4-1, plant P specifies two groupings of vector inputs
and outputs. Such systems or transfer matrices are referred to as extended
transfer matrices or systems. To enter these in Xmath requires a
modification of your existing system representation. The standard system
has the form y = G(s)u and can be described either in state-space form:
x' = Ax + Bu
y = Ax + Du
or as a transfer matrix:
G(s) = D + C(sI – A)–1B
G(s) can be described in Xmath using the state-space system object:
G = system(A,B,C,D)
There is, however, insufficient information in this form to distinguish
the input/output groupings in the extended system P in Figure 4-1.
The state-space form of P is:
·
x = Ax + B1w + B2u
e = C1x + D11w + D12u
y = C2x + D21w + D22u
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Equivalently, as a transfer matrix:
D11 D12
C1
C2
P(s) =
+
(sI – A)–1[B1 B2]
D21 D22
in Figure 4-1. The extended plant P can be constructed using the Xmath
interconnection functions, as shown in Example 4-1.
Building the Plant Model
The general form of the plant P is shown in Figure 4-2.
Plant P
v
z
w
e
y
Win
Wout
G
u
Figure 4-2. Construction of Plant P
The plant consists of three transfer matrices: Win and Wout, referred to as
weights, and G which can be interpreted as the system dynamics. Both P
and G distinguish inputs and outputs into two groups of variables.
The input/output variables are organized as follows:
•
•
•
actuator/sensor variables
u—vector of actuator (control) signals
y—vector of sensor (measured) and other accessible signals
exogenous inputs
v—vector of commands and disturbance
w—vector of “normalized” commands and disturbances
performance outputs
z—vector of critical performance signals (regulated variables)
e—vector of “normalized” critical performance signals
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The transfer matrix G can be viewed as a model of the underlying system
dynamics with v and u as generalized forces that produce effects in the
performance signals z and measured signals y.
The weight Win is used to model the exogenous input v by v = Winw.
form the normal critical variables e = Woutz.
In general, the input weight Win can be viewed as a dynamic model of the
exogenous inputs and the output weight Wout as the inverse of the desired
performance. As an illustration, consider the plant configuration in
Figure 4-3.
P
G
Win
Wout
Wreg
wdist
yreg
d
n
ereg
w
w
u
e
e
y
Wdist
wnoise
eact
u
Gdyn
Wnoise
Wact
ysens
Figure 4-3. Typical Plant Configuration
The exogenous input vectors d and n represent disturbances and sensor
noise, respectively. These are generated by passing normalized
unpredictable signals, ωdist and ωnoise, through stable transfer matrices,
Wdist and Wnoise, respectively. The critical performance variables are some
regulated variables yreg, as well as the actuator commands u. These are
variables ereg and eact. The sensed variables ysens are contaminated by
additive noise n to form the measured signal y. The transfer matrix Gdyn
represents the underlying system dynamics. Observe that the transfer
output/input connections among the variables n and u as depicted in
Figure 4-3. This is in the form of the familiar LQG setup, except that
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here the weighting matrices are transfer matrices, whereas in the LQG
setup they are constants.
A description of the plant in Figure 4-3 is as follows:
•
Dynamical system Gdyn:
·
x = Ax + Bdistd + Bact
yreg = Creg
ysens = Csens
u
x
x
•
•
Measured variables y = ysens + n:
Input weight Win:
wdist
wdist
d
n
Wdist
0
=
= Win
wnoise
wnoise
0
Wnoise
•
Output weight Wout:
ereg
yreg
u
yreg
u
Wreg
0
=
= Win
eact
0
Wact
Weight Selection
In the standard LQG formulation, the weighting functions (Wdist, Wnoise
,
Wreg, and Wact) are all constant matrices. In fact, if Qreg and Qact are positive
definite matrices in the LQ cost, for example,
∞
⎛
⎜
⎝
⎞
1
2
--
J =
x'Qregx + u'Qactu dt
⎟
∫
⎠
0
then the following apply:
1/2
1/2
Wreg = Q
Wact = Q
act
reg
Similarly, if Rproc and Rsens are positive definite matrices corresponding to
the process and measurement noise intensities, then the following apply:
1/2
1/2
Wdist = Q
Wnoise = Q
sens
proc
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Selecting these weights has much the same effect here. Specifically, let Hzv
be the closed-loop transfer matrix (with u = Kγ) from inputs:
d
v =
n
to outputs:
yreg
z =
u
Thus,
H
H
yreg
yreg
d
n
Hzv
=
Hud Hun
Suppose that the controller u = Ky approximates Equation 4-2. Thus,
Wout HzvWin ≈ γopt
∞
In many cases, this means that the maximum singular value of the
frequency response matrix (Wout HzvWin)( jω) is constant over all
frequencies. That is,
⎛
⎜
⎜
⎝
⎞
Wreg
H
dWdist Wreg
H
nWnoise
yreg
yreg
⎟
σmax
(jw) ≈ γopt
⎟
⎠
WactHudWdist WactHjnWnoise
An interpretation is that the weighting filters Win and Wout determine the
shape of the closed-loop frequency response Hzw( jω), and γopt determines
the peak value. This observation helps motivate the selection of the weights
so as to shape the closed-loop frequency response matrix Hzw( jω).
Observe, however, that the elements of the frequency response matrix,
(WoutHzvWin)( jω), need not be constant. Instead, the maximum singular
value of at least one of the four subblocks is within 3 dB of γopt. For all ω,
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where
and
M(ω) = max{m11(u),m12(u)m21(u)m22(u)}
m11 = σmax[WregHy dWdist
]
reg
m12 = σmax[WregHy nWnoise
]
reg
m21 = σmax[WactHudWdist
]
m22 = σmax[WactHunWnoise
]
The weights also can be viewed as “design knobs” (for example,
[ONR84]). In this view, the weights are not directly related to specific
disturbance or performance models but rather are used as a vehicle to obtain
a closed-loop transfer matrix, Hzv, from v to z with desired properties. For
every selection of weights Win and Wout, the closed-loop system has the
following property:
WinHzvWout ∞ ≤ γ
But Hzv has other properties, both good and bad. To some extent, these all
can be affected by varying the weights. An effective way to provide a rapid
evaluation of performance is with the function perfplots( ), as
described in the perfplots( ) section of Chapter 3, System Evaluation. With
a good indication of their effect on performance.
Restrictions on the Extended Plant
Not all choices of weights will result in an extended plant P = WoutGWin
that will solve Equation 4-1. The following conditions, established in
references [GD88,DGKF89], if satisfied, will result in a solution. If any
are not satisfied, an error condition occurs.
The conditions are:
•
•
•
(A, B2, C2) can be stabilized and detected
rank(D12) = NU (number of inputs)
rank(D21) = NY (number of outputs)
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•
For all ω ≥ 0,
A – jωI B2
rank
rank
= NS + NU
C1 D12
•
A – jωI B1
= NS + NY
C2 D21
Condition 1 is a standard condition to ensure the existence of a stabilizing
controller. Condition 2 ensures that the control signal u is contained in the
normalized error vector e (refer to Figure 4-3). Conversely, condition 3
ensures that some exogenous input (disturbance or noise) affects the
measured signals (refer to Figure 4-3). Conditions 4 and 5 ensure certain
minimal realizations of subblocks of the extended plant ([GD88]).
[SysC,Syszw]=hinfcontr(SysAug,gamma,nw,nz,{method})
The hinfcontr( )function designs an H∞ controller for an augmented
plant. The augmented plant should satisfy the five restrictions in the
Restrictions on the Extended Plant section. The hinfcontr( )function
tests for these restrictions and returns an error if they are violated.
Assuming the restrictions are not violated, a controller satisfying
Hew ∞ ≤ γ will exist if certain low-level conditions also hold. These
involve conditions for the solution of the underlying Riccati equations
and conditions for some other constraints. The details can be found in
[GD88,DGKF89] and are beyond the scope of this manual. If the low-level
conditions are violated, an error statement is displayed:
hinfcontr –>No stabilizing controller meets the spec!
Adjust gamma and try again!
When this occurs it means that the original gammais too small and a
larger gamma(for example, a looser spec) is required to eliminate the
error condition. If gammais too small, or any other necessary condition
is not met, the hinfcontr( )function returns a zero controller and the
closed-loop system is equal to the open-loop system:
SysC = system( [], [], [], 0 ),
Syszw = system(A, B1, C1, D11)
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If no error message occurs, then Hew ∞ ≤ γ is guaranteed. However,
this does not preclude the possibility that either Hew ∞ « γ or that
γ
opt « Hew ∞ .
For the former case, there are two checks:
•
•
Use the linfnorm( )function to compute Hew
.
∞
Compute the graph σmax[Hew(jω)] versus ω.
again.
When gammais very large, the specification (Equation 4-1) is easily
met. In this case, the hinfcontr( )function returns a controller that
approximately minimizes the H2 norm of Hew while satisfying
Equation 4-2. Gammacan be interpreted as a “knob” that smoothly
transforms the H2 optimal (LQG) controller, (with gammalarge), to a
H∞ optimal controller (with gamma≈ γopt ).
Similarly, for a large gamma, the controller has good RMS performance
with the noise spectra determined by the weights Wdist and Wnoise. For a
small gamma, the controller has good worst-case performance for noise
spectra that lay below the weights Wdist and Wnoise
.
Example 4-1
Example of hinfcontr( )
Referring to Figure 4-2, suppose G has the state space description,
·
x = x + d + u
y = x + n
where:
x
d
n
z =
v =
u
1. The extended system matrix for G is:
A = 1;
B1 = [1,0]; B2 = 1; B = [B1,B2];
C1 = [1;0]; C2 = 1; C = [C1;C2];
D11 = zeros(2,2); D12 = [0;1]; D21 = [0,1]; D22 = 0;
D = [D11,D12; D21,D22];
G = system(A,B,C,D);
nw = 2; nz = 2;
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Suppose the input/output weights are as follows:
1
s + 1
-----------
0
1
0
0
Win
=
Wout
=
0.1
0
0.1
2. Create the four weights:
Wdist = 1/makepoly([1,1],"s")
Wdist (a transfer function) =
1
-----
s + 1
Wnoise = 0.1;
Wreg=1/makepoly(1,"s");
Wact = 0.1;
3. Combine the weights in Win and Wout (refer to Figure 4-4):
Win = [Wdist,0,0;0,Wnoise,0;0,0,1];
Wout=[Wreg,0,0;0,Wact,0;0,0,1];
The resulting system, Pcan be obtained by putting in series the plant
G and the two weights:
P = Wout*G*Win;
Before using the hinfcontr( )function, you must decide on an
initial guess for gamma.
Win
Wout
d
n
u
u
u
u
y
y
y
1
s + 1
1
G
0.1
1
0.1
1
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4. For this example, you will start with gamma=1as the initial guess and
enter:
[K,Hew] = hinfcontr(P,1,2,2);
No error messages are reported. This means that a stabilizing
controller has been found such that Equation 4-1 holds. That is,
Hew ∞ ≤ 1 .
The actual H∞ norm is found from:
normHew=linfnorm(Hew)
normHew (a scalar) = 0.211984
The result is that on this first iteration:
gamma = 1 Æ normHew = 0.212
Continuing to iterate for the optimal gamma:
[K,Hew] = hinfcontr(P,.2,2,2);
normHew=linfnorm(Hew)
normHew (a scalar) = 0.173218
[K,Hew] = hinfcontr(P,.15,2,2);
normHew=linfnorm(Hew)
normHew (a scalar) = 0.147418
[K,Hew] = hinfcontr(P,.13,2,2);
normHew=linfnorm(Hew)
normHew (a scalar) = 0.13103
[K,Hew] = hinfcontr(P,.12,2,2);
normHew=linfnorm(Hew)
normHew (a scalar) = 2.02252
hinfcontr --> No stabilizing controller meets the
spec.!!
Adjust gamma and try again
The iterations establish that γopt lies between 0.12 and 0.13. Figure 4-5
shows the output of the perfplotsfunction on the closed-loop
system Hew for γ = 0.13.
[K,Hew] = hinfcontr(P,.13,2,2);
normHew = linfnorm(Hew, {tol=1e-3})
normHew (a scalar) = 0.129863
svHew = perfplots(Hew,1,1,{Fmin=0.01,Fmax=100});
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Figure 4-5. Perfplots for Hew
It also is useful to perform perfplots( )on the unweighted
closed-loop system, Hzv, which in this case is the closed-loop transfer
matrix from (d,n) into (x,u).
The following function calls produce Figure 4-6:
Hzv=clsys(G,K);
Hzv=perfplots(Hzv,1,1,{vF=domain(svHew)});
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Figure 4-6. Perfplots for Hzv
singriccati( )
[P, solstat] = singriccati(A,Q,R {method})
The singriccati( )function solves the Indefinite Algebraic Riccati
Equation (ARE):
A′P + PA – PRP + Q = 0
The ARE is solved by decomposing the Hamiltonian:
A –R
–Q –A′
The required decomposition of the Hamiltonian can be achieved using
method="eig"or method="schur". The Schur decomposition is
slower, but might handle some ill-conditioned problems. After solving the
ARE, singriccati( )calculates the residue (ATP + PA – PRP + Q).
A warning is displayed if the residue is large. For an example of this
function, refer to the Xmath Help.
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Chapter 4
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Linear-Quadratic-Gaussian Control Synthesis
The H2 Linear-Quadratic-Gaussian (LQG) control design methods are
based on minimizing a quadratic function of state variables and control
inputs. Conventionally, the problem is specified in the time domain.
By converting the LQG performance index into the frequency domain,
it becomes obvious that the conventional LQG places equal penalty on
states and control inputs at all frequencies. It is possible to realize
significant improvement in robustness and performance by making the
penalty weighting matrices functions of frequency.
LQG Frequency Shaping
Bryson’s rule [BH69] can be extended to initially select a frequency
shaping for a particular problem. For the control design problem, the
frequency-shaped weighting matrices should be large at frequencies where
control inputs are less desirable. For example, a large weighting on control
signals at high frequency would produce less control activity at those
frequencies, leading to a closed-loop system with lower bandwidth. Similar
ideas apply to selection of state weighting in control design and the
development of robust state estimators.
Three functions are available to solve the problem of frequency-shaped cost
functionals:
• fsregu( )—frequency-shaped regulator
• fsesti( )—frequency-shaped estimator
• fslqgcomp( )—frequency-shaped linear-quadratic-gaussian
compensator
fsregu( )
[SysC, SysCC, vEV] = fsregu(SysA, ns, RXXA, RUUA, {RXUA})
The fsregu( )function computes a frequency-shaped control law.
It assumes you start with:
·
x = Ax + Bu
and
1
2
[x*(jω)Q(jω)x(jω) + u*(jω)R(jω)u(jω)]dω
--
J =
∫
–∞
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This expression can be converted into the following form [Gu80]:
·
B1
B2
x
x
A11 A12
A21 A22
=
+
v
·
x′
x′
Ac
Bc
x
u =
+ D v
0 C12
x′
Cc
Dc
If R(jω) is not a function of frequency, then C12 = 0 and D = I.
Note The system has a new input v and the old input u is now the output of the system.
This structure is only used for computational convenience.
Define:
x
xA
=
x'
Thus,
·
xA = AcxA + Bcv
u = CcxA + Dcv
∞
1
2
(xA* R xA + v*Ruu v + 2xA* Rxu v)dω
--
J =
xxA
A
A
∫
After the system is put in this form, you are ready to use the fsregu( )
function. For more information on the fsregu( )syntax, refer to the
Xmath Help.
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fsesti( )
[SysF, vEV] = fsesti(SysA, ns, QWWA, QVVA, {QWVA})
The fsesti( )function computes a frequency-shaped state estimator.
The estimation problem is stated as follows:
·
x = Ax + Bu + w
y = Cx + Du + v
The frequency-shaped filter design problem is to minimize,
∞
1
2
(w*Qww(jω)w + v*Qvv(jω)v)dω
--
J =
∫
–∞
which can be written as:
or it can be written as:
·
xA = AexA + Beu + wA
·
x
x
B
A11 A12
A21 A22
=
+
u + wA
·
x′
x′
0
xA
Be
Ae
·
x
⎛
⎝
⎞
⎠
u =
+ Deu + vA
C1 C2
·
x′
Ce
∞
1
2
wA* Qww wA + v*AQvv vA + 2wA* Qwv v)dω
--
J =
A
A
A
∫
–∞
For more information on the fsesti( )syntax, refer to the Xmath Help.
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fslqgcomp( )
[SysCC, vEV] = fslqgcomp(SysF, SysC)
The fslqgcomp( )function combines filter and control law to compute a
controller from a control law and an estimator. For more information on the
fslqgcomp( )syntax, refer to the Xmath Help.
Frequency-Shaped Control Design Commands
This extended example uses the previously discussed functions to
demonstrate frequency-shaped control design techniques. A four-state,
poorly damped system is studied to demonstrate how robustness can be
attained using frequency shaping. The control law and filter are designed
on a reduced second order system with and without frequency shaping.
1. Create a full-order system:
a=[0,1,0,0;-1,-.01,0,0;0,0,0,1;0,0,-25,-.05];
b=[0,1,0,1]';
c=[0,1,0,-1]; d=0;
Sys=system(a,b,c,d);
2. Calculate open-loop eigenvalues:
eig(a)
ans (a column vector) =
-0.005 + 0.999987 j
-0.005 - 0.999987 j
-0.025 + 4.99994 j
-0.025 - 4.99994 j
3. Create a reduced order system by selecting only the first mode:
ar=a(1:2,1:2);br=b(1:2);cr=c(1:2);
Sysr=system(ar,br,cr,d);
4. Design an LQG compensator for the reduced-order system, without
using frequency shaping:
qxx=diagonal([0,1]);quu=1;
kr=regu(Sysr,qxx,quu);
ke=esti(Sysr,qxx,quu);
# Linear-Quadratic-Regulator
# Linear-Quadratic-Estimator
Sysc=lqgcomp(Sysr,kr,ke); # LQG Compensator
Syscl=feedback(Sysr,Sysc); # Reduced-order closed-loop
poles(Syscl)
ans (a column vector) =
-0.500025 + 0.866011 j
-0.500025 - 0.866011 j
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-0.500025 + 0.866011 j
-0.500025 - 0.866011 j
5. Try the LQG compensator with the full-order system:
[Syscl_fo]=feedback(Sys,Sysc);
poles(Syscl_fo)
ans (a column vector) =
-0.401519 + 0.864869 j
-0.401519 - 0.864869 j
-0.638796 + 0.855861 j
-0.638796 - 0.855861 j
0.0152647 + 4.90994 j
0.0152647 - 4.90994 j
6. Find a stabilizing reduced order controller using frequency shaping.
To stabilize this system, try a frequency-shaped control-law, where the
weighting on the control signal will be 1 + (ω)4.
First, form an augmented system, defining two additional states,
·
μ and u.
The augmented system will be:
0
0
0
0
0
1
xr
u
xr
u
Ar Br
d
dt
----
=
+
v = AaxA + Bav
(4-3)
(4-4)
0 0 0 1
0 0 0 0
·
·
u
u
xr
u =
+ (0)v = CaxA + Dav
0 0 1 0
u
·
u
Note The augmented system has two extra states implementing the frequency shaping on
the control system.
aa=[ar,br,[0;0];0,0,0,1;0,0,0,0];
bb=[0,0,0,1]';cc=[0,0,1,0];dd=0;
Sysa=system(aa,bb,cc,dd)
Sysa (a state space system) =
A
0
1
0
1
0
0
-1
-0.01
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0
0
0
0
0
0
1
0
B
0
0
0
1
C
0
D
0
0
1
0
X0
0
0
0
0
System is continuous
7. Frequency-weight the control signal.
Transfer the weight on Ufrom RUUto the third diagonal entry in RXXA.
Note In Equation 4-3, u is the third state of the augmented system. RUUweighs the new
frequency-shaped control signal v.
Design a frequency-shaped regulator:
rxxa=diagonal([0,1,1,0]);ruua=1;
[Sysfs_sr,,fs_evr]=fsregu(Sysa,2,rxxa,ruua)
Sysfs_sr (a state space system) =
A
0
1
-1.95171
-1.97571
B
0
0
0.951712
-0.228069
C
1
0
0
D
0
X0
0
0
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System is continuous
fs_evr (a column vector) =
-0.645263 + 0.587929 j
-0.645263 - 0.587929 j
-0.347592 + 1.09155 j
-0.347592 - 1.09155 j
8. Calculate the frequency-shaped estimator:
Sysaf=system(ar,br,cr,0);qwwa=qxx;qvva=quu;
[Sysfs_se,fs_eve]=fsesti(Sysaf,2,qwwa,qvva)
Sysfs_se (a state space system) =
A
0
1
-1
-1.00005
B
5.52357e-17
0.99005
0
1
C
1
0
0
1
D
0
0
0
0
X0
0
0
System is continuous
fs_eve (a column vector) =
-0.500025 + 0.866011 j
-0.500025 - 0.866011 j
The compensator should be structured as shown in Figure 4-7.
x
u
A
y
Sysfs_se
Sysfs_se
Figure 4-7. Frequency-Shaped Compensator
The fslqgcomp( )function can be used to develop the compensator.
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Chapter 4
Controller Synthesis
9. Design the LQG compensator.
[Sysfs_sc,fs_evc]=fslqgcomp(Sysfs_se,Sysfs_sr)
Sysfs_sc (a state space system) =
A
0
-1
0
1
0
0
-1.00005
0
1
0
0
1
0.951712
-0.228069
-1.95171
-1.97571
B
5.52357e-17
0.99005
0
0
C
0
0
1
0
D
0
X0
0
0
0
0
System is continuous
fs_evc (a column vector) =
-0.373302
-1.17564
-0.713411 + 1.33028 j
-0.713411 - 1.33028 j
Enforce negative feedback:
Sysfs_sc = -Sysfs_sc;
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Chapter 4
Controller Synthesis
10. Compute the closed-loop system for the reduced order plant and the
frequency-shaped compensator:
[Sysfs_scl]=feedback(Sysr,Sysfs_sc);
poles(Sysfs_scl)
ans (a column vector) =
-0.645263 + 0.587929 j
-0.645263 - 0.587929 j
-0.500025 + 0.866011 j
-0.500025 - 0.866011 j
-0.347592 + 1.09155 j
-0.347592 - 1.09155 j
11. Compute the closed-loop system for the full-order plant and the
frequency-shaped compensator.
Sysfs_scl_fo = feedback(Sys,Sysfs_sc);
poles(Sysfs_scl_fo)
ans (a column vector) =
-0.690216 + 0.522898 j
-0.690216 - 0.522898 j
-0.419783 + 0.892632 j
-0.419783 - 0.892632 j
-0.381722 + 1.10668 j
-0.381722 - 1.10668 j
-0.0261589 + 5.00027 j
-0.0261589 - 5.00027 j
The full-order closed-loop system is stable. The open-loop eigenvalues
of the unmodelled mode have not moved much, which is a sign of good
robustness. The eigenvalue of the unmodelled mode changed from
–.0250 5j to –0.0262 5j.
Loop Transfer Recovery (lqgltr)
Loop transfer recovery (LTR) is fully described in references [KS72,
DoS79,DoS81,SA88]. The properties of the recovery pertain to the LQG
feedback system as shown in Figure 4-8.
The parameter ρ (rho) can be manipulated by the user to obtain loop
transfer recovery through the regulator (lqrltr) or the estimator
(lqeltr).
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Chapter 4
Controller Synthesis
Plant
G(s)
u
y
x = Ax + Bu
x
y = Cx
Estimator
KR
x = Ax + Bu + KEε
ε = y – Cx
x
K(s)
Figure 4-8. LQG Feedback System for Loop Transfer Recovery
lqgltr( )
[SysC,EV,Kr] = lqgltr(Sys,Wx,Wy,K,rho,{keywords})
The lqgltr( )function designs an estimator or regulator which recovers
loop transfer robustness through the design parameter ρ (rho). For a
discussion of the syntax and a full listing of keywords, refer to the Xmath
Help.
The keyword recoverspecifies whether the recovery should be achieved
through regulator or estimator design.
If the keyword is set to recover="regulator", the loop-transfer is
recovered by designing a regulator with the following model:
·
x = Ax + w
y = Cx + v
and the objective function:
J =
∞(x'Qx + u'Ru)dt
1
2
--
∫
0
with:
Q = Rxx + ρC'C
R = Ruu
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Chapter 4
Controller Synthesis
Then ρ is increased so that pointwise in s:
K(s)G(s) → KR(sI – A)–1B
Regulator recovery is only guaranteed if G(s) is minimum-phase and there
are at least as many control signals u as measurements y.
If recover="estimator", the loop-transfer is recovered by designing an
estimator with the following model:
·
x = Ax + w
y = Cx + v
where w and v have noise intensities:
W = Qxx + ρBB'
V = Qyy
Then ρ is increased so that pointwise in s:
G(s)K(s) → C(sI – A)–1KE
Recovery is only guaranteed if G(s) is minimum-phase and there are at least
as many measurements y as there are control signals u.
If graphis on(default) the singular value loop transfer response is plotted.
The solid line represents the original loop transfer, and the dashed line
represents the loop transfer after recovery.
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A
Bibliography
[BBK88]
[BeP79]
[BoB90]
S. Boyd, V. Balakrishnan, and P. Kabamba. “A bisection method for computing the
L norm of a transfer matrix and related problems.” Mathematical Control Signals,
∞
Systems Vol. 2, No. 3, pp 207–219, 1989.
A. Berman and R.J. Plemmons. “Nonnegative Matrices in the Mathematical
Sciences.” Computer Science and Applied Mathematics Series, Academic
Press, 1979.
S. Boyd and V. Balakrishnan. “A regularity result for the singular values of a transfer
matrix and a quadratically convergent algorithm for computing its L∞ norm.”
Systems Control Letters, Vol. 15, pp 1–7, 1990.
[BoB91]
[BH69]
S. Boyd and C. Barratt. Linear Controller Design: Limits of Performance.
Prentice-Hall, 1991.
A.E. Bryson and Y.C. Ho. Applied Optimal Control, p 149. Blaisdell
Publishing Co., 1969.
[DoS79]
[DoS81]
J.C. Doyle and G. Stein. “Robustness with Observers.” IEEE Transactions on
Automatic Control, August 1979.
J.C. Doyle and G. Stein. “Multivariable Feedback Design: Concepts for a
Classical/Modern Synthesis.” IEEE Transactions on Automatic Control,
Vol. AC-26, pp 4–16, February 1981.
[Doy82]
J.C. Doyle. “Analysis of Feedback Systems with Structured Uncertainties.”
IEEE Proceedings, November 1982.
[DWS82]
J.C. Doyle, J.E. Wall, and G. Stein. “Performance and Robustness Analysis for
Structure Uncertainties.” Proceedings IEEE Conference on Decision and Control,
pp 629–636, 1982.
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Appendix A
Bibliography
[FaT88]
M.K. Fan and A.L. Tits. “m-form Numerical Range and the Computation of the
Structured Singular Value.” IEEE Transactions on Automatic Control, Vol. 33,
pp 284–289, March 1988.
[FaT86]
M.K. Fan and A.L. Tits. “Characterization and Efficient Computation of the
Structured Singular Value.” IEEE Transactions on Automatic Control, Vol. AC-31,
pp 734–743, August 1986.
[Fr87]
B. Francis. A Course in L Control Theory. Springer-Verlag,
Berlin-New York, 1987.
∞
[FPGM87]
D.S. Flamm, S. Boyd, G. Stein, and S.K. Mitter. “Tutorial Workshop on L∞ Control
Theory.” pre-conference workshop, Proceedings 26th IEEE Conference on Decision
and Control, December 1988.
[GD88]
K. Glover and J.C. Doyle. “State-space formulae for all stabilizing controllers that
satisfy an L norm bound and relations to risk sensitivity.” Systems and Control
∞
Letters, Vol. 11, pp 167–172, 1988.
[DGKF89]
J.C. Doyle, K. Glover, P.K. Khargonekar, and B. Francis. ‘‘State-space solutions to
standard H2 and L control problems.” IEEE Transactions on Automatic Control,
∞
Vol. AC-34, No. 8, pp 831–847, August 1989.
[Gu80]
N.K. Gupta. “Frequency Shaping of Cost Functionals: An extension of LQG Design
Methods.” AIAA Journal of Guidance and Control, Vol. 3, No. 6, December 1980.
[ONR84]
ONR/Honeywell Workshop on Advances in Multivariable Control. Lecture Notes,
Minneapolis, MN, 1984.
[Osb60]
[Saf82]
E.E. Osborne. “On Preconditioning of Matrices.” JACM, 7:338–345, 1960.
M.G. Safonov. “Stability Margins of Diagonally Perturbed Multivariable Feedback
Systems.” IEEE Proceedings, 129-D:251–256, November 1982.
[SD83]
[SD84]
M.G. Safonov and J.C. Doyle. “Optimal Scaling for Multivariable Stability Margin
Singular Value Computation.” Proceedings of MECO/EES 1983, Symposium, 1983.
M.G. Safonov and J.C. Doyle. “Minimizing Conservativeness of Robust Singular
Values.” Multivariable Control, pp 197–207, S.G. Tzafestas, editor. D. Reidel
Publishing Company, 1984.
[SLH81]
M.G. Safonov, A.J. Laub, and G.L. Hartmann. “Feedback Properties of
Multivariable Systems: The Role and Use of the Return Difference Matrix.”
IEEE Transactions on Automatic Control, Vol. AC-26, February 1981.
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Appendix A
Bibliography
[SA88]
[Za81]
[KS72]
G. Stein and M. Athans. “The LQG/LTR Procedure for Multivariable Control
Design.” IEEE Transactions on Automatic Control, Vol. AC-32, No. 2, pp 105–114,
February 1987.
G. Zames. “Feedback and optimal sensitivity: model reference transformations,
multiplicative semi-norms, and approximate inverses.” IEEE Transactions on
Automatic Control, Vol. AC-26, pp 301–320, 1981.
H. Kwakernaak and R. Sivan. Linear Optimal Control Systems. Wiley, 1972.
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B
Technical Support and
Professional Services
Visit the following sections of the National Instruments Web site at
ni.comfor technical support and professional services:
•
Support—Online technical support resources at ni.com/support
include the following:
–
Self-Help Resources—For answers and solutions, visit the
award-winning National Instruments Web site for software drivers
and updates, a searchable KnowledgeBase, product manuals,
step-by-step troubleshooting wizards, thousands of example
programs, tutorials, application notes, instrument drivers, and
so on.
–
Free Technical Support—All registered users receive free Basic
Service, which includes access to hundreds of Application
Engineers worldwide in the NI Discussion Forums at
ni.com/forums. National Instruments Application Engineers
make sure every question receives an answer.
For information about other technical support options in your
area, visit ni.com/servicesor contact your local office at
ni.com/contact.
•
•
Training and Certification—Visit ni.com/trainingfor
self-paced training, eLearning virtual classrooms, interactive CDs,
and Certification program information. You also can register for
instructor-led, hands-on courses at locations around the world.
System Integration—If you have time constraints, limited in-house
technical resources, or other project challenges, National Instruments
Alliance Partner members can help. To learn more, call your local
NI office or visit ni.com/alliance.
If you searched ni.comand could not find the answers you need, contact
your local office or NI corporate headquarters. Phone numbers for our
the Worldwide Offices section of ni.com/niglobalto access the branch
office Web sites, which provide up-to-date contact information, support
phone numbers, email addresses, and current events.
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Index
problem definition, 4-1
restrictions on extended plant, 4-7
weight selection, 4-5
A
Algebraic Riccati Equation (ARE), 4-13
H2 control synthesis, 4-14
H2 norm, 3-4
help, technical support, B-1
hinfcontr( ), 4-2, 4-8
C
clsys( ), 3-10
D
instrument drivers (NI resources), B-1
diagnostic tools (NI resources), B-1
documentation
NI resources, B-1
drivers (NI resources), B-1
L
E
L∞ norm, 3-3
examples (NI resources), B-1
linfnorm( ), 3-4
F
frequency-shaped
control law, 4-14
filter design, 4-16
fsesti( ), 4-16
fslqgcomp( ), 4-17
fsregu( ), 4-14
M
N
National Instruments support and services, B-1
NI support and services, B-1
nomenclature, 1-2
H
H∞ control synthesis, 4-1
building the plant model, 4-3
extended transfer matrix, 4-2
nominal closed-loop system, 2-2
creating, 2-4
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Index
nominal transfer function, 2-8
norm
software (NI resources), B-1
H2, 3-4
L∞, 3-3
stability margin, 2-3, 2-10
structured nonparametric uncertainty, 2-1
structured singular value, 2-11
support, technical, B-1
O
optscale( ), 2-17
osscale( ), 2-16
feedback connection, 2-2
nominal closed-loop, 2-1
stability margin, 2-3
P
perfplots( ), 3-7
pfscale( ), 2-16
programming examples (NI resources), B-1
technical support, B-1
training and certification (NI resources), B-1
R
reducibility, 2-17
nominal, 2-8
robust stability, 2-3
uncertain, 2-2
S
scaled singular values, 2-11
scaling
Optimal (Boyd), 2-15
Osborne, 2-15
Perron-Frobenius, 2-15
ssv( ), 2-15
wcgain( ), 2-19
Web resources, B-1
well-posed closed-loop system, 3-11
worst-case gain, 2-8
worst-case performance degradation, 2-8,
2-18
singriccati( ), 4-13
singular value Bode plots, 3-1
singular values, 2-11
smargin( ), 2-4
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