Gain Bounds
In this chapter we consider robustness specications, which limit the worst case
variation in the closed-loop system that can be caused by a specic set of plant
variations. We describe a powerful method for formulating inner approximations
of robustness specications as norm-bounds on the nominal closed-loop transfer
matrix. These specications are closed-loop convex.
In the previous chapter we studied the di erential sensitivity of the closed-loop
system to variations in the plant. Di erential sensitivity analysis often gives a
good prediction of the changes that occur in the closed-loop system when the plant
changes by a moderate (non-vanishing) amount, and hence, designs that satisfy
di erential sensitivity speci cations are often robust to moderate changes in the
plant. But di erential sensitivity speci cations cannot guarantee that the closed-
loop system does not change dramatically (e.g., become unstable) when the plant
changes by a non-vanishing amount.
In this chapter we describe robustness specications, which, like di erential sen-
sitivity speci cations, limit the variation in the closed-loop system that can be
caused by a change or perturbation in the plant. In this approach, however,
the sizes of plant variations are explicitly described, e.g., a particular gain
varies 1dB,
robustness speci cations limit the worst case change in the closed-loop system
that can be caused by one of the possible plant perturbations.
By contrast, in the di erential sensitivity approach,
the sizes of plant variations are not explicitly described they are vaguely
described as \small",
209
210
CHAPTER 10 ROBUSTNESS SPECIFICATIONS VIA GAIN BOUNDS
di erential sensitivity speci cations limit the rst order changes in the closed-
loop system that can be caused by the plant perturbations.
Robustness speci cations give guaranteed bounds on the performance deterio-
ration, even for \large" plant variations, for which extrapolations from di erential
sensitivity speci cations are dubious. O setting this advantage are some possible
disadvantages of robustness speci cations over di erential sensitivity speci cations:
It may not be possible to model the actual variations in the plant in the
precise way required by robustness speci cations. For example, we may not
know whether to expect a 1dB or a 0 5dB variation in a particular gain.
:
It may not be desirable to limit the worst case variation in the closed-loop
system, which results in a conservative design. A speci cation that limits the
typical variations in the closed-loop system (however we may de ne typical)
may better capture the designer's intention.
Robustness speci cations are often not closed-loop convex, just as the most gen-
eral speci cations that limit di erential sensitivity are not closed-loop convex. We
will describe a general small gain method for formulating convex inner approxima-
tions of robustness speci cations the Notes and References for this chapter describe
some of the attempts that have been made to make approximations of robustness
speci cations that are less conservative, but not convex. Since we will be describ-
ing convex approximations of robustness speci cations, we should add the following
item to the list of possible disadvantages:
The small gain based convex inner approximations of robustness speci cations
can be poor approximations. Thus, designs based on these approximations
can be conservative.
This topic is addressed in some of the references at the end of this chapter.
In the next section we give a precise and general de nition of a robustness
speci cation, which may appear abstract on rst reading. In the remainder of this
chapter we describe the framework for small gain methods, and then the small
gain methods themselves. The framework and methods are demonstrated on some
simple, speci c examples that are based on our standard example SASS 1-DOF
control system described in section 2.4. These examples continue throughout the
chapter.
10.1
Robustness Specifications
10.1.1
Some Definitions
In this section we give a careful de nition of a robustness speci cation we defer until
the next section examples of common robustness speci cations. Roughly speaking,
10.1 ROBUSTNESS SPECIFICATIONS
211
a robustness speci cation requires that some design speci cation must hold, even
D
if the plant is replaced by any pert from a speci ed set of possible perturbed
P
P
P
plants.
Let us be more precise. Suppose that is any set of ( + ) ( + )
P
n
n
n
n
w
u
z
y
transfer matrices. We will refer to as the perturbed plant set, and its elements as
P
perturbed plants. Let denote some design speci cation, i.e., a boolean function
D
on
transfer matrices, and let denote any
transfer matrix.
n
n
K
n
n
z
w
u
y
We say holds robustly for and if for each pert
,
De
nition
10.1:
D
K
P
P
2
P
D
holds for the transfer matrix pert + pert (
pert ) 1 pert.
;
P
P
K
I
;
P
K
P
z
w
z
u
y
u
y
w
In words, the design speci cation holds robustly for and if the controller
D
K
P
connected to any of the perturbed plants pert
yields a closed-loop system
K
P
2
P
that satis es . De nition 10.1 is not, by itself, a design speci cation: it is a
D
property of a controller and a set of transfer matrices. Note also that de nition 10.1
makes no mention of the plant .
P
Once we have the concept of a design speci cation holding robustly for a given
controller and perturbed plant set, we can de ne the notion of a robustness speci-
cation, which will involve the plant .
P
The robustness specication rob formed from , , and is
De
nition
10.2:
D
D
P
P
given by:
rob :
holds robustly for and
D
D
K
P
for every that satises
K
=
+
(
) 1
;
(10.1)
H
P
P
K
I
;
P
K
P
:
z
w
z
u
y
u
y
w
Thus a robustness speci cation is formed from a design speci cation , a per-
D
turbation plant set , and the plant . The reader should note that rob is indeed
P
P
D
a design speci cation: it is a boolean function of . We can interpret rob as fol-
H
D
lows: if satis es rob and is any controller that yields the closed-loop transfer
H
D
K
matrix (when connected to ), the closed-loop transfer matrix that results from
H
P
connecting to any pert
will all satisfy .
K
P
2
P
D
A sensible formulation of the plant will including signals such as sensor and
actuator noises and the sensor and actuator signals (recall chapter 7). In this case
the controller is uniquely determined by a closed-loop transfer matrix that
K
H
is realizable, since (
) 1 will appear as a submatrix of , and we can
;
I
;
P
K
H
y
u
determine from this transfer matrix. In these cases we may substitute \the "
K
K
for \every " in the de nition 10.2.
K
In many cases,
, and consists of transfer matrices that are \close" to .
P
2
P
P
P
In this context is sometimes called the nominal plant. In this case the robustness
P
speci cation rob requires that even with the worst perturbed plant substituted for
D
the nominal plant, the design speci cation will continue to hold.
D
212
CHAPTER 10 ROBUSTNESS SPECIFICATIONS VIA GAIN BOUNDS
If the design speci cation is stable, i.e., closed-loop stability (see chapter 7),
D
D
we call rob the robust stability design speci cation associated with and .
D
P
P
Throughout this chapter, is understood, so the robustness speci cation will
P
be written
rob(
)
D
P
D
:
The robust stability speci cation associated with the perturbed plant set will be
P
denoted
rob stab( ) = rob(
stable)
D
P
D
P
D
:
10.1.2
Time-Varying and Nonlinear Perturbations
It is possible to extend the perturbed plant set to include time-varying or nonlinear
systems, although this requires some care since many of our basic concepts and
notation depend on our assumption 2.2 that the plant is LTI. Such an extension
is useful for designing a controller for a nonlinear or time-varying plant nonlin.
K
P
The controller is often designed for a \nominal" LTI plant that is in some
K
P
sense \close" to nonlin nonlin is then considered to be a perturbation of .
P
P
P
In this section we brie y and informally describe how we may modify our frame-
work to include such nonlinear or time-varying perturbations. In this case the per-
turbed plant is a nonlinear or time-varying system with + inputs and +
n
n
n
n
w
u
z
y
outputs. The perturbed closed-loop system, obtained by connecting the controller
between the signals and of the perturbed plant, is now also nonlinear or time-
y
u
varying, so the perturbed closed-loop system cannot be described by an n n
z
w
transfer matrix, as in (10.1). Instead, the closed-loop system is described by the
nonlinear or time-varying closed-loop operator that maps into the resulting .
w
z
A design speci cation will simply be a predicate of the closed-loop system. The
only predicate that we will consider is closed-loop stability, which, roughly speaking,
means that is bounded whenever is bounded (the de nition of closed-loop
z
w
stability given in chapter 7 does not apply here, since it refers to the transfer matrix
). The reader can consult the references given at the end of this chapter for precise
H
and detailed de nitions of closed-loop stability of nonlinear or time-varying systems.
The robust stability speci cation rob stab will mean that when the (LTI) con-
D
troller , which is designed on the basis of the (LTI) plant , is connected to any
K
P
of the nonlinear or time-varying perturbed plants in , the resulting (nonlinear or
P
time-varying) closed-loop system is stable.
10.2
Examples of Robustness Specifications
In this section we consider some examples of robustness speci cations, organized
by their associated plant perturbation sets. Most of these robustness speci cations
10.2 EXAMPLES OF ROBUSTNESS SPECIFICATIONS
213
are not convex, but later in this chapter we describe a general method of forming
convex inner approximations of these speci cations.
10.2.1
Finite Plant Perturbation Sets
A simple but important case occurs when is a nite set:
P
= 1 ...
(10.2)
P
fP
P
g
:
N
Neglected Dynamics
Recall from chapter 1 that may be a simple (but not very accurate) model of
P
the system to be controlled. Our perturbed plant set might then be = cmplx ,
P
fP
g
where cmplx is a complex, detailed, and accurate model of the system to be con-
P
trolled. In this case, the robustness speci cation rob guarantees that the controller
D
we design using the simple model , will, when connected to cmplx, satisfy the
P
P
design speci cation .
D
As a speci c example, suppose that our plant is our standard numerical example,
the 1-DOF controller described in section 2.4, with plant
2
std
std 3
=
0
0 0
0
P
P
P
P
z
w
z
u
=
0
0 0
1
4
5
P
std
std :
P
P
y
w
y
u
0
1 1
0
;P
;
;P
The more detailed model of the system to be controlled might take into account a
high frequency resonance and roll-o in the system dynamics and some fast sensor
dynamics, neither of which is included in the plant model :
P
cmplx
cmplx
2
3
cmplx =
0
0
0
0
P
P
0
0
0
1
4
5
P
cmplx
cmplx
0
sens
sens 1
0
sens
;P
H
;H
;P
H
where
cmplx
std )
0
( ) =
0 (
P
s
sens( ) =
1
P
s
1 + 1 25( 100) + ( 100)2
H
s
1 + 80:
:
s=
s=
s=
This is shown in gure 10.1 below (c.f. gure 2.11).
For this example, the perturbed plant set is
= cmplx
(10.3)
P
fP
g:
The robust stability speci cation rob stab that corresponds to (10.3) requires
D
that the controller designed on the basis of the nominal plant will also stabilize the
P
complex model cmplx of the system to be controlled. Roughly speaking, rob stab
P
D
requires that the system cannot be made unstable by the high frequency resonance
and roll-o in the system dynamics and the dynamics of the sensor, which are
ignored in the model .
P
214
CHAPTER 10 ROBUSTNESS SPECIFICATIONS VIA GAIN BOUNDS
proc
p
sensor
u
n
y
n
+
+
+
r
q
r
q
sens
r
cmplx
r
H
K
;
+
0
P
+
The controller , which is designed on the basis of the model
Figure
10.1
K
std
0 , is connected to a more detailed model of the system to be controlled,
P
cmplx. The model cmplx includes a high frequency resonance and roll-o
P
P
in cmplx
0
, and the sensor dynamics (1 + 80) 1.
;
P
s=
Failure Modes
The perturbed plants in (10.2) may represent di erent failure modes of the system to
be controlled. For example, 1 might be a model of the system to be controlled after
P
an actuator has failed (i.e., 1 is , but with the column associated with the failed
P
P
actuator set to zero). In this case the speci cation of robust stability guarantees
that the closed-loop system will remain stable, despite the failures modeled by
1 ...
.
P
P
N
10.2.2
Parametrized Plant Perturbations
In some cases the perturbed plant set can be described by some parameters that
P
vary over ranges:
= pert( )
1
1
1 ...
P
P
L
U
L
U
:
k
k
k
In this case we often have
the corresponding parameter is called the nominal
P
2
P
parameter:
= pert( nom)
P
P
:
Parametrized plant perturbation sets can be used to model several di erent
types of plant variation:
Component tolerances. A single controller
is to be designed for many
K
plants, for example, a manufacturing run of the system to be controlled. The
controller is designed on the basis of a nominal plant the parameter variations
represent the (slight, one hopes) di erences in the individual manufactured
systems. Designing a controller that robustly achieves the design speci cations
avoids the need and cost of tuning each manufactured control system (see
section 1.1.5).
