6.241J Course Notes, Chapter 20: Stability Robustness
Lectures on Dynamic Systems and
Control
Mohammed Dahleh Munther A. Dahleh George Verghese
Department of Electrical Engineering and Computer Science
1
Massachuasetts Institute of Technology
1
c �
Chapter 20
Stability Robustness
20.1 Intro duction
Last chapter showed how the Nyquist stability criterion provides conditions for the stability
- p ossible to provide an extension of those conditions by gen robustness of a SISO system. It is
eralizing the Nyquist criterion for MIMO systems. This, however, turns out to b e unnecessary
derivation is p ossible through the small gain theorem, which will b e presented in and a direct
chapter. this
20.2 Additive Representation of Uncertainty
It is commonly the case that the nominal plant mo del is quite accurate for low frequencies
b ecause of parasitics, nonlinearities and/or time- but deteriorates in the high-frequency range,
varying e�ects that b ecome signi�cant at higher frequencies. These high-frequency e�ects may
have b een left unmo deled b ecause the e�ort required for system identi�cation was not justi�ed
by the level of p erformance that was b eing sought, or they may b e well-understo o d e�ects that
were omitted from the nominal mo del b ecause they were awkward and unwieldy to carry along
control design. This problem, namely the deterioration of nominal mo dels at higher during
extent by the fact that almost all physical systems have frequencies, is mitigated to some
prop er transfer functions, so that the system gain b egins to roll o� at high frequency. strictly
In the ab ove situation, with a nominal plant mo del given by the prop er rational matrix
P (s), the actual plant represented by P (s), and the di�erence P (s) ; P (s) assumed to b e
0 0
we may b e able to characterize the mo del uncertainty via a b ound of the form stable,
� [ P (j ! ) ; P (j ! )] � ` (! ) (20.1)
0 a max
where
(
\Small" � j! j � !
c
` (! ) � (20.2)
a
\Bounded" � j! j � !
c
This says that the resp onse of the actual plant lies in a \band" of uncertainty around that of
plant. Notice that no phase information ab out the mo deling error is incorp orated the nominal
into this description. For this reason, it may lead to conservative results.
The preceding description suggests the following simple additive characterization of the
uncertainty set:
� � fP (s) j P (s) � P (s) + W (s)�(s)g (20.3)
0
where � is an arbitrary stable transfer matrix satisfying the norm condition
k�k � sup � (�(j ! )) � 1 (20.4)
max 1
!
and the stable prop er rational (matrix or scalar) weighting term W (s) is used to represent
any information we have on how the accuracy of the nominal plant mo del varies as a function
frequency. Figure 20.1 shows the additive representation of uncertainty in the context of a of
servo lo op, with K denoting the comp ensator. standard
When the mo deling uncertainty increases with frequency, it makes sense to use a weight-
ing function W (j ! ) that lo oks like a high-pass �lter: small magnitude at low frequencies,
b ounded at higher frequencies. In the case of a matrix weight, a variation increasing but
� additive term W � is to use a term of the form W �W we leave you to on the use of the
1 2
how the analysis in this lecture will change if such a two-sided weighting is used. examine
- -
� W
� �
�
l l
- - - - -
.
y
+ + r
P K
0
;
6
Figure 20.1: Representation of the actual plant in a servo lo op via an additive p erturbation
plant. of the nominal
Caution: The ab ove formulation of an additive mo del p erturbation should not b e interpreted
saying that the actual or p erturb ed plant is the paral lel combination of the nominal system as
P (s) and a system with transfer matrix W (s)�(s). Rather, the actual plant should b e
0
b eing a minimal realization of the transfer function P (s), which happ ens to b e considered as
additive form P (s) + W (s)�(s). written in the
0
Some features of the ab ove uncertainty set are worth noting:
� The unstable p oles of all plants in the set are precisely those of the nominal mo del. Thus,
mo deling and identi�cation e�orts are assumed to b e careful enough to accurately our
p oles of the system. capture the unstable
� The set includes mo dels of arbitrarily large order. Thus, if the uncertainties of ma jor
were parametric uncertainties, i.e. uncertainties in the values of the concern to us
parameters of a particular (e.g. state-space) mo del, then the ab ove uncertainty set
would greatly overestimate the set of plants of interest to us.
The control design metho ds that we shall develop will pro duce controllers that are guar-
anteed to work for every member of the plant uncertainty set. Stated slightly di�erently,
metho ds will treat the system as though every mo del in the uncertainty set is a p ossible our
representation of the plant. To the extent that not all memb ers of the set are p ossible plant
mo dels, our metho ds will b e conservative.
20.3 Multiplicative Representation of Uncertainty
Another simple means of representing uncertainty that has some nice analytical prop erties is
multiplicative perturbation, which can b e written in the form the
� � fP j P � P (I + W �)� k�k � 1g: (20.5)
0 1
- -
� W
� �
�
m -
.
- -
+
P
0
Figure 20.2: Representation of uncertainty as multiplicative p erturbation at the plant input.
An alternative to this input-side representation of the uncertainty is the following output-
representation: side
� � � fP j P � (I + W �)P k�k � 1g: (20.6)
0 1
In b oth the multiplicative cases ab ove, W and � are stable. As with the additive represen-
tation, mo dels of arbitrarily large order are included in the ab ove sets. Still other variations
may b e imagined� in the case of matrix weights, for instance, the term W � can b e replaced
. by W �W
1 2
The caution mentioned in connection with the additive p erturbation b ears rep eating
ab ove multiplicative characterizations should not b e interpreted as saying that the here: the
plant is the cascade combination of the nominal system P and a system I + W �. actual
0
plant should b e considered as b eing a minimal realization of the transfer Rather, the actual
(s), which happ ens to b e written in the multiplicative form. function P
Any unstable p oles of P are p oles of the nominal plant, but not necessarily the other
way, b ecause unstable p oles of P may b e cancelled by zeros of I + W �. In other words,
0
plant is allowed to have fewer unstable p oles than the nominal plant, but all its the actual
p oles are con�ned to the same lo cations as in the nominal mo del. In view of the unstable
such cancellations do not corresp ond to unstable hidden caution in the previous paragraph,
mo des, and are therefore not of concern.
20.4 More General Representation of Uncertainty
Consider a nominal interconnected system obtained by interconnecting various (reachable and
observable) nominal subsystems. In general, our representation of the uncertainty regarding
any nominal subsystem mo del such as P involves taking the signal � at the input or output
0
\uncertainty blo ck" with transfer function of the nominal subsystem, feeding it through an
, where W �W each factor is stable and k�k � 1, and then adding the output W � or
1 2 1
� of this uncertainty blo ck to either the input or output of the nominal subsystem. The
additive and two multiplicative representations describ ed earlier are sp ecial cases of this one
p ossibilities with construction, but the construction actually yields a total of three additional
given uncertainty blo ck. Sp eci�cally, if the uncertainty blo ck is W �, we get the following a
feedback representations of uncertainty: additional
;1
� P � P (I ; W �P ) �
0 0
;1
� P � P (I ; W �) �
0
;1
. � P � (I ; W �) P
0
uncertainty representations itemized ab ove is that the unstable A useful feature of the three
p oles of the actual plant P are not constrained to b e (a subset of ) those of the nominal plant
. P
0
Note that in all six representations of the p erturb ed or actual system, the signals � and
� b ecome internal to the actual subsystem mo del. This is b ecause it is the combination of
P with the uncertainty mo del that constitutes the representation of the actual mo del P , and
0
mo del is only accessed at its (overall) input and output. the actual
In summary, then, p erturbations of the ab ove form can b e used to represent many typ es of
uncertainty, for example: high-frequency unmo deled dynamics, unmo deled delays, unmo deled
variations. sensor and/or actuator dynamics, small nonlinearities, parametric
20.5 A Linear Fractional Description
, and a We start with a given a nominal plant mo del P feedback controller K that stabilizes
0
. The robust P stability question is then: under what conditions will the controller stabilize
0
al l P 2 �� More generally, we assume we have an interconnected system that is nominally
internally stable, by which we mean that the transfer function from an input added in at
any subsystem input to the output observed at any subsystem output is always stable in
stability question is then: under what conditions will the the nominal system. The robust
interconnected system remain internally stable for all p ossible p erturb ed mo dels.
If the plant uncertainty is sp eci�ed (additively, multiplicatively, or using a feedback
representation) via an uncertainty blo ck of the form W �, where W and � are stable, then
(closed-lo op) system can b e mapp ed into the very simple feedback con�guration the actual
�
�
� �
-
G
z w
- -
Figure 20.3: Standard mo del for uncertainty.
shown in Figure 20.3. (The generalization to an uncertainty blo ck of the form W �W is
1 2
avoid additional notation.) trivial, and omitted here to
resp ectively denote the input and output As in the previous subsection, the signals � and �
uncertainty blo ck. The input w is added in at some arbitrary accessible point of the of the
interconnected system, and z denotes an output taken from an arbitrary accessible p oint. An
p oint in our terminology is simply some subsystem input or output in the actual accessible
p erturb ed system� the input � and output � of the uncertainty blo ck would not qualify as or
p oints. accessible
If we remove the p erturbation blo ck � in Fig. 20.3, we are left with the nominal closed-
lo op system, which is stable by hyp othesis (since the comp ensator K has b een chosen to
plant and is lump ed in G). Stability of the nominal system implies that stabilize the nominal
the transfer functions relating the outputs � and z of the nominal system to the inputs � and
Thus, in the transfer function representation w are all stable.
� ! � ! � !
�(s) M (s) N (s) �(s)
� (20.7)
Z (s) J (s) L(s) W (s)
each of the transfer matrices M , N , J , and L is stable.
Now incorp orating the constraint imp osed by the p erturbation, namely
� � (�) � (20.8)
and solving for the transfer function relating z to w in the perturbed system, we obtain
;1
G (s) � L + J �(I ; M �) N : (20.9)
w z
Note that M is the transfer function \seen" by the p erturbation �, from the input � that
imp oses on the rest of the system, to the output � that it measures from the rest of the it
p oints system. Recalling that w and z denoted arbitrary inputs and outputs at the accessible
closed-lo op system, we see that internal stability of the actual (i.e. p erturb ed) of the actual
closed-lo op system requires the ab ove transfer function b e stable for all allowed �.
20.6 The Small-Gain Theorem
;1
other than Since every term in G (I ; M �) is known to b e stable, we shall have stability of
w z
;1
is stable , and hence G guaranteed stability of the actual closed-lo op system, if (I ; M �)
w z
allowed �. In what follows, we will arrive at a condition | the smal l-gain condition for all
;1
. It can also b e shown (see App endix) that if guarantees the stability of (I ; M �) | that
;1
and k�k � 1 such that (I ; M �) this condition is violated, then there is a stable � with
1
;1
is unstable for some are unstable, and G choice of z and w . �(I ; M �)
w z
Theorem 20.1 (\Unstructured" Small-Gain Theorem) De�ne the set of stable pertur-
4
;1 ;1
6
bation matrices � � f� j k�k � 1g. If M is stable, then (I ; M �) and �(I ; M �)
1
6
are stable for each � in � if and only if kM k � 1.
1
Pro of. The pro of of necessity (see App endix) is by construction of an allowed � that causes
;1 ;1
is unstable. and �(I ; M �) to b e unstable if kM k � 1, and ensures that G (I ; M �)
w z 1
For here, we fo cus on the pro of of su�ciency. We need to show that if kM k � 1 then
1
;1
6
has no p oles in the closed right half-plane for any � 2 (I ; M �) �, or equivalently that
� 0 and any s in the closed right half-plane M � has no zeros there. For arbitrary x 6 I ;
+
b oth M and � are well-de�ned throughout the CRHP, we (CRHP), and using the fact that
can deduce that
k[I ; M (s )�(s )]xk � kxk ; kM (s )�(s )xk
+ + + + 2 2 2
� kxk ; � [M (s )�(s )]k xk
max + + 2 2
� kxk ; kM k k�k kxk
1 2 1 2
� 0 (20.10)
The �rst inequality ab ove is a simple application of the triangle inequality. The third inequal-
ity ab ove results from the Maximum Modulus Theorem of complex analysis, which says that
over a region of the complex plane is found on the the largest magnitude of a complex function
b oundary of the region, if the function is analytic inside and on the b oundary of the region.
0 0 0 0
M M q and q � �q are stable, and therefore analytic, in the CRHP, for b oth q In our case,
vectors q � hence their largest values over the CRHP are found on the imaginary axis. unit
inequality in the ab ove set is a consequence of the hyp otheses of the theorem, and The �nal
M � is nonsingular | and therefore has no zeros | in the CRHP. establishes that I ;
20.7 Stability Robustness Analysis
Next, we present a few examples to illustrate the use of the small-gain theorem in stability
robustness analysis.
Example 20.1 (Additive Perturbation)
For the con�guration in Figure 20.1, it is easily seen that
;1 ;1
M � ;K (I + P K ) W � ;(I + K P ) K W
0 0
Example 20.2 (Multiplicative Perturbation)
A multiplicative p erturbation of the form of Figure 20.2 can b e inserted into the
closed-lo op system at either the plant input or output. The pro cedure is then
identical to Example 20.1, except that M b ecomes a di�erent function. Again it
veri�ed that for a multiplicative p erturbation at the plant input, is easily
;1
M � ;(I + K P ) K P W� (20.11)
0 0
while a p erturbation at the output yields
;1
M � ;(I + P K ) P K W: (20.12)
0 0
What the ab ove examples show is that stability robustness requires ensuring the weighted
versions of certain familiar transfer functions have H norms that are less than 1. For
1
multiplicative p erturbation at the output as in the last example, what we instance, with a
stability robustness is kT W k � 1, where T is the complementary sensitivity require for
1
asso ciated with the nominal closed-lo op system. This condition evidently has the function
�avor as the conditions we discussed earlier in connection with nominal p erformance of same
closed-lo op system. the
The small-gain theorem fails to take advantage of any sp ecial structure that there might
6
b e in the uncertainty set �, and can therefore b e very conservative. As examples of the kinds
following two examples. of situations that arise, consider the
Example 20.3
Supp ose we have a system that is b est represented by the mo del of Figure 20.4.
have a blo ck-diagonal When this system is reduced to the standard form, � will
� -
W � � W
a a b b
6
6
� �
m m m
- - -
.
- - -
+ + P +
0 K
;
6
Figure 20.4: Plant with multiple uncertainties.
two p erturbations enter at di�erent p oints in the system: structure, since the
" #
� 0
a
� � (20.13)
0 �
b
Thus, there is some added information ab out the plant uncertainty that can-
not b e captured by the unstructured small-gain theorem, and in general, even if
kM k � 1 for the M that corresp onds to the � ab ove, there may b e no admissible
1
;1
. p erturbation that will result in unstable (I ; M �)
Example 20.4
Supp ose that in addition to norm b ounds on the uncertainty, we know that the
� �
� 30 ]. Again, even if kM k � p erturbation remains in the sector [;30 phase of the
1
corresp onds to the � for this system, there may b e no admissible 1 for the M that
;1
. p erturbation that will result in unstable (I ; M �)
In b oth of the preceding two examples, the unstructured small-gain theorem gives con-
servative results.
Relating Stability Robustness to the (SISO) Nyquist Criterion
Supp ose we have a SISO nominal plant with a multiplicative p erturbation, and a nominally
controller K . Then P � P (1 + W �), and the comp ensated op en-lo op transfer stabilizing
0
function is
K + P K � P P K W �: (20.14)
0 0
, Since P K , and W are known and j�j � 1 with arbitrary phase, we may deduce from (20.14)
0
any given frequency ! is contained in a region delimited by that the \real" Nyquist plot at
0
), with radius centered at P (j ! )K (j ! jP K W (j ! )j. This is illustrated in Figure a circle
0 0 0 0 0
Clearly, if the circle of uncertainty ever includes ;1, there is the p ossibility that the 20.5(a).
encirclement, and hence is unstable. We may relate this \real" Nyquist plot has an extra
stability problem as follows. From Example 20.2, the SISO system is robustly to the robust
by the small gain theorem if stable
� �
� �
K P
0
� �
W � 1� 8 ! : (20.15)
� �
K 1 + P
0
Equivalently,
jP K W j � j1 + P K j: (20.16)
0 0
The right-hand side of (20.16) is the magnitude of a translation of the Nyquist plot of the
lo op transfer function. In Figure 20.5(b), b ecause of the translation, encirclement nominal
Clearly, this cannot happ en if (20.16) is satis�ed. This of zero will destabilize the system.
makes the relationship of robust stability to the SISO Nyquist criterion clear.
Performance as Stability Robustness
Supp ose that, for some plant mo del P , we wish to design a feedback controller that not only
plant (�rst order of priority!), but also provides some p erformance b ene�ts, such stabilizes the
improved output regulation in the presence of disturbances. Given that something is known as ω ω ωo ω o Po KW( j o ) Po KW( j o )
Po K + 1
-1 1
Po K Po K + 1
(a) (b)
Figure 20.5: Relation of Nyquist criterion and robust stability.
ab out the frequency sp ectrum of such disturbances, the system mo del might lo ok like Figure
k� k � 1, and the mo deling �lter W can b e constructed to capture frequency 20.6, where
2
characteristics of the disturbance. Calculating the transfer function of this lo op from � to
;1
W � . We assume that the p erformance sp eci�cation will b e y , we have that y � (I + P K )
;1
W k � 1, which do es not restrict the problem, since W can always b e k(I + P K ) met if
1
p erformance sp eci�cation. This scaled to re�ect the actual magnitude of the disturbance or
formulation lo oks analogous to a robust stability problem, and indeed, it can b e veri�ed that
identical constraint the small-gain theorem applied to the system of Figure 20.7 captures the
into the standard form of Figure on the system transfer function. By mapping this system
;1
W , which is exactly the M that is needed if the small-gain we �nd that M � (I + P K ) 20.3,
condition is to yield the desired condition.
Finally, plant uncertainty has to b e brought into the picture simultaneously with the
�
�
W
d
�
- l - - - l -
.
r y + +
K P
6
;
Figure 20.6: Plant with disturbance.
�
� �
�
W
d
�
- l - - - l -
. .
y
r + +
K P
6
;
Figure 20.7: Mapping p erformance sp eci�cations into a stability problem.
p erformance constraints. This is necessary to formulate the performance robustness problem.
It should b e evident that this will lead to situations with blo ck-diagonal �, as was obtained
in the context of the last example in the previous subsection. The treatment of this case will
values, as we shall see in the next lecture. require the notion of structured singular
App endix
Necessity of the small gain condition for robust stability can b e proved by showing that if
)] � 1 for some , � [M (j ! ! we can construct a � of norm less than one, such that the
max 0 0
is unstable. This is done as closed-lo op map G follows. Take the singular value resulting
z v
), decomp osition of M (j !
0
2 3
�
1
6 7
0 0
.
.
M (j ! ) � U �V � U V : (20.17)
4 5
0
.
�
n
)] � 1, ) can Since � [M (j ! � � 1. Then �(j ! b e constructed as:
max 0 0 1
2 3
1��
1
6 7
0
6 7
0
6 7
�(j ! ) � V U (20.18)
0
.
6 7
.
.
4 5
0
) � 1. Clearly, � �(j ! We then have
max 0
2 3 2 3
1�� �
1 1
6 7 6 7
� 0
6 7 6 7
2
;1
0 0
6 7 6 7
(I ; M �) (j ! ) � I ; U V U V
0
. .
6 7 6 7
. .
. .
4 5 4 5
� 0
n
2 2 33
1
6 6 77
0
6 6 77
0
6 6 77
� U I ; U (20.19)
.
6 6 77
.
.
4 4 55
0
2 3
0
6 7
1
6 7
0
6 7
� U U
.
6 7
.
.
4 5
1
which is singular. Only one problem remains, which is that �(s) must b e legitimate as the
system, evaluating to the prop er value at s � j ! , and having transfer function of a stable
0
maximum singular value over all ! b ounded b elow 1. The value of the destabilizing its
p erturbation at ! is given by
0
1
0
� (j ! ) � v u
0 0 1
1
)) � (M (j !
max 0
0
Write the vectors v and u as
1
1
2 3
j �
1
�ja je
1
6 7
j �
2
h i
�ja je
6 7
2
0
j � j � j �
n
1 2
6 7
� v � u � �jb je �jb je � � � �jb je � (20.20)
.
1 1 2 n
1
6 7
.
.
4 5
j �
n
�ja je
n
where � and � b elong to the interval [0� � ). Note that we used � in the representation of
i i
0
so that we can restrict the angles � and � to the interval [0� � ). Now vectors v and u the
i i 1
1
� � we can cho ose the nonnegative constants � � � � � � � � and � � � � � � � � such that the
1 2 1 2 n n
s;�
s;�
i
i
phase of the function , and the phase of the function at s � j ! is � at s � j ! is
i 0 0
s+� s+�
i i
. � Now the destabilizing �(s) is given by
i
1
T
�(s) � g (s)h (s) (20.21)
� (M (j ! ))
max 0
where
2 3
2 3
s;�
1
s;�
1
�jb j
�ja j
1
1
s+�
1
s+�
1
6 7
6 7
s;�
s;�
2
2
6 7
�ja j
�jb j
6 7
2
2
s+�
s+�
6 7
2
2
6 7
g (s) � � h(s) � : (20.22)
6 7
.
.
6 7
.
6 7
.
4 5
.
.
4 5
s;�
n
s;�
n
�ja j
�jb j
n
n
s+�
n
s+�
n
Exercises
Exercise 20.1 Consider a plant describ ed by the transfer function matrix
� �
� 1
s;1 s;1
P (s) �
�
2s;1 1
s(s;1) s;1
where � is a real but uncertain parameter, con�ned to the range [0:5 � 1:5]. We wish to design a
feedback comp ensator K (s) for robust stability of a standard servo lo op around the plant.
(a) We would like to �nd a value of �, say �~ , and a scalar, stable, prop er rational W (s) such that the
p ossible plants P (s) is contained within the \uncertainty set" set of
�
P (s)[I + W (s)�(s)]
~ �
where �(s) ranges over the set of stable, prop er rational matrices with k�k � 1. Try and �nd
1
p ossible!) a suitable � and W (s), p erhaps by keeping in mind that ~ (no assurances that this is
we really want to do is guarantee what
;1
� fP (j ! )[P (j ! ) ; P (j ! )]g � jW (j ! )j
~ max � �
�~
What sp eci�c choice of �(s) yields the plant P (s) (i.e. the plant with � � 1) �
1
(b) Rep eat part (a), but now working with the uncertainty set
P (s)[I + W (s)�(s)W (s)]
~ � 1 2
where W (s) and W (s) are column and row vectors resp ectively, and �(s) is scalar. Plot the
1 2
upp er b ound on
;1
� fP (j ! )[P (j ! ) ; P (j ! )]g
~ max � �
�~
that you obtain in this case.
(c) For each of the cases ab ove, write down a su�cient condition for robust stability of the closed-lo op
involving the nominal complementary sensitivity system, stated in terms of a norm condition
;1
and W | or, in part (b), . � (I + K P ) K P W and W function T
~ ~ � � 2 1
Exercise 20.2 It turns out that the small gain theorem holds for nonlinear systems as well. Con-
sider a feedback con�guration with a stable system M in the forward lo op and a stable, unknown
p erturbation in the feedback lo op. Assume that the con�guration is well-p osed. Verify that the closed
lo op system is stable if kM kk�k � 1. Here the norm is the gain of the system over any p-norm. (This
b oth DT and CT systems� the same pro of holds). result is also true for
Exercise 20.3 The design of a controller should take into consideration quantization e�ects. Let us
variable in the closed lo op which is sub ject to quantization is the output of the assume that the only
plant. Two very simple schemes are prop osed:
-
P
0
� �
Q
K
Figure 20.8: Quantization in the Closed Lo op.
-
P
0
� �
f
K
6
n
Figure 20.9: Quantization Mo deled as Bounded Noise.
1. Assume that the output is passed through a quantization op erator Q de�ned as:
jxj
Q(x) � ab csg n(x)� a � 0
:5 + a
where br c denotes the largest integer smaller than r . The output of this op erator feeds into
controller as in Figure 20.8. Derive a su�cient condition that guarantees stability in the the
Q. presence of
2. Assume that the input of the controller is corrupted with an unknown but b ounded signal, with
b ound as in Figure 20.9. Argue that the controller should b e designed so that it do es a small
not amplify this disturbance at its input.
Compare the two schemes, i.e., do they yield the same result� Is there a di�erence� MIT OpenCourseWare http://ocw.mit.edu
6.241J / 16.338J Dynamic Systems and Control Spring 2011
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