Learning, control and hybrid systems: Festschrift in honor of Bruce Allen Francis and Mathukumalli Vidyasagar on the occasion of their 50th birthdays

441 Pages·1999·18.71 MB·English

by  M. Vidyasagar (auth.)

dezimodnaR Algorithms rof
Robust rellortnoC Synthesis Using
Statistical gninraeL Theory
.M Vidyasagar*
Abstract
By now it is known that several problems in the robustness analysis
and synthesis of control systems are NP-complete or NP-hard. These
negative results force us to modify our notion of "solving" a given prob-
lem. If we cannot solve a problem exactly because it is NP-hard, then
we must settle for solving it approximately. If we cannot solve all in-
stances of a problem, we must settle for solving "almost all" instances
of a problem. An approach that is recently gaining popularity is that of
using randomized algorithms. The notion of a randomized algorithm as
defined here is somewhat different from that in the computer science lit-
erature, and enlarges the class of problems that can be efficiently solved.
We begin with the premise that many problems in robustness analysis
and synthesis can be formulated as the minimization of an objective
function with respect to the controller parameters. It is argued that,
in order to assess the performance of a controller as the plant varies
over a prespecified family, it is better to use the average performance
of the controller as the objective function to be minimized, rather than
its worst-case performance, as the worst-case objective function usually
leads to rather conservative designs. Then it is shown that a prop-
erty from statistical learning theory known as uniform convergence of
empirical means (UCEM) plays an important role in allowing us to con-
struct efficient randomized algorithms for a wide variety of controller
synthesis problems. In particular, whenever the UCEM property holds,
there exists an efficient (i.e., polynomial-time) randomized algorithm.
Using very recent results in VC-dimension theory, it is shown that the
UCEM property holds in several problems such as robust stabilization
and weighted H2/Hoo-norm minimization. Hence it is possible to solve
such problems efficiently using randomized algorithms.
*Centre for Artificial Intelligence and Robotics, Raj Bhavan Circle, High Grounds, Ban-
galore 065 001, India. E-Mail: sagar~cair.ernet.in
4 Vidyasagar
1 Introduction
During recent years it has been shown that several problems in the robustness
analysis and synthesis of control systems are either NP-complete or NP-hard.
See 16, 15, ,1 2 for some examples of such NP-hard problems. In the face
of these and other negative results, one is forced to make some compromises
in the notion of a "solving" a problem. An approach that is recently gaining
popularity is the use of randomized algorithms, which are not required to work
"all" of the time, only "most" of the time. Specifically, the probability that
the algorithm fails can be made arbitrarily small (but of course not exactly
equal to zero). In return for this compromise, one hopes that the algorithm
is effeient, i.e., runs in polynomial-time. The idea of using randomization
to solve control problems is suggested, among other places, in 17, 13. In
10, 19, randomized algorithms are developed for a few problems such as: (i)
determining whether a given controller stabilizes every plant in a structured
perturbation model, (ii) determining whether there exists a controller of a
specified order that stabilizes a given fixed plant, and so on.
The objective of the present paper is to demonstrate that some recent ad-
vances in statistical learning theory can be used to develop efficient random-
ized algorithms for a wide variety of controller analysis/synthesis problems.
For such problems, it is shown that whenever a property known as uniform
convergence of empirical means (UCEM) holds, there exists an efficient ran-
domized algorithm for an associated function minimization problem. Some
specific problems that fall within this framework include: robust stabilization
of a family of plants by a fixed controller belonging to a specified family of
controllers, and the minimization of weighted H2/Hoo-norms. Note that what
is developed here is a broad framework that can accommodate a wide variety
of problems. The few specific problems solved here are meant only to illustrate
the power and breadth of this framework. No doubt other researchers would
be able to solve many more problems using the same general approach.
2 Paradigm of Controller Synthesis Problem
Suppose one is given a family of plants {G(x), x E X} parametrized by x, and
a family of controllers {K(y), y E Y} parametrized by y. The objective is to
find a a single fixed controller K(yo), oY E Y that performs reasonably well for
almost all plants G(x). By choosing an appropriate performance index, many
problems in controller synthesis can be covered by the above statement. The
objective of this section is to put forward an abstract problem formulation
that makes the above statement quite precise, and which forms the "universe
of discourse" for the remainder of the paper. In particular, it is argued that,
to avoid overly conservative designs, the performance of a controller should be
taken as its average performance as the plant varies over a prespecified family,
Randomized Algorithms for Robust Controller Synthesis 5
and not its worst-case performance.
Suppose r .) is a given cost function. Thus r K) is a measure of
the performance of the system when the plant is G and the controller is K.
The phrase "cost function" implies that lower values of r are preferred. For
instance, if the objective is merely to choose a stabilizing controller, then one
could define
,1 if the pair (G,K) is unstable,
(2.1)
r K) := 0, if the pair (G, K) is stable.
As a second example, one could choose
i, if the pair (G, K) is unstable,
r K) := J(G, K)/1 + J(G, K), if the pair (G, K) is stable,
(2.2)
where
J(a, K) 1= W(I + ag) 1- 2
if one wishes to study filtering problems, and
J(G, K) :=II W(I + GK) -1 261rII
if one wishes to study problems of optimal rejection of disturbances. Note
that W is a given weighting matrix. Two points should be noted in the above
definition: (i) The usual weighted//2 or H~-norm denoted by J(G, K) takes
values in ,0 c~). For purely technical reasons that will become clear later,
this cost function is rescaled by defining r = J/(1 + J), so that r K) takes
values in ,0 .1 (ii) To guard against the possibility that W(I+GK) -1 belongs
to/-/2 even though the pair (G, K) is unstable, 1 the cost function r K) is
explicitly defined to be 1 (corresponding to J = oo), if the pair (G, K) is
unstable.
The preceding discussion pertains only to quantifying the performance of a
single plant-controller pair. However, in problems of robust stabilization and
robust performance, the cost function should reflect the performance of a fixed
controller for a variety of plants. Since G = G(x) and K = K(y), let us define
g(x, y) := r g(y).
Note that g depends on both the plant parameter x E X and the controller
parameter y E Y. As such, g maps X x Y into ,0 .1 The aim is to define an
objective function of y alone that quantifies the performance of the controller
K(y), so that by minimizing this objective function with respect to y one could
find an "optimal" controller.
As a first attempt, one could choose
h(y) := sup g(x, y) = sup r g(y)l. (2.3)
xEX xEX
1For example, consider the case where W = 1/(s + 1) 2 and (1 + GK) -1 is improper and
behaves as O(s) as s ~-- oo, but has all of its poles in the open left half-plane.
6 Vidyasagar
Thus h(y) measures the worst-case performance of a controller K(y) as the
plant varies over {G(x), x E X}. For instance, if one chooses r as in (2.1),
then h(y) = 0 if and only if the controller K(y) stabilizes every single plant in
{G(x), x E X}. If K(y) fails to stabilize even a single plant, then h(y) = .1
Thus minimizing the present choice of h(.) corresponds to solving the robust
(or simultaneous) stabilization problem. Similarly, if r K) is chosen as in
(2.2), then minimizing the associated h(-) corresponds to achieving the best
possible guaranteed performance with robust stabilization.
It is widely believed that methods such as Hoo-norm minimization for
achieving robust stabilization, and #-synthesis for achieving guaranteed per-
formance and robust stabilization, lead to overly conservative designs. Much
of the conservatism of the designs can be attributed to the worst-case nature
of the associated cost function. It seems much more reasonable to settle for
controllers that work satisfactorily "most of the time." One way to capture
this intuitive idea in a mathematical framework is to introduce a probability
measure P on the set X, that reflects one's prior belief on the way that the
"true" plant G(x) is distributed in the set of possible plants {G(x), x E X}.
For instance, in a problem of robust stabilization, Go can be a nominal, or
most likely, plant model, and the probability measure P can be "peaked"
around Go. The more confident one is about the nominal plant model Go, the
more sharply peaked the probability measure P can be. Once the probability
measure P is chosen, the objective function to be minimized can be defined as
f(y) --: Epg(x, y) = Epr g(y)). (2.4)
Thus/(y) is the expected or average performance of the controller K(y) when
the plant is distributed according to the probability measure P. The expected
value type of objective function captures the intuitive idea that a controller
can occasionally be permitted to perform poorly for some plant conditions,
provided these plant conditions are not too likely to occur.
While the worst-case objective function defined in (2.3) is easy to under-
stand and to interpret, the interpretation of the expected-value type of ob-
jective function defined in (2.4) needs a little elaboration. Suppose r is
defined as in (2.1). Then f(y) is the measure (or "volume") of the subset of
{G(x), x e X} that fails to be stabilized by the controller K(y). Alternatively,
one can assert with confidence 1 -/(y) that the controller K(y) stabilizes a
plant G(x) selected at random from {G(x), x e X} according to the probabil-
ity measure P. More generally, suppose r (or equivalently g) assumes values in
,0 )oo (and not just ,0 1 as assumed elsewhere). Then a routine computation
shows that, for each 7 >/(y), we have
P{x (cid:12)9 X: r >-y} = P{x (cid:12)9 X: g(x,y) _> 7} _< I(Y)/'Y.
Hence, given any ,~ > f(y), it can be asserted with confidence 1 - f(y)/~/that
r K(y) < ,q for a plant G(x) chosen at random from {G(x), x (cid:12)9 X}
according to the probability measure P.
Randomized Algorithms for Robust Controller Synthesis 7
3 Various Types of "Near" Minima
Suppose Y is a given set, t : Y ~ ~ is a given function, and that it is desired
to minimize f(Y) with respect to y. There are many problems, such as those
mentioned in Section 1, in which finding the minimum value
f* := inf f(y)
yEY
is NP-hard. More precisely, given a number t0, it is NP-hard to determine
whether or not to >_ f*- In such cases, one has to be content with "nearly"
minimizing f('). The objective of this section is to introduce three different
definitions of "near minima."
Definition 1 Suppose f : Y ~ ~ and that e > 0 is a given number. A number
to E ~ is said to be a Type 1 near minimum of t(') to accuracy e, or an
approximate near minimum of t(') to accuracy e, if
f(y)-e<fo f(y)+e, )1.3(
inf < inf
yEY -- -- yEY
or equivalently
fo-- fniyEy t(Y) _ < e.
Definition 2 Suppose f : Y +-- ~, that Q is a given probability measure on
Y, and that a > 0 is a given number. A number fo E ~ is said to be a Type
2 near minimum of f(-) to level a, or a probable near minimum of
f(.) to level a, if to _> f*, and in addition
Q{y E Y: t(y) < to} < a.
The notion of a probable near minimum can be interpreted as follows: f0
is a probable near minimum of t(') to level a if there is an "exceptional set"
S with Q(S) _< a such that
In other words, t0 is bracketed by the infimum of t(') over all of Y, and the
infimum of f(') over "nearly" all of Y.
It is important to note that even if t0 is a probable minimum of f(')
to some level a, however small a might be, the difference t0 - f* could be
arbitrarily large, or even infinite. In fact, it is possible for a finite number
to be a probable near minimum of a function that is unbounded from below.
Thus a probable near minimum to level a can be interpreted as follows: If one
person gives the number t0 to the adversary, and challenges the adversary to
8 Vidyasagar
"beat" f0 by producing a y E Y such that f(y) < f0, and if the adversary
tries to produce a suitable y by choosing y E Y at random according to the
probability measure Q, then his/her chances of winning are no more than a.
However, if the adversary does succeed in producing a y such that f(y) < fo,
then the difference fo - f(Y) could be arbitrarily large.
Definition 3 Suppose f : Y ~ ~, that Q is a given probability measure
on Y, and that e,a > 0 are given numbers. A number fo E ~ is said to
eb a Type 3 near minimum of f(.) to accuracy e and level a, or a
probably approximate near minimum of f(.) to accuracy e and level
a, if fo _> f* e, and in addition
-
Q{y E Y: f(y) < fo - e} ~ .~t
Another way of saying this is that there exists an "exceptional set" S _C Y
with Q(S) _< a such that
inf f(y)-e< fo < inf f(y)+e. (3.3)
yet
-- -- y E Y \ S
A comparison of (3.1), (3.2) and (3.3) brings out dearly the spihsnoitaler
neewteb the suoirav types of near .aminim
4 A General Approach to Randomized Algo-
rithms
In this section, a general approach is outlined that could be used to develop
randomized algorithms for minimizing an objective function of the type (2.4).
Subsequent sections contain a study of some specific situations in which this
general approach could be profitably applied.
4.1 The UCEM Property
Let us return to the specific problem of minimizing the type of objective func-
tion introduced in (2.4), namely
f(y) : EBb(X, .)Y
In general, evaluating an expected value exactly is not an easy task, since an
expected value is just an integral with respect to some measure. However, it is
possible to approximate an expected value to arbitrarily small error, as follows:
Let us for the time being ignore the y variable, and suppose a : X -+ ,0 1 is
a measurable function. Then
Ep(a) = Ix a(x) P(dx).
Randomized Algorithms for Robust Controller Synthesis 9
To approximate Ep(a), one generates i.i.d, samples xl,... ,x,, E X dis-
tributed according to P, and defines
1 m
k(a;x) := - a(xj).
m
j=l
The number E(a; x) is referred to as the empirical mean of the function a(.)
based on the multisample x := Ix1 ... x,~ t E X m. Note that/~(a; x) is itself
a random variable on the product space X '~. Now one can ask: How good an
approximation is E(a; x) to the true mean Ep(a)? As estimate is given by a
well-known bound known as Hoeffding's inequality 7, which states that
for each e > 0, we have
P"~{x E X m : IE(a; x) - Ep(a) > e} <_ 2 exp(-2me2).
All the material presented thus far in this subsection is standard and clas-
sical. Now we come to some recent ideas. Suppose .4 is a family of measurable
functions mapping X into 0, 1. Note that .4 need not be a finite family.
For each function a E .4, one can form an empirical mean /~(a; x) using a
multisample x E X m, as described above. Now let us define
q(m,e;.4) := Pm{x E Xm: sup J/~(a;x) - Ep(a) > e},
aEA
or equivalently,
q(m,e;-4) := Pm{x E X m : 3a E A s.t. /~(a;x) - Ep(a) > e}.
Then, after m i.i.d, samples have been drawn and an empirical mean/~(a; x) is
computed for each function a E ,4, it can be said with confidence 1 - q(m, ;e -4)
that every single empirical mean/~(a; x) is within e of the corresponding true
Ep(a). The family -4 is said to have the property of uniform convergence
of empirical means (UCEM) if q(m, ;e )4- ~-- 0 as m ~-- r
Note that if the family .4 is finite, then by repeated application of Hoeffd-
ing's inequality it follows that
q(m, e;,4) < 21AI exp(-2me2). (4.1)
Hence every finite family of functions has the UCEM property. However, an
infinite family need not have the UCEM property. During the last twenty five
years or so, many researchers have studied this property. A standard reference
for some of the early results is 20, while a recent and thorough treatment can
be found in 23. In particular, Section 3.1 of 23 contains a detailed discussion
of the UCEM property as well as several examples.
10 Vidyasagar
4.2 An Approach to Finding Approximate Near Minima
with High Confidence
The notion of UCEM (uniform convergence of empirical means) introduced
in the preceding subsection suggests the following approach to finding an ap-
proximate near minimum of the objective function (2.4). Let us introduce the
notation
gy(x) := g(x,y), Vx e X, Vy E Y.
Thus for each y E Y, the function gy(.) maps X into ,0 .1 Now define the
associated family of functions G := (gy(.), y e Y}. Suppose now that x :=
lX ... x,~ t e X m is a collection of i.i.d, samples. For each function 9~(.) E ~,
one can define its empirical mean based on the multisample x as
k(g,;x) := ~ ~g,(m x,) m
= ~ ~g(x~,y), y e y.
j=l j=l
As before, let
q(m, ;e ~) := P'~{x E X 'n : sup I~(g~;x) - EP(gy) > e}. (4.2)
~(Ey9
Observe that an equivalent way of writing q(m, e; ~) is as follows:
q(m,e;G) := Pm{x E X'~: sup I/~(gy;x) - f(y) > e}.
yeg
The family G has the UCEM property if and only if q(m, ;e ~) ~ 0 as m -~ oo
for each e > 0.
Suppose the family G does indeed have the UCEM property, and consider
the following approach: Given e, 5 > 0, choose the integer m large enough that
q(m, ;e ~) < .5
Then, using one's favourite algorithm, find a value of y E Y that exactly
minimizes the function/~(gy; x). In other words, choose 0Y E Y such that
E(g~0 ;x) : min/~(g~; x).
yEY
Then it can be said with confidence 1- 5 that y is an approximate near
minimizer of the original objective function f(.) to accuracy e.
The claim made above is easy to establish. Once the integer m is chosen
large enough that q(m, ;e G) < 5, it can be said with confidence 1 - 5 that
)yCII -/~(gy;x)l <_ ,e yV e Y.
In other words, the function /~(g.; x) is a uniformly close approximation to
the original objective function f(-). Hence it readily follows that an exact
Randomized Algorithms for Robust Controller Synthesis 11
minimizer of/~(g.; x) is also an approximate near minimizer of f(.) to accuracy
Note that the above approach is an example of a randomized algorithm,
in the sense that there is a nonzero probability (namely, q(m, e; G)) that the
algorithm may fail to produce an approximate near minimum of f(-). By
increasing the integer m of x-samples used in computing the empirical mean
/~(gy; x), this failure probability can be made arbitrarily small, but it can never
be made exactly equal to zero.
4.3 A Universal Algorithm for Finding Probable Near
Minima
Suppose h : Y -+ ~, Q is a probability measure on Y, and that h is measurable
with respect to the a-algebra that underlies Q. Then the following algorithm
produces, with arbitrarily high confidence, a probable near minimum of h(.)
to a specified level.
Algorithm 1 Given
* A probability measure Q on Y,
(cid:12)9 A measurable function h : Y -+ ~,
(cid:12)9 A level parameter a E (0, 1), and
(cid:12)9 A confidence parameter 5 E (0, 1).
Choose an integer n such that
lg(1/5) (4.3)
(1 - a) n _< 5, or equivalently n > lg1/(1 - a)"
Generate independent identically distributed (i.i.d.) samples Yl,... ,yn E Y
distributed according to Q. Define
h := min h(yi).
l<i<n
Then it can be said with confidence at least 1 - ~ that h is a probable near
minimum of h(.) to level .~(
This claim is proved in 23, Lemma 11.1, p. 357.
12 Vidyasagar
4.4 An Algorithm for Finding Probably Approximate
Near Minima
The ideas in the preceding two subsections can be combined to produce a
randomized algorithm for finding a probably approximate (or Type 3) near
minimum of an objective function f(.) of the form (2.4). Actually two distinct
algorithms are presented. The first is "universal," while the second algorithm
is applicable only to situations where an associated family of functions has
the UCEM property. The sample complexity estimates for the first "univer-
sal" algorithm are the best possible, whereas there is considerable scope for
improving the sample complexity estimates of the second algorithm.
Suppose real parameters ,~ a, 5 > 0 are given; the objective is to develop a
randomized algorithm that constructs a probably approximate (Type 3) near
minimum of
f(y) ----: Epg(x,y)
to accuracy e and level a, with confidence 1- 6. In other ~vords, the probability
that the randomized algorithms fails to find a probably approximate near
minimum to accuracy e and level a must be at most 6.
Algorithm 2 Given
(cid:12)9 Sets X, Y,
* Probability measures P on X and Q on Y,
(cid:12)9 A measurable function g : X x Y ~ 0, 1, and
(cid:12)9 An accuracy parameter e E (0, 1), a level parameter a E (0, 1), and a
confidence parameter 6 E (0, 1).
Define
f(y) := Epb(x,y)
Choose integers
lg(2/6) and m > 1 4n (4.4)
n __> lg1-7~- --_ or)' _ ~ In .-~--
Generate i.i.d. samples Yx,...,Yn E Y according to Q and Xl,... ,xm E X
according to P. Define
1 ~"
i := -- Eg(xj,Yl), i= 1,...,n, and
m
j=l
fo := min .~.
1<i<.
Then with confidence 1 - 6, it can be said that o is a probably approximate
(Type 3) near minimum of f(.) to accuracy e and level a.