Stability Radii of Systems with Stochastic Uncertainty and Their Optimization by Output Feedback.
We consider linear plants controlled by dynamic output feedback which are subjected to blockdiagonal stochastic parameter perturbations. The stability radii of these systems are characterized, and it is shown that, for real data, the real and the complex stability radii coincide. A corresponding result does not hold in the deterministic case, even for perturbations of single-output feedback type. In a second part of the paper we study the problem of optimizing the stability radius by dynamic linear output feedback. Necessary and sufficient conditions are derived for the existence of a compensator which achieves a suboptimal stability radius. These conditions consist of a parametrized Riccati equation, a parametrized Liapunov inequality, a coupling inequality, and a number of linear matrix inequalities (one for each disturbance term). The corresponding problem in the deterministic case, the optimal $\mu$-synthesis problem, is still unsolved.
Introduction
One of the main purposes of feedback control is to ensure satisfactory behaviour of a dynamical
system in the presence of unforeseen disturbances. This classical problem which was central
to the work of Bode and Nyquist has seen a vigorous renaissance over the past decade and
recent developments in control theory have been strongly influenced by it. The focus has been
on deterministic disturbances: either unstructured (additive or multiplicative) perturbations of
the plant's transfer function or structured perturbations of the parameters of a given nominal
state space model. As examples we mention two approaches, H 1 and stability radii. H 1
theory (see [6]) deals with the problem of minimizing (by feedback compensation) the effect
of deterministic disturbances on the to be controlled variables. The results can be applied
to maximize robustness of stability with respect to unstructured perturbations of the transfer
matrix. On the other hand the theory of stability radii determines precise robustness measures
for stable linear state space systems subject to different classes of structured parameter perturbations
[14]. Surprisingly there is a close relationship between the two theories for the special
case where stability radii with respect to complex perturbations of single output feedback type
are considered. In fact, in this case the problem of optimizing the stability radius by feedback
control is equivalent to a singular H 1 control problem [13].
In this paper we use the framework of stability radii to study robust stability and robust stabilization
problems for systems with stochastic uncertainty. Because of the close relationship
between the theories of stability radii and H 1 control our results can be regarded as an extension
of H 1 control theory to systems with stochastic uncertainty.
We consider the system:
where the matrices A; B; C; are given and the processes w i are independent scalar
We view the above equations as describing a linear
deterministic differentiable system (A; B; C) perturbed by stochastic multi-perturbations
(t). The family (D of matrix pairs describe the structure of these
perturbations while are unkown Lipschitzian nonlinearities. We assume that
all the Wiener processes w i have zero mean. In other words, the nominal model (A; B; C) is
assumed to be exact in the mean. If the system matrix A is also subject to deterministic parameter
perturbations the problems of robust stability and robust stabilization are more involved
and so far only estimates are available for the corresponding stability radii, see [15].
Many authors have studied stability and stabilization problems for systems with state dependent
noise, see for example [17]. The quadratic optimal control problem was solved in [19]
and a collection of papers concerning Liapunov exponents for such systems can be found in
[1]. However, there are few papers dealing with robustness issues for this class of systems. An
important reference is [18] which, in our terminology, derives necessary and sufficient conditions
under which infinite or arbitrarily large stability radii can be achieved by state feedback. Some
results on stochastic stability radii defined via Liapunov exponents can be found in [4, Section
7]. A characterization of the stability radius in the special case where all the E i are equal was
given in [7] (the mathematical development is essentially the same as in the single perturbation
case
Here we study the robust stability and robust stabilization problems under multi-perturbations.
For deterministic systems the development of such stability radii requires the use of -analysis
[14] and it is well known that in the presence of more than three perturbation terms (N ?
scaling techniques only yield upper estimates for the complex stability radius. In contrast we
will derive a precise characterization via scaling techniques. This is based on the analysis of an
associated minimax problem for quadratic forms. Moreover we will show that the real and the
complex stability radii coincide for multi-perturbations of the above kind.
The second main contribution of this paper concerns the problem of optimizing the stability
radius of systems of the form (1) by dynamic output feedback. We characterize the supremal
stability radius by combining a Riccati inequality with a Liapunov inequality, a coupling condition
and a number of additional linear matrix inequalities. Moreover we give explicit formulae
for suboptimal controllers. These results are obtained by using an inequality approach to deterministic
theory developed by Gahinet and his co-workers, see [10], [11]. Whereas
in the deterministic case the suboptimal controllers can be characterized by a pair of Riccati
equations and a coupling condition, it is not possible in the stochastic case to replace both the
Riccati and the Liapunov inequalities by equalities. This will be illustrated by an example.
We proceed as follows. In the next section we give some results on a minimax problem for
quadratic forms (the proofs are in an Appendix). These results will be instrumental for our
characterization of the stability radius relative to stochastic multi-perturbations in Section 3.
In Section 4 the problem of optimizing the stability radius by (linear) feedback is studied and it
is shown that the supremal stability radius can be determined via matrix inequalities. Finally,
in Section 5 we show that in this characterization the Riccati inequality may be replaced by
a Riccati equation whereas the Liapunov inequality cannot be transformed into an equality.
Moreover the corresponding results for state feedback are derived as corollaries of the previous
results on dynamic output feedback.
problem for quadratic forms
Throughout the paper we use the following notation. K is either the field R of real numbers
or the field C of complex numbers. For any integer ' - 1, H ' (K ) is the real vector space of
Hermitian matrices in K '\Theta' and H
is the convex cone of positive
semidefinite matrices in H ' (K ). By h\Delta; \Deltai we denote the usual inner product on K '
by kHk the associated operator norm or spectral norm of H 2 H ' (K
We suppose that N 2 N is given and
(K
is a given family of nonnegative ' j \Theta ' j Hermitian matrices. For any set C, we denote by (0; 1) C
the set of all mappings from C to (0; 1), the set of all positive real numbers. If C is finite
the elements of (0; 1) C are represented by finite families In particular the set
(0; 1) N consists of all N-tuples In our later analysis
the H ij will be given by
Z 1D
and the ff 0
scaling parameters which are used to improve a lower bound for the stability
radius. In fact we will see that the scaling technique applied to multi-perturbations leads to
the following optimization problem:
Minimize
with respect to
Let
then the optimization problem (3) is equivalent to minimizing the function f(\Delta) on the positive
orthant (0; 1) N . The optimal value of (3) is denoted by
In this section, in order to maintain the flow of the paper, we only give the pertinent results.
The proofs are relegated to the appendix, nevertheless we would like to stress that these proofs
are an important part of the overall proof of our main results in x3 - x5.
The solution of (3) depends strongly on the zero block pattern of the compound matrix
i;j=1 . To capture this pattern we denote by G the directed graph [3] with node set
set of directed arcs
G is said to be strongly connected if every node of G is connected to every distinct node of G by
a directed path in G.
Theorem 2.1 Suppose that G is strongly connected, then there exists a subset J ae N and a
vector -
-, and
The theorem shows that in the strongly connected case the optimal value of the minimization
problem (3) is equal to the optimal value of the subproblem
Minimize
subject to ff 2 (0; 1) J (7)
for which only the data play a role. According to (6) the reduced scaling vector
is a minimizer for (7) and each of the norms
is equal to -
-.
A natural question is under which conditions is the subset . The following proposition
gives a sufficient condition.
Proposition 2.2 Suppose G is strongly connected and for every nonempty subset J ae N ; J 6=
there exists j 2 N n J such that
Then
for all -
-.
Remark 2.3 Suppose each Hermitian matrix is either invertible or zero. Then
assumption (8) is satisfied if G is strongly connected. In particular (9) always holds in the
scalar case strongly connected.
If G is not strongly connected there will not, in general, exist a minimum of f . To deal with
this case we introduce the following notation. Let C be the node sets of the
strongly connected components [3] of G ordered in such a way that for 1 there is
no directed arc (i; A such that
it follows from (10) that
The next theorem shows that problem (3) can be solved by restricting our considerations to
the strongly connected components of G.
Theorem 2.4 Let
then
Moreover, if - k satisfies - there exist a subset J ae C - k and, for every
vector ff 2 (0; 1) N such that
We see that the solution of problem (3) can always be reduced to the solution of a subproblem
corresponding to a strongly connected component of G.
3 Characterization of stability radii
Suppose that A 2 K n\Thetan is a given matrix with spectrum oe(A) in the open left half-plane
0g. Let N 2 N and let ((D be a given family of matrices
We will consider uncertain systems described by Ito
stochastic differential equations of the form
are unkown Lipschitzian nonlinearities satisfying
(w are independent zero mean Wiener processes on a probability space
relative to an increasing family of oe-algebras F t ae F . Thus, if - i denotes
the variance of (w i (t)) t2R+ , we have
E(w
is the Kronecker symbol. For each the Euclidian norm is taken on
. The disturbances \Delta i vary in
and the size of each measured by the Lipschitz norm
The unknown in the state dependent gains through which the stationary
white noise processes dw i (t) affect the evolution of the system. The matrix family
determines the structure of the perturbations and oe ? 0 indicates the overall
level of the stochastic disturbances. Altogether (14) with constraints (15) describes a set of
stochastic systems parametrized by
the space of square-integrable K m -valued functions (modulo equivalence)
on the probability
We denote by L 2
the space of nonanticipa-
tive stochastic processes with respect to e.g. [9]) satisfying
For arbitrary any initial state x 0 2 K n there exists a unique
solution of (14) on e.g. [9]). x(\Delta) is a
continuous nonanticipative stochastic process with L 2 second moments on every finite interval
Many concepts of stability have been studied for stochastic systems. In this paper we consider
Definition 3.1 The system (14) is said to be L 2 -stable if, for every x 0 2 K n , the unique
solution x(\Delta) of (14) on
Our aim is to determine which bounds oe on the perturbations \Delta i ensure the stability of the
deterministic system -
preserved under additive stochastic perturbations of the
denote the combined perturbation operator
The Lipschitz norm of \Delta is given by
Note that because \Delta i
The maximum oe ? 0 for which all the systems in (14) are L 2 -stable is called the stability radius
of (14).
Definition 3.2 The stochastic stability radius of A 2 K n\Thetan with respect to the perturbation
structure and the Wiener processes (w i ) i2N is
r w
that (14) is not L 2 -stableg: (19)
Remark 3.3 (i) We have chosen x 0 2 K n since we regard (14) as a stochastic perturbation
of a deterministic system. However it is straightforward to extend the theory to any F 0
measurable initial state x 0 2 L
(ii) A stability radius with respect to linear perturbations can be defined analogously by
restricting the perturbations \Delta i in (19) to be linear, i.e. It is an open
question whether this restriction leads to a different stability radius.
(iii) If the data A; D are real, two stability radii are obtained according to whether one
chooses (only real perturbations) in (19). In a
deterministic framework the real and the complex stability radii are, in general, distinct,
see [14]. We will show later that they are equal in the present stochastic framework.
In order to characterize the stochastic stability radius we need the following lemmata.
Lemma 3.4 Suppose
Moreover,
where
Proof: The first part is a standard result for stochastic integrals [9]. Now since
the following integrals are well defined and we have
By Fubini's theorem
s
E(kEe A(t\Gammas) D i v i (s)k 2 )dtds
s
The result follows since the unique solution of (22) is given by
Consider the map
defined by
by the previous lemma.
Lemma 3.5 The linear
defined by (24) has
the operator norm
where P satisfies (22).
Proof: If v(\Delta) 2 L 2
we have by Lemma 3.4
where P satisfies (22). Hence
is achieved for
and
chosen arbitrarily. Then
This completes the proof.
Theorem 3.6 Suppose A is stable and there exist
ing
A   P
I ' j
Proof: Let x
suppose that
are such that (26), (27) hold. The unique solution x(\Delta) of (14) with
initial condition satisfies the scaled integral equation
Z te A(t\Gammas) D ff i
where
The input-output operator
of the scaled system
(D ff i
Z t2
Let u ff i
y ff N
then (28) implies
Z tE(ff)e A(t\Gammas) D ff i
For every T ? 0, define the truncations u ff i
Then
Z T/ N
Z
\Omega
Z T/ N
Z
\Omega
dt
Let y ff
T denote the output of the scaled system (A; (D ff
generated by the input u ff
y ff
E(ff)
Z te A(t\Gammas) D ff i
It follows from (31), (32), (33) that
By Lemma 3.5
where P satisfies (26). So from (27) we have kL ff k - oe \Gamma1 and since k\Delta ff k oe, the
operator a contraction on L 2
1. Hence from
(34) for all T ? 0,
Therefore y ff (\Delta) 2 L 2
Applying
Lemma 3.4 (with
i instead of D i and u ff (\Delta) instead of v(\Delta)) it follows from (28) that
this completes the proof.
Remark 3.7 Suppose the \Delta i (t; y) are time-varying Lipschitzian nonlinearities, measurable in
Then the previous proof carries through showing that no time-varying Lipschitzian perturbations
of Lipschitz norm smaller than oe can destabilize the system.
If in the previous theorem condition (27) is satisfied with - instead of - then clearly
r w
Similarly, the equality in (26) may be replaced by an inequality
Corollary 3.8 Suppose A is stable and there exist ff 2 (0; 1) N ,
A   P
I ' j
Proof: Let P (ff) denote the solution of the Liapunov equation (26) then
Hence P (ff) satisfies (26) and (27) and r w
follows from Theorem 3.6.
For any J ae N and any scaling vector ff J 2 (0; 1) J , define
Z 1e A   -
i.e. P (ff J
is the unique solution of
A   P
Then
Z 1D
where
Z 1D
We are now in a position to prove our main theorem by applying the results of x2 to the family
of positive semidefinite matrices (40).
Theorem 3.9 Given (A; (D in (14) the associated stability
radius is determined by
r w
where P (ff) is the unique solution of (26). If r w
exists a minimum
norm destabilizing perturbation
K . Moreover, there exist a
subset J ae N and a scaling vector ff J 2 (0; 1) J such that
r w
where P (ff J
is the unique solution of (38).
Proof: By (39)
If -
Theorem 3.6, hence (41) is satisfied. Moreover, in
this trivial case,
so that (42) is satisfied for every singleton
Now assume that -
let oe(ff) be the largest oe for which (26) and
(27) have a joint solution
(see (39)). Whence sup ff2(0;1) N oe(ff)
. By Theorem 2.4 there exist J ae N and a vector
where P (ff J
is the unique solution of (38). Hence there are

Setting -
oe for
We will show that for this \Delta (14)
cannot be stable. Assume the contrary, then, for all x 0 2 K n , the solution x(\Delta) of (14) with
must satisfy
satisfies the reduced scaled integral equation
Z te A(t\Gammas) D ff j
where D ff j
j (\Delta) are defined by (29). By assumption y ff j
Defining y ff J
(\Delta) and E ff J
by
y ff J
we get
y ff J
e At x
oe
Z te A(t\Gammas) D ff j
Application of Lemma 3.4 to (49) yields
Z 1kE ff J
e At x
where P (ff J
is the unique solution of (38). But then by (46), we have
Z 1kE ff J
e At x
Z 1kE ff J
e At x
for all x 0 2 K n . This would imply
contrary
to our assumption. Therefore there exists x 0 2 K n such that
neither
of the two stochastic systems (14) and (48) are L 2 -stable. It follows that
r w
On the other hand, for every oe ! -
there exists ff 2 (0; 1) N such that
But this implies oe - r w
Theorem 3.6 and concludes the proof.
Note that the destabilizing disturbance \Delta defined by (47) is real when the data (A; (D
are real. In this case we can choose K = R in (41) and obtain a formula for the real stability
radius. However, we may also choose so that we obtain the same formula for the complex
stability radius since the RHS of (41) does not depend on the choice of the field K . Thus
Corollary 3.10 Under the conditions of Theorem 3.9, if the data (A; (D are real then
the complex and the real stability radii coincide:
r w
R
Remark 3.11 In the deterministic case stability radii for complex and real multi-perturbations
are, in general, not the same, not even in the single perturbation case
Moreover, the scaling technique does not provide a characterization of the complex stability
radius, but yields only an upper bound, see [16]. In the stochastic case, however, the scaling
technique works and we have - as a consequence of (41) -
r w
r w
for . The application of the theorems of x2 to obtain this result is based on the
simple formula (25) for the norm of L. The corresponding characterization in the deterministic
case is much more complicated and involves a parametrized Riccati equation (see [14]) instead
of the single Liapunov equation (22) without parameters.
One reason for the basic difference between the deterministic and the stochastic case lies in the
fact that there is no deterministic counterpart to the fundamental equation (20) (on which all
our results are built).
For later use we note the following characterization of the stability radius in terms of strict
inequalities.
Corollary 3.12 Given (A; (D in (14) the following statements are
equivalent for oe - 0:
(ii) there exists
I ' i
Proof: Suppose (i) and choose oe 0 2 (oe; r w
Theorem 3.9 there exists
such that
-Z 1e A   - E(ff)   E(ff)e A- d-
But then
-Z 1e A   - E(ff)   E(ff)e A- d-
and hence there exists " ? 0 such that
Setting
Hence
Conversely, (ii) ) (i) follows first from the fact that (50) implies and then from
application of Corollary 3.8.
Remark 3.13 Let G denote the directed graph with node set N and set of directed arcs
0g where the Hermitian matrices
are defined by (40).
By Theorem 2.4
- is defined by (43) and C are the strongly connected components of G. Since
we have
r w
r w
Thus the L 2 -stability of the uncertain stochastic system (14) is equivalent to the L 2 -stability
of each uncertain stochastic system corresponding to the connected components of G:
This reduces our original problem to the separate investigation of K uncertain stochastic systems
with strongly connected perturbation structures. In particular, the subset J in Theorem
3.9 can be chosen in a strongly connected component C k with - -. The question of determining
conditions under which no further reduction beyond the connected components is
possible, i.e. has been dealt with in Proposition 2.2.
4 Maximizing the stability radius by dynamic output
feedback
In this section we investigate how the stability radius of a stochastically perturbed system can
be improved by dynamic output feedback. For this we introduce a control term into the system
equation (14) and add a measurement equation. We consider controlled stochastic systems
described by Ito equations of the form
are the input and output matrices and the other variables and
matrices are of the form specified in the previous section.
Remark 4.1 Extensions of this problem including control dependent noise and noise corrupting
the output should be considered in the framework of a general stochastic H 1 control.
While the development of a comprehensive H 1 control theory for stochastic systems requires
substantial new work, it can be built on the results presented here. This will be the subject of
future work.
The compensator takes the form
where (H; G; F; K) 2 K -
n \Theta - n \Theta K - n \Thetap \Theta K m\Theta-n \Theta K m\Thetap and the dimension -
arbitrary. The
resulting overall system is
A +BKC BF
We will use the notation
A +BKC BF
and
x
x
Then the above system can be written
For all compensators (53), arbitrary exists
a unique solution of (54) on R+ such that
Our aim is to determine conditions for the existence of dynamic compensators of the form (53)
that stabilize the system and achieve a stability radius r w
oe ? 0. We follow an approach based on inequalities similar to the one that Gahinet developed
in his approach to the H 1 control problem, see [10], [11]. We proceed in two steps. First we
derive some necessary conditions and then we show that these conditions are also sufficient for
building a stabilizing compensator of dimension n which achieves the required stability radius.
We will make use of the following criterion for the positive definiteness of Hermitian block
matrices.
Lemma 4.2 Let
Theorem 4.3 Given oe ? 0 and a compensator (53) such that
r w
there exist ff 2 (0; 1) N ,
I
where (see (30)) E(ff)
Proof: By Corollary 3.12 there exists ff 2 (0; 1) N ,
I ' i
Writing
R M
with
Now
Hence
because of (60). Moreover,
I n R
S I n
Multiplying (59) on the left by
I n 0
R M
and on the right by
I n R
yields
I n 0
A +BKC BF
I n R
I n 0
R M
S I n
I n 0
R M
E(ff)   E(ff) 0
I n R
I n 0
R M
I n R
for some
Writing out these equations, we obtain
R
(64) is equivalent to
A   S
And (65) is equivalent to
AR+RA   +RE(ff)   E(ff)R \GammaBB
R
Note that [R M ] has full row rank so that [R M ]\Pi
R
Therefore, choosing fl and ffi
sufficiently small we obtain (55) and (56). These inequalities and (61) are still satisfied if for
sufficiently small we replace S by S . Denoting the modified S by the same symbol
we get 0 OE R \Gamma1 OE S and thus (55) - (58) are satisfied.
Remark 4.4 (55) and (56) hold for some fl; ffi ? 0 if and only if
AR +RA   +RE(ff)   E(ff)R OE 0 on ker B   ;
so it is possible to state an equivalent theorem which does not involve fl and ffi.
Theorem 4.5 Suppose (55) - (58) hold for some ff 2 (0;
there exists an n-dimensional compensator (H; G; F; K) 2 K n\Thetan \Theta K n\Thetap \Theta K m\Thetan \Theta
K m\Thetap such that
Proof: Choose
An easy calculation shows that (64) and (65) hold. Moreover, it follows from (55) and (56) that
22
22 \Pi 11 - 0, hence
Finally, we have by assumption 0 OE R \Gamma1 OE S so that are
Therefore
is the unique H 2 K n\Thetan satisfying (66) (with the specifications (67) - (69)). Altogether we see
that, with the above choices, equation (63) holds with \Pi - 0. Now
I n R
I n R
is
invertible. Multiplication of (63) on the left by
I n 0
R M
and on the right by
I n R
yields equation (59) with
S I n
I n R
Remark 4.6 (i) The above theorems show that if a compensator of any order -
stabilizes
the system with a stability radius greater than oe, then this can always be achieved by
a compensator of order n. Moreover, for this compensator the feedthrough matrix K
may be taken to be zero. We do not address the problem of reduced order observers, but
expect that a development similar to [11] (in the deterministic case) is possible.
(ii) For oe - 0, let A oe denote the set of all pairs (R;
. By the construction in the proof
Theorem 4.3, for any stabilizing compensator (H; G; F; K) which achieves a stability
radius r w
there exists a (R; S) 2 A oe obtained from the solutions X
of the matrix inequalities (59) and (60). Conversely by the construction in the proof of
Theorem 4.5 for every given (R; S) 2 A oe there is an associated stabilizing compensator
G; F; K) which achieves a stability radius greater than oe.
For later use we add another remark in which an alternative formula for the system matrix H
of the observer is derived, cf. (70).
Remark 4.7 The gap between the Riccati inequality (55) and the corresponding Riccati equation
is measured by the operator
Using the specifications (67) - (69) we get
and so
A   R
Using this equation and N to transform (70) we get
It follows from (72) and
(R
Thus H is the sum of the usual observer system matrix A a correction term
(R depending on the gap between the Riccati inequality (55) and the
corresponding equality.
Theorems 4.3 and 4.5 together yield a complete characterization of the stability radii which can
be achieved by dynamic output feedback applied to the system (52). We conclude this section
by discussing this point.
For any ff 2 (0;
A oe;fl
A oe;fl
A oe;fl ;ffi;ff ; and oe(fl;
In the following remark we collect some properties of A oe;fl ;ffi;ff and oe(fl; ffi).
Remark 4.8 A oe;fl ;ffi;ff is an open subset of H
. If ff 2 (0; 1) N is fixed, the set A oe;fl ;ffi;ff increases as the parameters oe; fl;
decrease:
As a consequence we have
Definition 4.9
the compensator (H; G; F; K) is stabilizingg
is said to be the supreme stability radius for the uncertain stochastic system (52).
As a consequence of the previous two theorems we obtain the following characterization of the
supreme stability radius.
Corollary 4.10
For the computation of oe(fl; ffi) the following description of the sets A oe;fl ;ffi;ff in terms of linear
matrix inequalities is useful.
Lemma 4.11 Given ff 2 (0; , the set A oe;fl ;ffi;ff consists of all
Hermitian matrix pairs (R; which satisfy the following linear matrix
inequalities
E(ff)R \GammaI q
R I n
I n S
I
In particular, A oe;fl ;ffi;ff is convex.
Proof: The equivalence of (55) - (58) and (74) - (77) is obtained by application of Lemma
4.2. The convexity of A oe;fl ;ffi;ff then follows.
For a discussion of linear matrix inequalities and their numerical solution see [2].
Remark 4.12 In the case of single complex perturbations the deterministic counter-part
of the problem considered in this section leads to a singular H 1 -optimal control problem
which can be solved via Riccati equations, see [13], or linear matrix inequalities, see [11]. The
maximization of the real stability radius by dynamic output feedback is still an unsolved prob-
lem, even for 1. In the (complex) multi-perturbation case the deterministic version of
our problem leads to an optimal -synthesis problem since the stability radius with respect
to multi-perturbations can be characterized via the -function [14]. To our knowledge this
problem is still unsolved. Our solution of the stochastic problem is based on the fact that
the scaling technique works and yields a characterization of the stochastic stability radius in
terms of matrix inequalities, see Theorem 3.9. A similar result is not available for deterministic
multi-perturbations, see Remark 3.11.
Replacing Riccati inequalities by Riccati equations
In this section we explore the possibility of replacing the Riccati inequality (55) by a Riccati
equation. Throughout the section we will assume that (A; B) is stabilizable and (A; C) is detectable

As starting point we take the Riccati inequality
Since (A; B) is assumed to be stabilizable this inequality always has positive definite solutions.
Moreover, every such solution P solves (78) if and only if
solves (55). We will also consider the usual linear-quadratic control Riccati equation:
and its "-approximations:
The following lemma summarizes some useful and well known properties of these equations,
see e.g. [12], [5].
Lemma 5.1 Suppose (A; B) is stabilizable and fl ? 0; ff 2 (0; 1) N . Then
(i) For each " ? 0 (79) has a unique solution X ff;fl (") in H
For every
solution of (78) there exists " such that X ff;fl (") OE P .
(ii) (ARE ff;fl ) has a unique maximal solution X
this solution is characterized
amongst all other Hermitian solutions of (ARE ff;fl ) by the property
Remark 5.2 It can be shown that the maximal solution X ff;fl of (ARE ff;fl ) is stabilizing if and
only if
ker
Consider the following set of conditions for S 2 H n (K
I
where X ff;fl denotes the maximal solution of (ARE ff;fl ). For oe - 0;
A R
A R
A R
The following proposition is derived by means of Lemma 5.1.
Proposition 5.3 Suppose (A; B) is stabilizable and fl ? 0;
Proof: Let oe ! oe(fl; ffi), then there exist ff 2 (0; 1) N , (R;
oe;fl ;ffi and thus oe - oe R (fl; ffi). This shows that oe R (fl; ffi) - oe(fl; ffi).
Conversely, suppose oe ! oe R (fl; ffi), then there exist ff 2 (0; 1) N , S 2 H n (K ) such that (81)
5.1 and (82) there exists " ? 0 such that Hence
This shows that oe(fl; ffi) - oe R (fl; ffi) and concludes the
proof.
For any oe ! oe R (fl; ffi) we now construct, via the Riccati equation (ARE ff;fl ), a compensator
of order n so that the stability radius of the overall system is greater than oe. For this, the
maximal solution of (ARE ff;fl ) must be stabilizing, i.e. condition (80) must be satisfied.
Proposition 5.4 Suppose (80), (81) - (83) hold for some
. Then the compensator (53) defined by
achieves a stability radius r w
Proof: Replacing R \Gamma1 by X ff;fl in the formulae (67) and (73), then using equation (72) for
R , we see that the compensator (H; G; F; K) in (84) coincides with the compensator defined
in the proof of Theorem 4.5 by (67) and (70). Let
then exactly the same calculation as in the proof of Theorem 4.5 yields
where now
but note that in contrast to the development in the proof of Theorem 4.5 we only
have
with
We will now show that . Suppose there exist x 2 C 2n ; x 6= 0 and - 2 C with
Re- 0 such that Multiplying (85) on the right by x and on the left by x   , we
obtain
This can only be the case if x 2 ker \Pi, i.e.
with But then the first
component of the equation contradicts
assumption which implies that and we conclude
from Corollary 3.8 and
I
that r w
In the deterministic case the Liapunov inequality (81) takes the form of a Riccati inequality
and it is possible to replace this by a Riccati equation. Here this is not possible, in general,
since the requirements S
against each
other as S increases or decreases. So this is an essential difference between the deterministic
and stochastic cases. We illustrate it in the following example.
Example 5.5 Consider the perturbed stochastic system
i.e.
"1
"0
"1
Note that (A; B) is controllable, (A; C) is detectable and ker (-!I
r
s
Equality in (56) or (81) takes the form
We see that, for arbitrary parameters there does not exist any solution
Therefore we cannot replace the inequalities in (56) or (81) by
equalities.
We will now determine for the present example and we do this
via inequalities rather than the Riccati equation (ARE ff;fl ) and (81)-(83). Suppose oe ! r.
Since
"0
is a basis for both ker B   and ker C, by Theorem 4.3 and Remark 4.4, there exist
Therefore
We will now demonstrate that equality holds. For this let - 1 oe
must show that there exist ff 1 ? 0; ff 2 ? 0 such that (89) and (90) hold and S
Normalizing ff 2
Choose
Then it is easy to see that (91) is satisfied. Since
then holds. So all we need to show is S
0:
We conclude the paper with a proposition concerning the state feedback case.
Proposition 5.6 Suppose and C is invertible. Then the following conditions are equivalent
for oe - 0:
(i) There exists a static feedback matrix K 2 K m\Thetan such that
r w
(ii) there exists a dynamic output feedback of the form (53) such that
r w
(iii) there exist ff 2 (0;
I
(iv) there exist ff 2 (0; 1) N , fl ? 0 such that the maximal solution X ff;fl of (ARE ff;fl ) satisfies
I
Proof: follows from Theorem 4.3 and (iii) ) (iv) follows from
Lemma 5.1. Now suppose (iv). Select an S 2 H n (K ) such that S - X ff;fl and (83) are satisfied.
choose sufficiently small so that (81) holds. Then (81) - (83) hold and (ii) follows
from Proposition 5.3 and Theorem 4.3. Hence (ii), (iii) and (iv) are equivalent.
It remains to prove e.g. (iii) ) (i). Suppose (iii), then there exist ff 2 (0; 1) N , fl ? 0,
I ' i
From the first inequality it follows that
Applying Corollary 3.12 we get r w
oe. Thus is suffices to choose
in order to obtain (i).
In particular, the proposition shows that it is not possible to obtain a larger stability radius by
dynamic state feedback than that which can be achieved by static state feedback.
In [8] the equivalence (i) , (iv) was proved for the special case
6

Appendix


We use the notation introduced in Section 2. In order to prove Theorem 2.1 we need the
following lemmata. First note that the functions f in (4) are constant on rays in
(0;
Lemma 6.1 Suppose that (ff k ) k2N is a sequence in S N
can be reached from i 2 N via a directed path in G then
lim
Proof: By induction it suffices to prove (92) for the case where (i; A. But in this case
thus (92) follows from the boundedness of (f(ff k
Applying this lemma we obtain the following existence result.
Proposition 6.2 Suppose that G is strongly connected. Then there exists -
such that
Proof: Since f is constant on rays it suffices to consider f on S N
. Let (ff k ) be a minimizing
sequence for f on S N
which converges towards some limit -
ff in the closure of S N
. We only
have to prove that -
since
(f(ff k )) is bounded and j can be reached from every i 2 N via a directed path in G, we must
have -
by the previous lemma. On the other hand -
sequence in
. The contradiction shows that -
Lemma 6.3 Suppose H
are such that
Then
be such that hv; (H
Hence increasing and
hence the result follows.
Proof of Theorem 2.1 By Proposition 6.2 there is a vector z 2 (0; 1) N such that
-:
Amongst all these minimizing vectors we choose one, denoted by -
z, for which the number of
is minimal. Let J be the set of these j 2 N . Then
For r 2 [0; 1], define - z(r) 2 (0; 1) N by setting - z j
For r sufficiently close to 1, say r the inequalities (95) still hold when -
z is
replaced by -
z(r). For these r,
But by the minimality assumption on J , none of the above inequalities can be strict for r 2 [-r; 1].
Applying Lemma 6.3 with H
ij we conclude that
Setting -
z it follows from (96), (95) that (6) is satisfied.
Proof of Proposition 2.2: Let -
ff be a minimum such that
-g has a
minimum number of elements and suppose J 6= N . Arguing as in the proof of Theorem 2.1 we
obtain (see
If -
trivially satisfied. Therefore we may assume -
such that fl fl fl fl fl fl

Then

0:
It follows that v j 2
J and this contradicts assumption
(8). Thus and the proposition is proved.
In the case where G is not strongly connected, for every " ? 0 we define the set
The proof of the following statement is straightforward.
Lemma 6.4 Suppose that, for each k 2 K, a positive vector z k 2 (0; 1) C k is given. Choose
K such that
and define ff 2 (0; 1) N by
Then
Lemma 6.5 Given a family of vectors z
(where - k is defined by (12)), there exists, for every ffi ? 0, a vector ff 2 (0; 1) N , such that
Proof: Suppose that z k 2 (0;
For every " ? 0, there exists, by Lemma 6.4, a vector such that (99) is
satisfied. It follows that
But for all " 2 (0; 1) we have
Choosing " 2 (0; 1) such that
we obtain
This concludes the proof.
Proof of Theorem 2.4: By Proposition 6.2 there always exists a family of vectors z
satisfying (100). Hence by Lemma 6.5 for ffi ? 0 there exists ff 2 (0; 1) N such that
But for every ff 2 (0; 1) N ,
we conclude that -
. The second part of Theorem 2.4
follows from this and Theorem 2.1.



--R


Linear Matrix Inequalities in System and Control Theory.
Applied Graph Theory.
Stability radii and Liapunov exponents.
A note on the analyticity of the Riccati metric.
State space solutions to standard H 2 and H 1 control problems.
Stability radius of linear systems with respect to stochastic perturbations.
A Riccati equation approach to maximizing the stability radius of a linear system by state feedback under structured stochastic Lipschitzian perturbations.
Stochastic Differential Equations and Applications.
A convex parametrization of suboptimal H1 controllers.
A linear matrix inequality approach to H1 control.
On the Hermitian solutions of the symmetric algebraic Riccati equation.
Riccati equation approach to maximizing the complex stability radius by state feedback
a survey.
Stability margins for systems with deterministic and stochastic uncertainty
The complex structured singular value.
Feedback stabilizability of stochastic systems with state and control dependent noise
Robust stabilization of uncertain systems
Optimal stationary control of a linear system with state dependent noise
Model reference transformations
--TR
dynamic output feedback;linear matrix inequalities;multiperturbations;stability radius;state-dependent noise;riccati inequalities;stochastic systems;scaling