Hirsch M.J., Pardalos P.M., Murphey R. Dynamics of Information Systems: Theory and Applications
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46 K.D. Pham
Fig. 3.1 An architecture for
risk-averse based large-scale
interconnected systems
chapter is focused on two related perspectives: the performance-information analy-
sis perspective and the distributed feedback stabilization perspective. These seem to
be the two dominant perspectives in stochastic control of large-scale interconnected
systems. Specifically, the following two problems are associated with adaptive con-
trol decision. The first is the problem of designing a performance-information sys-
tem that can motivate and screen appropriate adaptive control decision by an inter-
connected system in charge of local operation in a stochastic and dynamic environ-
ment. The second is the problem of emphasizing the importance of adaptive control
decision in stochastic control under environmental uncertainty and exploring the
characteristics of an interconnected system conducive to adaptive control decision
by the autonomous system to whom operating authority is delegated.
What framework can be used to view that process in distributed stabilization for
large-scale interconnected systems? To be useful, perhaps, such a framework must
be simple, but at the same time it must be comprehensive enough that most of the
essential elements of a distributed control process can be discussed. The control
process envisaged here is a dynamic process surrounding performance assessment,
feedback and corrective action. Diagrammatically, the entire relationship may be
depicted as shown in Fig. 3.1. Each autonomous system has some goal in terms
of the performance riskiness of the operating process it has to control. The perfor-
mance uncertainty is affected not only by the control decision or action but also
by the environment in which the system (the local control decision and the local
operating process) must operate. The basic reason for the existence of control deci-
sion is uncertainty or incomplete knowledge on the part of the decision policy about
(1) the mechanism of the operating process and (2) the environmental conditions
3 Performance-Information Analysis and Distributed Feedback Stabilization 47
that should prevail at any point in time. In this diagram, each interconnected system
recognizes the need for action in congruence with the expected standard of per-
formance, e.g., risk aversion for performance against all random realizations from
environmental disturbances is considered herein. The autonomous system will use
the standard as a guideline to determine whether its action is called for or not. Its
recognition comes as a result of filtering feedback information through some criteria
such as risk modeling and risk measures for making judgment.
From this point of view, all respective measures of performance for intercon-
nected systems are viewed as random variables with mixed random realizations
from their own uncertain environments. Information about the states of environ-
ments is assumed to be common knowledge. Furthermore, it is assumed here that
each interconnected system will choose control decisions which will be best for
its own utility, whether it is an internalized goal of either (1) performance prob-
ing via an effective knowledge construct that can be able to extract the knowledge
of higher-order characteristics of performance distribution; (2) performance caution
that mitigates performance riskiness with multiple attributes beyond performance
averaging as such variance, skewness, flatness, etc., just to name a few; or (3) both.
The final process execution has only one category of factors. It is the implementation
of a risk-averse decision strategy where explicit recognition of performance uncer-
tainty brings to what constitutes a “good” decision by the interconnected system.
To the best of the author’s knowledge, the problem of performance risk congruence
has not attracted much academic or practical attention in stochastic multi-agent sys-
tems until very recently [8, 9] and [10]. Failure to recognize the problem may not
be harmful in many operating decision situations when alternative courses of ac-
tion may not differ very much in their risk characteristics. But in cases of important
and indispensable consideration in reliability-based design [3] and incorporation of
aversion to specification uncertainty [5], which may involve much uncertainty and
alternatives whose risk characteristics are wide ranged, consideration of the inter-
connected system’s goal toward risk may be a crucial factor in designing a high
performance system.
The chapter is organized as follows. Section 3.2 puts the distributed control
of adaptive control decisions for interconnected systems in uncertain environ-
ments into a consistent mathematical framework. In Sect. 3.3, the performance-
information analysis is a focal point for adaptive control decisions of interconnected
systems. The methodological characteristics of moment and cumulant generating
models are used to characterize the uncertainty of performance information with
respect to uncertain environments. Some interesting insights into how the value of
performance information are affected by changes in the attributes of a performance-
information system. Section 3.4 will develop a distributed feedback stabilization for
a large-scale interconnected system with adaptive control decision using the recent
statistical control development. Another main feature of this section is the explo-
ration of risk and preference for a stochastic control system. A generalization of
performance evaluation is suggested. The risk of a performance measure is express-
ible as a linear combination of the associated higher-order statistics. From Sect. 3.5,
construction of a candidate function for the value function and the calculation of
48 K.D. Pham
decentralized efficient risk-averse decision strategies accounting for multiple at-
tributes of performance robustness are discussed. Finally, some concluding remarks
are drawn in Sect. 3.6. These final remarks will help put the novelty of this research
in perspectives of distributed control and performance-information analysis.
3.2 Problem Formulation
Before going into a formal presentation, it is necessary to consider some conceptual
notations. To be specific, for a given Hilbert space X with norm ·
X
,1≤p ≤∞
and a,b ∈R such that a ≤b, a Banach space is defined as follows
L
p
F
(a, b;X)
φ(·) =
φ(t,ω) :a ≤t ≤b
such that φ(·) is an X-valued
F
t
-measurable process on [a,b] with E
b
a
φ(t,ω)
p
X
dt
< ∞
(3.1)
with norm
φ(·)
F,p
E
b
a
φ(t,ω)
p
X
dt
1/p
(3.2)
where the elements ω of the filtered sigma field F
t
of a sample description space
Ω that is adapted for the time horizon [a,b] are random outcomes or events. Also,
the Banach space of X-valued continuous functionals on [a,b] with the max-norm
induced by ·
X
is denoted by C(a,b;X). The deterministic version of (3.1) and
its associated norm (3.2) is written as L
p
(a, b;X) and ·
p
.
More specifically, consider a network of N interconnected systems, or equiva-
lently, agents numbered from 1 through N. Each agent i,fori ∈
N {1, 2,...,N}
operates within its local environment modeled by the corresponding filtered proba-
bility space (Ω
i
, F
i
, {F
i
}
t≥t
0
>0
, P
i
) that is defined with a stationary p
i
-dimensional
Wiener process w
i
(t) w
i
(t, ω
i
) :[t
0
,t
f
]×Ω
i
→ R
p
i
on a finite horizon [t
0
,t
f
]
and the correlation of independent increments
E
w
i
(τ ) −w
i
(ξ)
w
i
(τ ) −w
i
(ξ)
T
=W
i
|τ −ξ |,W
i
> 0
Assume that agent i generates and maintains its own states based on information
concerning its neighboring agents and local environment. The update rule of agent
i evolves according to a simple nearest-neighbor model
dx
i
(t) =
A
i
(t)x
i
(t) +B
i
(t)u
i
(t) +C
i
(t)z
i
(t) +E
i
(t)d
i
(t)
dt
+G
i
(t) dw
i
(t)
x
i
(t
0
) =x
0
i
(3.3)
3 Performance-Information Analysis and Distributed Feedback Stabilization 49
where all the coefficients A
i
∈ C(t
0
,t
f
;R
n
i
×n
i
), B
i
∈ C(t
0
,t
f
;R
n
i
×m
i
), C
i
∈
C(t
0
,t
f
;R
n
i
×q
i
), E
i
∈ C(t
0
,t
f
;R
n
i
×r
i
), and G
i
∈ C(t
0
,t
f
;R
n
i
×p
i
) are deter-
ministic matrix-valued functions. Furthermore, it should be noted that x
i
∈
L
2
F
i
(t
0
,t
f
;R
n
i
) is the n
i
-dimensional state of agent i with the initial state x
0
i
∈R
n
i
fixed, u
i
∈ L
2
F
i
(t
0
,t
f
;R
m
i
) is the m
i
-dimensional control decision, and d
i
∈
L
2
(t
0
,t
f
;R
r
i
) is the r
i
-dimensional known disturbance.
In the formulation the actual interaction, z
i
∈L
2
F
i
(t
0
,t
f
;R
q
i
) represented by the
q
i
-dimensional process is of interest to agent i where L
ij
represent time-varying
interaction gains associated with its neighboring agents
z
i
(t) =
N
j=1,j =i
L
ij
(t)x
ij
(t), i ∈N (3.4)
For practical purposes, the process z
i
is however not available at the level of agent i.
It is then desired to have decentralized decision making without intensive commu-
nication exchanges. An approach of interaction prediction is therefore proposed to
resolve the communication difficulty by means of a crude model of reduced order
for the interactions among the neighboring agents that can exchange information
with agent i. In this work, the actual interaction z
i
(·) is now approximated by an
explicit model of the type
dz
mi
(t) =
A
zi
(t)z
mi
(t) +E
zi
(t)d
mi
(t)
dt +G
zi
(t) dw
mi
(t), z
mi
(t
0
) =0
(3.5)
where A
zi
∈ C(t
0
,t
f
;R
q
i
×q
i
) is an arbitrary deterministic matrix-valued function
which describes a crude model for the actual interaction z
i
(·). In particular, A
zi
can be chosen to be the off-diagonal block of the global matrix coefficient A corre-
sponding the partition vector z
i
(·). Other coefficients E
zi
∈ C(t
0
,t
f
;R
q
i
×r
mi
) and
G
zi
∈ C(t
0
,t
f
;R
q
i
×p
mi
) are deterministic matrix-valued functions of the stochas-
tic differential equation (3.5). The approximate interaction z
mi
(·) produced by the
model is affected not only by the known disturbance d
mi
∈L
2
(t
0
,t
f
;R
r
mi
) but also
by another local uncertainty w
mi
(t) w
mi
(t, ω
mi
) :[t
0
,t
f
]×Ω
mi
→R
p
mi
which is
an p
mi
-dimensional stationary Wiener process defined on a complete filtered prob-
ability space (Ω
mi
, F
mi
, {F
mi
}
t≥t
0
>0
, P
mi
) over [t
0
,t
f
] with the correlation of in-
dependent increments
E
w
mi
(τ ) −w
mi
(ξ)
w
mi
(τ ) −w
mi
(ξ)
T
=W
mi
|τ −ξ |,W
mi
> 0
With the approach considered here, there is a need to treat the actual interaction z
i
(·)
as a control process that is supposed to follow the approximate interaction process
z
mi
(·). Thus, this requirement leads to certain classes of admissible control laws as-
sociated with (3.3) to be denoted by U
i
×Z
i
⊂L
2
F
i
(t
0
,t
f
;R
m
i
) ×L
2
F
mi
(t
0
,t
f
;R
q
i
),
e.g., given (u
i
(·), z
i
(·)) ∈ U
i
×Z
i
, the 3-tuple (x
i
(·), u
i
(·), z
i
(·)) shall be referred
to as an admissible 3-tuple if x
i
(·) ∈ L
2
F
i
(t
0
,t
f
;R
n
i
) is a solution of the stochastic
50 K.D. Pham
differential equation (3.3) associated with u
i
(·) ∈ U
i
and z
i
(·) ∈ Z
i
. Furthermore,
the nearest neighbor model of local dynamics (3.3) in the absence of its known dis-
turbances and local environment is assumed to be uniformly exponentially stable.
That is, there exist positive constants η
1
and η
2
such that the pointwise matrix norm
of the closed-loop state transition matrix associated with the local dynamical model
(3.3) satisfies the inequality
Φ
i
(t, τ )
≤η
1
e
−η
2
(t−τ)
∀t ≥τ ≥t
0
The pair (A
i
(t), [B
i
(t), C
i
(t)]) is pointwise stabilizable if there exist bounded
matrix-valued functions K
i
(t) and K
zi
(t) so that the closed-loop system dx(t) =
(A
i
(t) +B
i
(t)K
i
(t) +C
i
(t)K
zi
(t))x
i
(t) dt is uniformly exponentially stable.
Under the assumptions of x
i
(t
0
) ∈ R
n
i
, z
mi
(t
0
) ∈ R
q
i
, u
i
(·) ∈ U
i
, and z
i
(·) ∈
Z
i
, there are tradeoffs among the closeness of the local states from desired states,
the size of the local control levels, and the size of local interaction approximates.
Therefore, agent i has to carefully balance the three to achieve its local performance.
Stated mathematically, there exists an integral-quadratic form (IQF) performance
measure J
i
:R
n
i
×R
q
i
×U
i
×Z
i
→R
J
i
x
0
i
,z
0
mi
;u
i
(·), z
i
(·)
=x
T
i
(t
f
)Q
f
i
x
i
(t
f
) +
t
f
t
0
x
T
i
(τ )Q
i
(τ )x
i
(τ ) +u
T
i
(τ )R
i
(τ )u
i
(τ )
+
z
i
(τ ) −z
mi
(τ )
T
R
zi
(τ )
z
i
(τ ) −z
mi
(τ )
dτ (3.6)
associated with agent i wherein Q
f
i
∈ R
n
i
×n
i
, Q
i
∈ C(t
0
,t
f
;R
n
i
×n
i
), R
i
∈
C(t
0
,t
f
;R
m
i
×m
i
), and R
zi
∈ C(t
0
,t
f
;R
q
i
×q
i
) representing relative weightings for
terminal states, transient states, decision levels, and interaction mismatches are de-
terministic and positive semidefinite with R
i
(t) and R
zi
(t) invertible.
The description of the framework continues with yet another collection of ran-
dom processes {y
i
(t)}
N
i=1
that carry critical information about, for instance, the
agents’ states and their underlying dynamic structures, e.g.,
dy
i
(t) =H
i
(t)x
i
(t) dt +dv
i
(t), i ∈N (3.7)
which is a causal function of the local process x
i
(t) corrupted by another stationary
s
i
-dimensional Wiener process v
i
(t) v
i
(t, ω
i
) :[t
0
,t
f
]×Ω
i
→ R
s
i
on a finite
horizon [t
0
,t
f
] and the correlation of independent increments
E
v
i
(τ ) −v
i
(ξ)
v
i
(τ ) −v
i
(ξ)
T
=V
i
|τ −ξ |,V
i
> 0
Within the chosen framework for decentralized decision making, each agent is fur-
ther supposed to estimate its evolving process that depends on both its own local
measurements and interaction approximates from other agents. Given observations
y
i
(τ ), t
0
≤τ ≤t , the state estimate at agent i is denoted by ˆx
i
(t). Further, the state
estimate error covariance matrix of the state estimation error ˜x
i
(t) x
i
(t) −ˆx
i
(t)
3 Performance-Information Analysis and Distributed Feedback Stabilization 51
is described by Σ
i
(t) E{[x
i
(t) −ˆx
i
(t)][x
i
(t) −ˆx
i
(t)]
T
}. With the additional as-
sumption of (A
i
,H
i
) uniformly detectable, it can then be shown that the estimate
ˆx
i
(t) for each agent i is readily generated by the solution of the stochastic differen-
tial equation (Kalman-type filter)
d ˆx
i
(t) =
A
i
(t)x
i
(t) +B
i
(t)u
i
(t) +C
i
(t)z
i
(t) +E
i
(t)d
i
(t)
dt
+L
i
(t)
dy
i
(t) −H
i
(t) ˆx
i
(t) dt
, ˆx
i
(t
0
) =x
0
i
(3.8)
where ˜x
i
(t) is the state estimation error obtained from the error process
d ˜x
i
(t) =
A
i
(t) −L
i
(t)H
i
(t)
˜x
i
(t) dt +G
i
(t) dw
i
(t) −L
i
(t) dv
i
(t)
˜x
i
(t
0
) =0
(3.9)
and for each agent i it is remarked that the Kalman gain is given by
L
i
(t) =Σ
i
(t)H
T
i
(t)V
−1
i
(3.10)
The error covariance matrix Σ
i
(t) is the solution of the matrix Riccati differential
equation
d
dt
Σ
i
(t) = A
i
(t)Σ
i
(t) +Σ
i
(t)A
i
(t) +G
i
(t)W
i
G
T
i
(t)
−Σ
i
(t)H
T
i
(t)V
−1
i
H
i
(t)Σ
i
(t), Σ
i
(t
0
) =0 (3.11)
The aggregate autonomy-interaction arrangement that locally depicts the aspirations
and interaction details of each agent i then requires the following augmented sub-
system variables and parameters
x
ai
(t)
⎡
⎣
ˆx
i
(t)
˜x
i
(t)
z
mi
(t)
⎤
⎦
; x
0
ai
⎡
⎣
x
0
i
0
0
⎤
⎦
; w
ai
(t)
⎡
⎣
w
i
(t)
v
i
(t)
w
mi
(t)
⎤
⎦
(3.12)
The respective tradeoff interaction levels and local incentive (3.6) for agent i is
rewritten as follows
J
i
x
0
ai
;u(·), z
i
(·)
=x
T
ai
(t
f
)Q
f
ai
x
ai
(t
f
) +
t
f
t
0
x
T
ai
(τ )Q
ai
(τ )x
ai
(τ ) +u
T
i
(τ )R
i
(τ )u
i
(τ )
+z
T
i
(τ )R
zi
(τ )z
i
(τ ) −2x
T
ai
(τ )S
ai
(τ )z
i
(τ )
dτ (3.13)
where some of the local weightings are given by
Q
f
ai
⎡
⎣
Q
f
i
Q
f
i
0
Q
f
i
Q
f
i
0
000
⎤
⎦
; Q
ai
⎡
⎣
Q
i
Q
i
0
Q
i
Q
i
0
00R
zi
⎤
⎦
; S
ai
⎡
⎣
0
0
R
zi
⎤
⎦
(3.14)
52 K.D. Pham
Fig. 3.2 Risk-averse control
process: performance-
information analysis and
risk-averse decision policies
subject to the local dynamics for strategic relations
dx
ai
(t) =
A
ai
(t)x
ai
(t) +B
ai
(t)u
i
(t) +C
ai
(t)z
i
(t) +D
ai
(t)
dt
+G
ai
(t)dw
ai
(t)
x
ai
(t
0
) =x
0
ai
(3.15)
with the corresponding system parameters
A
ai
⎡
⎣
A
i
L
i
H
i
0
0 A
i
−L
i
H
i
0
00A
zi
⎤
⎦
; B
ai
⎡
⎣
B
i
0
0
⎤
⎦
; C
ai
⎡
⎣
C
i
0
0
⎤
⎦
(3.16)
D
ai
⎡
⎣
E
i
d
i
0
E
zi
d
mi
⎤
⎦
; G
ai
⎡
⎣
0 L
i
0
G
i
−L
i
0
00G
zi
⎤
⎦
; W
ai
⎡
⎣
W
i
00
0 V
i
0
00W
mi
⎤
⎦
(3.17)
Thus, far only the structure and the circumstantial setting of a large-scale intercon-
nected system have been made clear. The next task is a discussion of a risk-averse
control process which can be divided into two phases: (1) performance-information
analysis and (2) risk-averse decision policy. The essence of these phases is, in a
sense, the setting of a standard. Standards in performance-information analysis pro-
vide a yardstick of performance evaluation and standards in risk-averse decision
policy phase communicate and coordinate acceptable levels of risk aversion efforts
to the interconnected systems. Essentially, Fig. 3.2 shows a model of the standards
in a risk-averse control system serving two purposes: (1) as targets for the intercon-
nected systems to strive for and (2) as an evaluation yardstick.
3.3 Performance-Information Analysis
Performance-information analysis affects the consequence factor to agent i at two
levels. First, it specifies the variable of performance measure to be looked at and thus
determines an important element in the consequence of control decision at agent i.
Second, performance risk analysis implies the existence of standards of compari-
son for the performance variables specified. This implies that some notions of risk
modeling and measures for which standards have great significance may be in need.
3 Performance-Information Analysis and Distributed Feedback Stabilization 53
As a basic analytical construct, it is important to recognize that, within the view
of the linear-quadratic structure (3.15) and (3.13), the performance measure (3.13)
for agent i is clearly a random variable with Chi-squared type. Hence, the degree
of uncertainty of (3.13) must be assessed via a complete set of higher-order perfor-
mance statistics beyond the statistical averaging. The essence of information about
the states of performance uncertainty regards as a source of information flow which
will affect agent i’s perception of the problem and the environment. More specifi-
cally, for agent i, first and second characteristic functions of the Chi-squared random
variable (3.13), denoted respectively, by ϕ
i
(s, x
s
ai
) and ψ
i
(s, x
s
ai
) which are utilized
to provide a conceptual framework for evaluation of performance information, are
given by
ϕ
i
s,x
s
ai
;θ
i
E
e
θ
i
J
i
(s,x
s
ai
)
(3.18)
ψ
i
s,x
s
ai
;θ
i
ln
ϕ
i
s,x
s
ai
;θ
i
(3.19)
where small parameter θ
i
is in an open interval about 0 for the information sys-
tems ϕ
i
(s, x
s
ai
) and ψ
i
(s, x
s
ai
) whereas the initial condition, (t
0
,x
0
ai
) is now param-
eterized as the running condition, (s, x
s
ai
) with s ∈[t
0
,t
f
] and x
s
ai
x
ai
(s).From
the definition, there are three factors as the determinants of the value of perfor-
mance information: (i) the characteristics of the set of actions; (ii) the characteris-
tics of consequence of alternatives; and (iii) the characteristics of the utility func-
tion.
The first factor relates to the action set, denoted by U
i
via state-estimate feed-
back strategies to maintain a fair degree of accuracy on each agent behaviors and
the corresponding utility. Since the significance of (3.15) and (3.13) is the linear-
quadratic nature, the actions of agent i should therefore be causal functions of
the local process ˆx
i
(t). The restriction of decision strategy spaces can be justi-
fied by the assumption that agent i participates in the large-scale decision mak-
ing where it only has access to the current time and local state estimates of the
interaction. Thus, it amounts to considering only those feedback strategies which
permit linear feedback syntheses ˆγ
i
:[t
0
,t
f
]×L
2
F
i
(t
0
,t
f
;R
n
i
) →L
2
F
i
(t
0
,t
f
;R
m
i
)
and ˆγ
z
i
:[t
0
,t
f
]×L
2
F
i
(t
0
,t
f
;R
n
i
) →L
2
F
i
(t
0
,t
f
;R
q
i
)
u
i
(t) =ˆγ
i
t, ˆx
i
(t)
K
i
(t) ˆx
i
(t) +p
i
(t) (3.20)
z
i
(t) =ˆγ
z
i
t, ˆx
i
(t)
K
zi
(t) ˆx
i
(t) +p
zi
(t) (3.21)
where matrix-valued gains K
i
∈ C(t
0
,t
f
;R
m
i
×n
i
) and K
zi
∈ C(t
0
,t
f
;R
q
i
×n
i
) to-
gether with vector-valued affine p
i
∈C(t
0
,t
f
;R
m
i
) and p
zi
∈C(t
0
,t
f
;R
q
i
) will be
appropriately defined, respectively.
The importance of the second factor comes from the consequence of actions,
denoted by the outcome function which maps the triplets of outcomes, actions and
54 K.D. Pham
random realizations of the underlying stochastic process into outcomes, e.g.,
dx
ai
(t) =
F
ai
(t)x
ai
(t) +B
ai
(t)p
i
(t) +C
ai
(t)p
zi
(t) +D
ai
(t)
dt
+G
ai
(t) dw
ai
(t)
x
ai
(s) =x
s
ai
(3.22)
where the time-continuous composite state matrix, F
ai
is given by
F
ai
⎡
⎣
A
i
+B
i
K
i
+C
i
K
zi
L
i
H
i
0
0 A
i
−L
i
H
i
0
00A
zi
⎤
⎦
The characteristics of the utility function J
i
(s, x
s
ai
) which maps outcomes into utility
levels can be influenced the value of performance information in various ways, e.g.,
J
i
(s, x
s
ai
) = x
T
ai
(t
f
)Q
f
ai
x
ai
(t
f
) +
t
f
s
x
T
ai
(τ )N
ai
(τ )x
ai
(τ ) +2x
T
ai
(τ )O
ai
(τ )
dτ
+
t
f
s
p
T
i
(τ )R
i
(τ )p
i
(τ ) +p
T
zi
(τ )R
zi
(τ )p
zi
(τ )
dτ (3.23)
where
N
ai
⎡
⎣
Q
i
+K
T
i
R
i
K
i
+K
T
zi
R
zi
K
zi
Q
i
0
Q
i
Q
i
0
−2R
zi
K
zi
0 R
zi
⎤
⎦
O
ai
⎡
⎣
K
T
i
R
i
p
i
+K
T
zi
R
zi
p
zi
0
−R
zi
p
zi
⎤
⎦
The next result investigates the perception of performance riskiness from the stand-
point of higher-order characteristics pertaining to probability distributions with re-
spect to all random realizations from the underlying stochastic environment. It pro-
vides a quantitative explication of the concept of performance risk which is express-
ible as a linear combination of its higher-order statistics, i.e., performance-measure
statistics whose mathematical descriptions are given below.
Theorem 1 (Performance-measure statistics) For each agent i, let the pairs
(A
i
,B
i
) and (A
i
,C
i
) be uniformly stabilizable and the pair (A
i
,H
i
) be uniformly
detectable on [t
0
,t
f
]in the autonomous system governed by (3.22) and (3.23). Then,
for any given k
i
∈Z
+
, the k
i
-th cumulant associated with (3.23) is given by
κ
i
k
i
=
x
0
ai
T
H
i
(t
0
,k
i
)x
0
ai
+2
x
0
ai
T
˘
D
i
(t
0
,k
i
) +D
i
(t
0
,k
i
) (3.24)
where the cumulant variables {H
i
(s, r)}
k
i
r=1
, {
˘
D
i
(s, r)}
k
i
r=1
and {D
i
(s, r)}
k
i
r=1
eval-
uated at s =t
0
satisfy the supporting matrix-valued differential equations (with the
3 Performance-Information Analysis and Distributed Feedback Stabilization 55
dependence of H
i
(s, r),
˘
D
i
(s, r), and D
i
(s, r) upon the admissible K
i
, K
zi
, p
i
, and
p
zi
suppressed)
d
ds
H
i
(s, 1) =−F
T
ai
(s)H
i
(s, 1) −H
i
(s, 1)F
ai
(s) −N
ai
(s) (3.25)
d
ds
H
i
(s, r) =−F
T
ai
(s)H
i
(s, r) −H
i
(s, r)F
ai
(s)
−
r−1
v=1
2r!
v!(r −v)!
H
i
(s, v)G
ai
(s)W
ai
G
T
ai
(s)H
i
(s, r −v),
2 ≤r ≤k
i
(3.26)
d
ds
˘
D
i
(s, 1) =−F
T
ai
(s)
˘
D
i
(s, 1) −H
i
(s, 1)
B
ai
(s)p
i
(s) +C
ai
(s)p
zi
(s) +D
ai
(s)
−O
ai
(s) (3.27)
d
ds
˘
D
i
(s, r) =−F
T
ai
(s)
˘
D
i
(s, r) −H
i
(s, r)
B
ai
(s)p
i
(s) +C
ai
(s)p
zi
(s) +D
ai
(s)
,
2 ≤r ≤k
i
(3.28)
d
ds
D
i
(s, 1) =−2(
˘
D
i
)
T
(s, 1)
B
ai
(s)p
i
(s) +C
ai
(s)p
zi
(s) +D
ai
(s)
−Tr
H
i
(s, 1)G
ai
(s)W
ai
G
T
ai
(s)
−p
T
i
(s)R
i
(s)p
i
(s) −p
T
zi
(s)R
zi
(s)p
zi
(s) (3.29)
d
ds
D
i
(s, r) =−2(
˘
D
i
)
T
(s, r)
B
ai
(s)p
i
(s) +C
ai
(s)p
zi
(s) +D
ai
(s)
−Tr
H
i
(s, r)G
ai
(s)W
ai
G
T
ai
(s)
, 2 ≤r ≤k
i
(3.30)
where the terminal-value conditions H
i
(t
f
, 1) =Q
f
ai
, H
i
(t
f
,r)=0 for 2 ≤r ≤k
i
;
˘
D
i
(t
f
,r)=0; and D
i
(t
f
,r)=0 for 1 ≤r ≤k
i
.
To anticipate for a well-posed optimization problem that follows, some sufficient
conditions for the existence of solutions to the cumulant-generating equations in the
calculation of performance-measure statistics are now presented in the sequel.
Theorem 2 (Existence of solutions for performance-measure statistics) Let (A
i
,B
i
)
and (A
i
,C
i
) be uniformly stabilizable. Also let (A
i
,H
i
) be uniformly detectable.
Then, any given k
i
∈ Z
+
, the time-backward matrix differential equations (3.25)–
(3.30) admit unique and bounded solutions {H
i
(s, r)}
k
i
r=1
, {
˘
D
i
(s, r)}
k
i
r=1
and
{D
i
(s, r)}
k
i
r=1
on [t
0
,t
f
].
Proof Under the assumptions of stabilizability and detectability, there always ex-
ist some feedback control and filter gains, K
i
(s), K
zi
(s) and L
i
(s) such that the