New results on robust state bounding estimation for discrete time markovian jumb stochastic control systems
TNU Journal of Science and Technology
226(06): 67 - 72
NEW RESULTS ON ROBUST STATE BOUNDING ESTIMATION FOR
DISCRETE TIME MARKOVIAN JUMB STOCHASTIC CONTROL SYSTEMS
Nguyen Truong Thanh*, Nguyen Thu Hang, Pham Ngoc Anh
Hanoi University of Mining and Geology
ARTICLE INFO
Received: 05/3/2021
Revised: 26/5/2021
Published: 27/5/2021
ABSTRACT
The paper deals with the robust state bounding estimation problem of
stochastic control systems with discrete time Markovian jump. By
using Lyapunov functional method and probability theory, we
propose new sufficient conditions to guarantee robust state
boundedness for the stochastic control systems. The conditions are
derived in terms of linear matrix inequalities, which is simple and
convenient for testing and application. Unfortunately, difficulties arise
when one attempts to derive the sufficient conditions and to extract
the controller parameters for these systems. In fact, we have to cope
with stochastic process and disturbance. Indeed, the Lyapunov
functional method is a powerful tool to stability analysis of
differential systems. However, this method is not effectively applied
for stochastic systems because we do not know how to construct
suitable Lyapunov functions and use them in these systems. To
overcome the difficulties, we first introduced basic concepts of
probability theory. Next, a new sufficient conditions of robust state
boundedness for unforced stochastic systems was established. Finally,
the result was applied to design controllers to guarantee robust state
boundedness for stochastic control systems.
KEYWORDS
Markov chain
Complete probability space
Linear matrix inequality
Discrete-time system
Stochastic control
MỘT VÀI KẾT QUẢ MỚI VỀ ƯỚC LƯỢNG TÍNH BỊ CHẶN BỀN VỮNG
CỦA HỆ ĐIỀU KHIỂN NGẪU NHIÊN VỚI XÍCH MARKOVIAN RỜI RẠC
Nguyễn Trường Thanh*, Nguyễn Thu Hằng, Phạm Ngọc Anh
Trường Đại học Mỏ - Địa chất
THÔNG TIN BÀI BÁO
Ngày nhận bài: 05/3/2021
Ngày hoàn thiện: 26/5/2021
Ngày đăng: 27/5/2021
TÓM TẮT
Bài báo này ước lượng tính bị chặn bền vững của hệ điều khiển ngẫu
nhiên với xích Markovian rời rạc. Bằng cách sử dụng phương pháp
hàm Lyapunov và lí thuyết xác suất, chúng tôi đề xuất một số điều
kiện đủ mới để đảm bảo tính bị chặn bền vững cho hệ điều khiển
ngẫu nhiên. Các điều kiện trên là các bất đẳng thức ma trận tuyến
tính có thể kiểm tra và dễ dàng sử dụng trong thực tế. Thật không
may mắn, có nhiều khó khăn nảy sinh khi nghiên cứu các hệ này khi
chúng ta phải đối mặt với các quá trình ngẫu nhiên và nhiễu không
mong muốn. Thêm vào đó, phương pháp hàm Lyapunov là một công
cụ đầy hiệu quả khi nghiên cứu tính ổn định của các hệ phương trình
vi phân. Tuy nhiên, phương pháp này không hiệu quả khi áp dụng
cho các hệ ngẫu nhiên do chúng ta không biết làm thế nào để cấu trúc
các hàm Lyapunov phù hợp và làm thế nào để sử dụng các hàm này
cho các quá trình ngẫu nhiên. Để vượt qua các khó khăn trên, đầu
tiên, chúng tôi giới thiệu các khái niệm cơ bản của lí thuyết ngẫu
nhiên. Tiếp đó, chúng tôi thiết lập một điều kiện đủ mới về tính bị
chặn bền vững cho hệ ngẫu nhiên không có điều khiển. Cuối cùng,
kết quả này được áp dụng cho việc thiết kế điều khiển cho hệ ngẫu
nhiên có điều khiển.
TỪ KHÓA
Xích Markov
Không gian xác suất đầy đủ
Bất đẳng thức ma trận tuyến tính
Hệ rời rạc
Điều khiển ngẫu nhiên
67
TNU Journal of Science and Technology
226(06): 67 - 72
1. Introduction
Markovian jump systems belong to a significant class of hybrid systems which are described
by the switches between subsystems and a finite-state Markov chain according to a known
Markov law. Discrete-time Markovian jump systems arise in many practical processes subjected
to random abrupt changes in the inputs, internal variables [1]-[4]. Examples of such systems with
Markov chain are solar thermal central receivers, economic systems and manufacturing systems.
Therefore, Markovian jump systems have attracted a lot of attention in many applications of
signal processing, control theory and communications because of their flexibility to model real
world phenomena [5], [6].
In recent years, considerable interest has been focused on many important problems in
systems and control theory of Markovian jump systems. Based on Lyapunov’s second method,
some sufficient conditions in terms of linear matrix inequalities for following problems are
proposed as mean square stability of linear discrete-time systems [7], state bounding problem [8],
[9], stabilisation for Markovian jump systems with state and input delays [10], and H control
problems for linear Markovian jump systems with mode-dependent/independent delays [11] etc.
Besides, disturbances is inherent characteristic of many physical systems and ubiquitous in
dynamic systems. These disturbances are the unavoidable source of instability and poor
performance. Hence, the problem of state bounding for Markovian jump systems with
disturbances is a key topic in control engineering. However, to the best knowledge of authors,
there are very few results on the problem for stochastic control system with disturbance input and
discrete-time Markovian chain. This has motivated our research.
The paper is organized as follows. Section 2 presents definitions and some well-known
technical lemmas needed for the proof of the main results. A main result for designing controllers
to ensure
mean square boundedness is presented in Section 3. The paper ends with
−
conclusions.
2. Preliminaries and problem statement
The following notations will be used throughout this paper:
denotes the set of all non-
N
negative integer numbers
R
+ stands for the set of all non-negative real numbers;
R
n denotes the n-
dimensional space with the scalar product (x, y) = xT y and the vector norm
;
AT
T
| x|= x x
denotes the transpose of the matrix
and n denotes the identity matrix in Rn
;
I
max (A),
A
min (A)
stand for the maximal and the minimal real part of eigenvalues of , respectively;
A
xT Qx 0
Q 0 means that
Q
is positive definite, .i.e.,
for all
A B means A− B 0
;
x 0;
, F,P .
is the expectation operator with respect to some complete probability space
The
symmetric terms in a matrix are denoted by *.
Consider the discrete time stochastic control system described by the following equations:
(
)
x(k +1) = A(r )x(k) + B(r )w(k) + C(r )u (k), k N,
k
k
k
r
k
(2.1)
x(0) = 0,
x(k)Rn
w(k)Rs
is the disturbance input;
u (k)
where
is the state vector;
is the control;
rk
M = 1,2,
r
be a discrete Markov chain with state space
The transition probabilities
k k=0
m
p 0
of
r
are given by P r = j | r =i = p , where
and
p =1, i, jM.
k k=0
(
)
ij
ij
k+1
k
ij
j=1
A(r ), B(r ),C(r )
k
k
k
are system matrices of appropriate dimensions in the finite set of
68
TNU Journal of Science and Technology
226(06): 67 - 72
A := A(i), B := B(i),Ci := C(i),iM.
with
A ,B ,C ,iM
i
i
i
i
i
The disturbance w(k) satisfies the condition
b 0: w(k) 2 b, kN.
Definition 2.1. ([8]) For a given 0 , system (2.1) is said to be
(2.2)
mean square bounded
−
if every trajectory x(k) of (2.1) satisfies
2
E x(k) | F , k N,
0
F = x(0),r
where
be the σ-algebra generated by
x(0) = 0,r .
0
(
)
0
0
Lemma 2.1. ([12]) Give constant matrices Y =YT 0 and X,Z . Then
X
Z
ZT
X + ZTY−1Z 0
0.
−Y
3. Main results
In this section, we will give sufficient conditions of
mean square boundedness for (2.1).
−
We first establish a sufficient condition for unforced system (2.1). Then, we design controllers
based on LMIs to guarantee
mean square boundedness for (2.1).
−
Let us set
m
Q = Q p , Q = Q( j), i, j M;
i
j
ij
j
j=1
(1−)
= min (Q)
.
b
Theorem 3.1. Give 0, (0,1), and a symmetric positive definite matrix Q. If there exist
Q , iM,
symmetric positive definite matrices
satisfying the following LMI conditions for all
i
iM,
T
A Q A −Q
AT Q B
i
i
i
i
i
i
i
0,
(3.1)
(3.2)
*
BT Q B − I
s
i
i
i
Q Q, iM,
i
then unforced system (2.1) is
mean square bounded.
−
Proof. Consider the following function
V(x(k),r ) = x(k)T Q(r )x(k), k N.
k
k
r = i.
rk+1 = j
pij .
Suppose that
Then, at time
the mode
with probability
k +1,
k
Firstly, we estimate
V (x(k +1),r ) |x(k),r
k
k+1
r = i,
as follows. It is easy to see that if we fix x(k) and
the value of
k
x(k +1) = A x(k) + B w(k)
i
i
x(k),w(k) .
is only dependent on
Then
(
)
V (x(k +1),r ) |x(k),r = i
k+1
k
r .
is dependent on
It leads to
k +1
69
TNU Journal of Science and Technology
226(06): 67 - 72
T
V (x(k +1),r ) |x(k),r = i = x(k +1) Q(r )x(k +1) |x(k),r = i
k+1
k
k+1
k
m
m
=
x(k +1)T Q x(k +1)P(r = j | r = i) = x(k +1)T Q x(k +1)p = x(k +1)T Q x(k +1)
j
k+1
k
j
ij
i
j=1
j=1
= Ax(k) + B w(k) T Q Ax(k) + B w(k) −x(k)T Q x(k) +V(x(k),r =i)
i
i
i
i
i
i
k
T
AT Q B
x(k)
A Q A −P
T
+ w(k) 2 +V(x(k),r = i).
i
i
i
i
i
i
i
= x(k),w(k)
k
w(k)
*
BT Q B − I
s
i
i
i
From (3.1), we have
V(x(k +1),r )|x(k),r =i w(k) 2 +V(x(k),r =i), iM.
k+1
k
k
Hence, because of (2.2), we obtain
V (x(k +1),r ) |x(k),r b +V (x(k),r ).
(3.3)
(3.4)
k
k+1
k
Next, we use the inequality (3.3) to evaluate
V (x(k),r ) |x(0),r , k N.
0
k
From (3.3), we have the followings.
For k = 0,
V (x(1),r ) |x(0),r b +V (x(0),r ) = b +x(0)T Q(r )x(0) = b.
0
1
0
0
For k =1,
V (x(2),r ) |x(1),r b +V (x(1),r ).
1
2
1
Consequently,
V(x(2),r ) |x(1),r |(x(0),r ) b +V(x(1),r ) |(x(0),r ).
1
2
0
1
0
Taking expectation both sides of the above inequality, we have
V(x(2),r ) |x(0),r = V(x(2),r ) |x(1),r |(x(0),r )
0 1
(
)
2
2
0
b + V(x(1),r ) |(x(0),r ) b +b = b 1+ .
(
)
(
)
1
0
Similarly, we obtain
V(x(k),r )|x(0),r b 1+ + = b
1−k−1
1−
, k N.
0
(
)
k
Hence,
b
1−
V(x(k),r )|F = V(x(k),r )|x(0),r
0 0
Since the condition (3.2), the following inequality holds
, k N.
(3.5)
k
k
2
V(x(k),r =i) = x(k)T Q x(k) x(k)T Qx(k) min (Q) x(k) .
k
i
Moreover,
V(x(k),r ) min (Q) x(k) 2 , k N.
(3.6)
(3.7)
k
2
Using (3.6) and the monotonicity of the operator
, we have
2
0
V(x(k),r )|F min (Q) x(k) |F = min (Q) x(k) |F .
0
k
0
Combining (3.5) and (3.7), we obtain
b
1
2
x(k) |F
= ,
0
1− min (Q)
which completes the proof of the theorem.
70
TNU Journal of Science and Technology
226(06): 67 - 72
ur (k) = Lr x(k)
The remaining of this section is to design controllers in the form of
such
k
k
that the system (2.1) is
Let us denote
mean square bounded.
−
m
Q = Q p , Q = Q( j), i, j M;
i
j
ij
j
j=1
T
= AT Q A + AT K C + K C A −Q ;
i
11
i
i
i
i
i
i
i
i
i
12 = A Q B + CiT Ki Bi ; 22 = BiT Q Bi − I.
i
i
i
Theorem 3.2. Given 0, (0,1), and a symmetric positive definite matrix Q. If there
Qi , i M,
Ki , i M,
satisfying the
exist symmetric positive definite matrices
following LMI conditions for all
and matrices
iM,
T
11 12
K C
i
i
(3.8)
(3.9)
*
*
22
0
0,
*
−min (Q)I
Q Q,
i
then system (2.1) is
mean square bounded with the controller.
−
−1
ui (k) = Q
Ki x(k).
i
Proof.
From (3.2) and
m
m
Q = Q p ,
p =1, p 0,
i
j
ij
ij
ij
j=1
j=1
we have
m
m
Q = Q p Qp = Q Q I 0,
min ( )
i
j
ij
ij
j=1
j=1
it leads to
.
min Q Q 0
min ( )
i
Consequently,
1
1
−1
Q
i
−1
Q
max
I =
I
I.
i
min
Q
( )
min
Q
i
Use the above inequality and (3.8) and Schur complement lemma (Lemma 2.1) with
−1
L = Q
Ki , iM,
i
i
we have
A + LC T Q A + LC −Q
A + LC Q B
T
T
−1
+ K C
Qi KiC 12
i i
i
i
i
*
i
i
i
i
i
i
i
i
i
11
i
=
BT Q B − I
*
22
i
i
i
(3.10)
1
T
11 +
K C KiC 12
i
i
min (Q)
0, iM.
*
22
From (3.9) and (3.10), using Theorem 3.1, system (2.1) is
completes the proof of the theorem.
mean square bounded. This
−
71
TNU Journal of Science and Technology
226(06): 67 - 72
4. Conclusion
In this paper, we have studied the robust state bounding estimation problem of stochastic
control systems with discrete time Markovian jump. The proposed analytical tools used in the
proof are based on Lyapunov functional method. The sufficient conditions for
mean square
−
boundedness have been established in terms of LMIs.
Acknowledgments
The authors would like to thank the anonymous reviewers for their valuable comments and
suggestions which allowed them to improve the paper.
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