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Networked guaranteed cost control for a class of industrial processes with state delay.

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ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING
Asia-Pac. J. Chem. Eng. 2007; 2: 650–658
Published online 13 September 2007 in Wiley InterScience
(www.interscience.wiley.com) DOI:10.1002/apj.092
Research Article
Networked, guaranteed cost control for a class of industrial
processes with state delay
Chen Peng1,2 *
1
2
School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing, Jiangsu, 210042, P. R. China
Faculty of Information Technology, Queensland University of Technology, Brisbane QLD 4001, Australia
Received 4 February 2007; Revised 15 May 2007; Accepted 8 August 2007
ABSTRACT: This paper addresses a class of industrial processes with state delay and controlled over communication
networks. With consideration of network-induced delay and data packet dropout, a model is proposed for networked
control systems (NCSs). With the introduction of free weighting matrices, a method for suboptimal guaranteed cost
controller design is proposed in the form of matrix inequalities. A nonconvex minimization problem is then formulated
for finding suboptimal allowable equivalent delay bound (SAEDB) and the feedback gain of a memoryless controller.
The proposed method is demonstrated through networked injection velocity control in thermoplastic injection molding.
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
KEYWORDS: guaranteed cost; state delay; NCSs; LMIs; suboptimal allowable equivalent delay bound
INTRODUCTION
Many industrial processes have input and/or output
delay. Basically, there are two methods to describe
those industrial processes: the first- or second-order
plus delay model, and the state-space model with state
delay. To deal with those systems with delay, many control schemes have been presented, such as the double
controller scheme (Tian and Gao, 1999), the model predictive control approach (Tellez et al ., 2005), and the
fault-tolerant control approach (Su et al ., 2006). However, most of them are local control systems (LCSs),
implying that the physical architecture of communication networks for process control is point-to-point with
asynchronous communications. The advantages of this
architecture are obvious, such as small transmission
delay (which usually is neglected in system analysis
and design), high reliability, and deterministic communication behavior. However, with the growing complexity of industrial processes and increasing demands
on optimization of conflicting objectives, safe operation, and minimum environment requirements, more
and more components and devices are being introduced
into communication networks of control systems (Tian
et al ., 2006a), which are termed networked control systems (NCSs). NCSs involve communication patterns in
*Correspondence to: Chen Peng, School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing, Jiangsu,
210042, P. R. China. E-mail: pchme@163.com
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
which both informational and physical control loops are
closed through a real-time network.
Recently, much attention has been paid to the study
of stability analysis and control design of NCSs (Zhang
et al ., 2001; Yue et al ., 2005; Peng and Tian, 2006a)
owing to their low cost, reduced weight and power
requirements, simple installation and maintenance, and
high reliability. Compared to LCS (Han, 2004; Jiang
and Han, 2006), one of the important issues in NCSs is
the effect of network-induced delay on the system performance. Performance of the NCSs is directly dependent upon the network-induced delay. Time-varying
characteristics of the network-induced delay not only
degrade the control performance but also introduce distortion of the controller signal (Hong, 1995).
There has been much effort to assign certain performance criteria when designing a controller, such
as quadratic cost minimization, pole placement, and
H2 /H∞ norm minimization. Among those methods, the
guaranteed cost control aims to stabilize the systems
while maintaining an adequate level of performance
represented by the quadratic cost. The existing guaranteed cost control design methods can be classified into
two types: delay-independent stabilization (Moheimani
and Moheimani, 1997; Yu and Gao, 2001) in which a
state feedback controller is obtained irrespective of the
size of the delay, and delay-dependent stabilization (Lee
et al ., 2001; Yue et al ., 2006) in which the controller
is designed with consideration of the size of the delay.
Most of the existing reports on guaranteed cost control
of NCSs in the open literature have dealt with signal
Asia-Pacific Journal of Chemical Engineering
COST CONTROL FOR A CLASS OF INDUSTRIAL PROCESSES
quantization in Yue et al . (2006), networked control for
T–S fuzzy systems in Zhang et al . (2007), discretetime model with structural uncertainties in Huang and
Nguang (2006) and the existence of a state feedback
controller in terms of the solvability of bilinear matrix
inequalities (BMIs) in Li et al . (2004). To the best of
our knowledge, no results have been found on the statedelay-guaranteed cost control of NCSs, an important
issue to practical industrial processes with delay. This
has greatly motivated this research.
This paper is structured as follows: Modeling of
NCSs and the definition of suboptimal allowable equivalent delay bound (SAEDB) are presented in Section 2.
The SAEDB and guaranteed cost controller design for
NCSs with state delay are considered and an optimal
problem is formulated in Section 3. Numerical examples are given to demonstrate the effectiveness of our
method in Section 4. Section 5 concludes the paper.
Notation: n denotes the n-dimensional Euclidean
space, n×m is the set of n × m real matrices, I is the
identity matrix of appropriate dimensions, || · || stands
for the Euclidean vector norm or the induced matrix 2norm as appropriate. The notation X > 0 (respectively,
X ≥ 0), for X ∈ n×n means that the matrix X is a
real symmetric positive definite (respectively, positive
semidefinite). λmax (P )(λmin (P )) denotes the maximum
(minimum) of eigenvalue of real symmetric matrix P .
For an arbitrary
matrix
B and two symmetric matrices
A B
A and C ,
denotes a symmetric matrix, where
∗ C
∗ denotes the entries implied by symmetry.
PROBLEM STATEMENTS
u(t + ) = u(ik h + τik ), t ∈ [ik h + τik , ik +1 h + τik +1 )
(4)
where h is the sampling period, ik (k = 1, 2, 3, . . .) are
some integers and {i1 , i2 , i3 , . . .} ⊂ {0, 1, 2, 3, . . .}. τik
is the network-induced delay, which denotes the time
from the instant ik h when sensor nodes sample from the
plant to the instant when actuator nodes transmit data to
the plant. It follows that ∪∞
k =1 [ik h + τik , ik +1 h + τik +1 ] =
+ τisc
, where τica
[t0 , ∞), t0 ≥ 0. Assume that τik = τica
k
k
k
sc
is the sensor to controller delay, τik is controller to
actuator delay and computational and overhead delays
.
are included in τisc
k
Further assume that there is no error code in the
transmission and the state feedback controller is chosen.
From above analysis, Eqn (4) can be replaced by
u(t + ) = Kx (t − τik ), t ∈ {ik h + τik , k = 1, 2, . . .}
(5)
where K is the state feedback gain.
According to Eqn (5), the real input u(t) realized
through the zero-order hold is a piecewise constant
function and the real control system can be modeled
as
ẋ (t) = (A + A)x (t) + (A1 + A1 )x (t − d)
Consider the following system with state delay and
parameter uncertainties
ẋ (t) = (A + A)x (t) + (A1 + A1 )x (t − d)
+ (B + B )u(t)
(1)
x (t) = ϕ(t), t ∈ [−d, 0]
(2)
where x (t) ∈ n and u(t) ∈ m are the state vector
and the control input vector, respectively. A, A1 , and
B are constant matrices with appropriate dimensions.
A, A1 and B are uncertain matrices which can be
time varying. d is a scalar representing the delay in the
industry system.
Assume that the parameter uncertainties A, A1 ,
and B of system model are norm-bounded and satisfy
[A A1 B ] = DF (t)[E1 E2 E3 ]
Also assume that the sensors are clock-driven and
controllers and actuators are event-driven; the data
is transmitted with a single-packet and the full state
variables are available for measurements; the real input
u(t) realizes through zero-order hold. The time interval
between the instant i(k ) h + τ(i(k ) ) , at which a packet
arrives at the actuator, and the next arrival instant
i(k +1) h + τ(i(k +1) ) is the effect duration of the Zero-hold
operation, which can be described as
(3)
where F (t) ∈ i ×j is an uncertain matrix which satisfies F T (t)F (t) ≤ I . D, E1 , E2 and E3 are constant
matrices of appropriate dimensions.
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
+ (B + B )u(t)
(6)
u(t + ) = Kx (t − τik ), t ∈ {ik h + τik , k = 1, 2, . . .} (7)
The closed-loop system (Eqn (6)) with network communication can be expressed as
ẋ (t) = (A + A)x (t) + (A1 + A1 )x (t − d)
+ (B + B )Kx (ik h), t ∈
[ik h + τik , ik +1 h + τik +1 )
(8)
Remark 1 In Eqn (8), {i1 , i2 , i3 , . . .} is a subset of
{0, 1, 2, 3, . . .}. Moreover, it is not required that ik +1 >
ik . If |ik +1 − ik | = 1, it means no data packet dropout in
the transmission. If ik +1 > ik + 1, it means some data
packet dropout. If ik +1 < ik , it means that unordered
data arrival sequence occurs. Therefore, Eqn (8) can be
viewed as a general form of NCSs, in which the effect
of the networked-induced delay, data packet dropout,
and unordered data arrival sequence are simultaneously
considered. Detailed explanation can be found in Peng
Asia-Pac. J. Chem. Eng. 2007; 2: 650–658
DOI: 10.1002/apj
651
652
C. PENG
Asia-Pacific Journal of Chemical Engineering
and Tian (2006a), Yue et al . (2004), Peng and Yue
(2006b).
are satisfied, where
1 + M
1T ,
11 = S + AX T + XAT + M
2T ,
12 = A1 X T + λ2 XAT + M
1 + M
3T ,
13 = BY + λ3 XAT − M
Given positive definite symmetric matrices Q1 and
R1 , we will consider the cost function
∞
[x T (t)Q1 x (t) + u T (t)R1 u(t)]dt
(9)
J =
4T ,
14 = P + λ4 XAT − X T + M
22 = −
S + λ2 A1 X T + λ2 XAT1 ,
2 ,
23 = λ2 BY + λ3 XAT1 − M
0
Associated with the cost function (9), the guaranteed
cost controller is defined as follows:
Definition 1 Consider the uncertain time-delay system of NCSs (8). If there exists a control law u(t) and
a positive scalar γ , such that for all admissible uncertainties the closed-loop system is stable and the value
of the cost function (9) satisfies J ≤ γ , then γ is said to
be a guaranteed cost and u(t) is said to be a guaranteed
cost controller for the system (8).
To facilitate development, we first introduce the
following definition.
Definition 2 SAEDB of NCSs, denoted by η, which
satisfies (ik +1 − ik )h + τik +1 ≤ η, k = 1, 2, 3, . . .
Remark 2 In Definition 2, |(ik +1 − ik )| > 1 implies
that there are some packet dropouts. If ik +1 < ik , there
are non-ordered data sequences. SAEDB is directly
related with the number of data packet dropouts
(|(ik +1 − ik )| − 1), network-induced delay τik +1 , and the
sampling period h. Therefore, the SAEDB η can be
used as a parameter for co-design of NCS controller
and network QoS.
MAIN RESULTS
24 = −λ2 X T + λ4 XAT1 ,
3 − M
3T ,
33 = λ3 BY + λ3 Y T B T − M
4T ,
34 = −λ3 X T + λ4 Y T B T − M
44 = η
R − λ4 X T − λ4 X ,
then the system (8) is asymptotically stable with the
feedback gain K = YX −T , the cost function J in
(9) satisfies the following bound:
P X −T ϕ(0) +
J ≤ϕ T (0)X −1 +

1
12 13 14 ηM
X
2
0
22 23 24 ηM
 ∗

3
33 34 ηM
∗
0
 ∗
 ∗
4
44 ηM
∗
∗
0

 ∗
∗
∗
∗
η
R
0

 ∗
∗
∗
∗
∗
−Q1−1
∗
∗
∗
∗
∗
∗
((ik +1 − ik )h + τik +1 ) ≤ η, k = 1, 2, . . .
11
0 
0 

YT 
0 
 < 0 (10)
0 

0 
−R1−1
(11)
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
−η
0
0
S X −T ϕ(t)dt
ϕ T (t)X −1
−d
RX −T ϕ̇(v )dvds
ϕ̇ T (v )X −1
(12)
s
Proof: We set y(t) = ẋ (t) and construct a Lyapunov–
Krasovskii functional candidate as
(13)
V (xt ) = V1 (xt ) + V2 (xt ) + V3 (xt )
t
where V1 (xt ) = x T (t)Px (t), V2 (xt ) = t−d x T (v )Sx
t t T
(v )dv , V3 (xt ) = t−η s y (v )Ry(v )dvds, and P > 0,
S > 0 and R > 0. Taking the time derivative of V (xt )
along the state trajectory (8) yields that, for t ∈ [ik h +
τik , ik +1 h + τik +1 ),
V̇1 (xt ) = 2x T (t)Py(t)
In this section, we will develop some practically computable criteria to obtain the SAEDB and relevant controller for system (8) based on a Lyapunov–Krasovskii
functional method. First, let us consider the simple case
without the parameter uncertainties A, A1 and B .
Theorem 1 For given scalars η and λi (i = 2, 3, 4), if
i (i =
there exist matrices P, S and R > 0, X ,Y and M
1, 2, 3, 4) with appropriate dimensions such that
0
(14)
V̇2 (xt ) = x (t)Sx (t) − x (t − d)Sx (t − d) (15)
t
T
V̇3 (xt ) = ηy (t)Ry(t) −
y T (s)Ry(s)ds (16)
T
T
t−η
the Newton–Leibnitz formula x (t) − x (ik h) −
Using
t
y(s)ds
= 0 and Eqn (8), for arbitrary matrices Ni
ik h
and Mi (i = 1, 2, 3, 4) of appropriate dimensions, we
have
t
T
T
y(s)ds = 0
(17)
ξ (t)M x (t) − x (ik h) −
ik h
and
ξ T (t)N T [Ax (t) + A1 x (t − d) + BKx (ik h) − y(t)] = 0
(18)
Asia-Pac. J. Chem. Eng. 2007; 2: 650–658
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
COST CONTROL FOR A CLASS OF INDUSTRIAL PROCESSES
where ξ T (t) = [x T (t)x T (t − d)x T (ik h)y T (t)], M =
[M1T M2T M3T M4T ], N = [N1T N2T N3T N4T ].
From (11), it can be seen that, when t ∈ [ik h +
τik , ik +1 h + τik +1 ),
t
t
y T (s)Ry(s)ds ≤ −
y T (v )Ry(v )dv
(19)
−
t−η
ik h
and
− 2ξ (t)M
T
+
t
T
y(s)ds ≤ ηξ T (t)M T R −1 M ξ(t)
Defining N2 = λ2 N1 , N3 = λ3 N1 , N4 = λ4 N1 and Y =
P = XPX T , R = XRX T , S =
KX T , defining X = N −1 , T
T
XSX and Mi = XMi X (i = 1, 2, 3, 4) then pre-postmultiplying both sides of (24) with diag(X , X , X , X )
and its transpose,respectively, (23) can be derived by
the (24).
 + Q
11
1
∗


∗

∗
∗
12
22
∗
∗
∗
13
23
33 + K T R1 K
∗
∗
14
24
34
44
∗
ηM1 
ηM2 
ηM3 
<0
ηM4
−ηR
(24)
ik h
t
y T (v )Ry(v )dv
(20)
ik h
Then combining Eqns (17)–(20) and considering the
performance index (9) for t ∈ [ik h + τik , ik +1 h + τik +1 ),
we obtain


11 12 13 14
 ∗
22 23 24 
V̇ (xt ) ≤ ξ T (t) 
∗
∗
33 34 
∗
∗
∗
44
× ξ(t) + ηξ T (t)M T R −1 M ξ(t)

11 + Q1 12
13
∗
22
23

T
≤ ξ (t) 
∗
∗
33 + K T R1 K
∗
∗
∗
× ξ(t) + ηξ T (t)M T R −1 M ξ(t)
(21)

14
24 
34 
44
(22)
From (24) and by Schur complements, we have for
t ∈ [ik h + τik , ik +1 h + τik +1 )
V̇ (xt ) ≤ −x T (t)Q1 x (t) − x T (ik h)K T R1 Kx (ik h) < 0
Since ∞
k =1 [ik h + τik , ik +1 h + τik +1 ) = [t0 , ∞) and x (t)
is continuous in t, t0 ≥ 0 and V (xt ) is continuous
in t ∈ [t0 , ∞) (Yue et al ., 2005). Therefore by using
Lyapunov–Krasovskii theorem, the closed-loop system
(8) is asymptotically stable. Furthermore, by integrating
both sides of (25) from 0 to T and using the initial
condition, we obtain
T
[x T (t)Q1 x (t) + u T (t)R1 u(t)]dt
−
0
≥ x T (T )Px (T ) − x T (0)Px (0)
0
T
x T (v )Sx (v )dv −
x T (v )Sx (v )dv
+
T −d
where
11 = S + N1 A + AT N1T + M1 + M1T
12 = N1 A1 + AT N2T + M2T
13 = N1 BK + A
T
N3T
− M1 +
M3T
44 =
12
22
∗
∗
∗
y T (v )Ry(v )dvds
−η
−
M4T
13
23
33 + Y T R1 Y
∗
∗
(26)
s
T
T →∞ T −d
By Schur complements, we can show that the solvability of (10) implies that of (23).

11 + XQ1 X T
∗


∗


∗
∗
0
lim
33 = N3 BK + K B N3T − M3 − M3T
34 =
s
0
lim x T (T )Px (T ) = 0,
AT1 N4T
T T
−N3 + K B N4T
ηR − N4 − N4T
T −η
T →∞
23 = N2 BK + AT1 N3T − M2
T
T
As the closed-loop system (8) is asymptotically stable,
22 = −S + N2 A1 + AT1 N2T
T
−d
y T (v )Ry(v )dvds
−
14 = P + AT N4T − N1 + M4T
24 = −N2 +
+
T
14
24
34
44
∗
1 
ηM
2 
ηM
3 
ηM
<0
4 
ηM
−η
R
(23)
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
T
x T (v )Sx (v )dv = 0,
lim
T →∞ T −η
T
y T (v )Ry(v )dvds = 0
(27)
s
Hence we obtain
∞
[x T (t)Q1 x (t)
0
+ u T (t)R1 u(t)]dt ≤ ϕ T (0)P ϕ(0)
0
ϕ T (v )S ϕ(v )dv
+
+
−d
0
0
ϕ̇ T (v )R ϕ̇(v )dvds
−η
(28)
s
Asia-Pac. J. Chem. Eng. 2007; 2: 650–658
DOI: 10.1002/apj
653
654
C. PENG
Asia-Pacific Journal of Chemical Engineering
From the definition P = XPX T , R = XRX T , S = XSX T ,
−1 −T
−1 −T
S
we have P = X P X , R = X RX , S = X −1
X −T . Substituting them into (28), we obtain (12). It
follows that this sufficient condition for the existence
of guaranteed cost controllers is satisfied from the
Definition 2. This completes the proof.
Remark 3 From the proof of Theorem 1, we define
N2 = λ2 N1 , N3 = λ3 N1 and N4 = λ4 N1 to derive condition (10), showing that (24) and (10) are not strictly
equivalent. Therefore, it can be expected that the controller designed by condition (24) can guarantee the
asymptotic stability of system (8) with a larger bound
η than one derived on the basis of (10).
where
P X −T ϕ(0),
J1 = ϕ T (0)X −1 0
S X −T ϕ(v )dv ,
ϕ T (v )X −1
J2 =
J3 =
+
0
−d
ϕ̇ T (v )R ϕ̇(v )dvds
−η
Algorithm 1 Obtain η:
1. Given αi and βi as upper and lower bounds of λi ,
respectively, and ζi (i = 1, 2, 3) as step increment.
Set η0 = 0 and K = 0.
2. According to Theorem 1, under the constraint of ζi ,
αi , and βi , group different λi to obtain corresponding
η and K based on LMI, subject to (10) and (11). If
η > η0 , set η = η0 and K = K0 , else return to step 2.
3. Based on Corollary 1, substitute K = K0 into (24)
and find the maximum allowable bound of η.
4. Output η and stop.
However, it is worth mentioning that the corresponding upper bound (12) for the cost function (9) is not
a convex function in P, S , and R. In the following,
we propose a method to find a suboptimal value for
this upper bound. In order to achieve the least guaranteed cost value γ among all possible choices of P, S,
i (i = 1, 2, 3, 4), we have to solve the
R, X , Y , and M
following minimization problem:
Min J1 + J2 + J3
subject to (10) and (11)
3 =
(30)
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
RX −T ϕ̇(v )dvds.
ϕ̇ T (v )X −1
s
−d
0
0
ϕ̇ T (v )ϕ̇(v )dvds
−η
(31)
s
According to tr(AB ) = tr(BA), the following relationships hold:
J1 = ϕ T (0)X −1 P X −T ϕ(0) = tr(1 X −1 P X −T )
0
ϕ T (v )X −1
S X −T ϕ(v )dv = tr(2 X −1
S X −T )
J2 =
J3 =
To obtain the η, we present following search algorithm:
−η
0
(29)
s
1 = ϕ T (0)ϕ(0),
0
ϕ T (v )ϕ(v )dv ,
2 =
−d
0
0
Before moving on, let us define matrix values 1 ,
2 , and 3 such that
From the proof of Theorem 1, we have the following
corollary.
Corollary 1 For given scalars η and K , if there exist
matrices P , S and R > 0, X , Ni and Mi (i = 1, 2, 3, 4)
with appropriate dimensions, such that (11) and (24)
are satisfied, then system (8) with the feedback gain
K is asymptotically stable, and the cost function J in
(9) satisfies the following bound:
0
T
ϕ T (t)S ϕ(t)dt
J ≤ ϕ (0)P ϕ(0) +
−d
0
−η
0
ϕ̇ T (v )X −1
RX −T
s
RX −T )
ϕ̇(v )dvds = tr(3 X −1
(32)
Therefore, the minimization of the upper bound of the
cost in (30) is formulated as follows:
P X −T ) + tr(2 X −1
S X −T )
Min tr(1 X −1 RX −T )
+ tr(3 X −1
subject to (10) and (11)
(33)
Let us derive the upper bounds on the cost function
(9). Assume that there exist new variables 1 = 1T ,
2 = 2T , 3 = 3T which satisfy
X −1 P X −T < 1 , X −1
S X −T
< 2 , X −1
RX −T < 3
(34)
By Schur complement, (34) is equivalent to (35).
1
X −T
3
X −T
2
X −1
>
0,
X −T
P −1
X −1
>0
−1
R
X −1
> 0,
S −1
(35)
Asia-Pac. J. Chem. Eng. 2007; 2: 650–658
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
COST CONTROL FOR A CLASS OF INDUSTRIAL PROCESSES
By introducing new variables L, T , W and H , and
P −1 , W = S −1 and H = R −1 ,
defining L = X −1 , T = we can replace condition (35) by (36).
2 L
3 L
1 L
>
0,
>
0,
> 0 (36)
LT T
LT W
LT H
For finding solution of (39), we present the following
iterative algorithm, which are similar to those in Ghaoui
et al . (1997) and Lee et al . (2001). Here, the (34) is
used as a termination criterion in the algorithm 2.
Assuming (36) is satisfied, we can conclude that the
following inequalities hold:
1. Choose different λi (i = 2, 3, 4) to obtain the maximum η based on Algorithm 1, such that there exists a
feasible solution to LMI conditions in (10) and (11).
2. Choose a sufficiently large initial γ0 , search a feai (i = 1, 2, 3, 4), 1 , 2 ,
sible set (
P,
S,
R, X , Y , M
3 , L, T , W , H ) satisfying LMI in (39). Set k = 1,
γ = γ0 .
3. Solve the following LMI problem for the variables
(L, X , T , P, W ,
S, H,
R)
0
−η
P X −T ϕ(0) < tr(1 1 )
ϕ T (0)X −1 0
−d
0
ϕ T (v )X −1
S X −T ϕ(v )dv < tr(2 2 )
ϕ̇ T (v )X −1
RX −T ϕ̇(v )dvds < tr(3 3 ) (37)
s
Algorithm 2
Pk T + Tk P + Wk S
Min tr(Xk L + Lk X + Rk H + Hk R)
(41)
+
Sk W + For some constant γ > 0, assume
tr(1 1 ) + tr(2 2 ) + tr(3 3 ) < γ
(38)
Therefore, we can construct a feasibility problem as
follows:
Given η and λi (i = 2, 3, 4),
i
Find P,
S,
R, X , Y , M
(i = 1, 2, 3, 4), 1 , 2 , 3 , L, T , W , H
subject to : P > 0, S > 0, R > 0, T > 0, W > 0, H > 0
and (10), (11), (36), (38)
(39)
If the above problem has a solution, we can say
that there exists a feedback gain K = YX −T which
guarantees the cost function (9) with the cost less than
γ . But (36) includes nonlinear conditions. It cannot be
solved directly by using LMIs. However, using the idea
of cone complementary linearization algorithm in Lee
et al . (2001), Ghaoui et al . (1997), the above feasibility
problem can be solved iteratively. So we present the
following nonlinear minimization problem instead of
the original nonconvex minimization in (30)
Min tr(XL + PT + SW +
RH )
subject to : P > 0, S > 0, R > 0, T > 0, W > 0, H > 0
and (10), (11), (36), (38),
X I
P I
S I
> 0,
> 0,
> 0,
I L
I T
I W
R I
>0
I H
(40)
Although it is only a suboptimal solution of (33), it
is easier to solve than the original nonconvex minimization problem (33). We can easily find a suboptimal
minimum of the guaranteed cost based on the cone complementary linearization method (Ghaoui et al ., 1997).
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Obtain the suboptimal upper bound γ :
subject to LMI in (40)
Set Xk +1 = X , Lk +1 = L, Tk +1 = T , Wk +1 = W ,
P, Sk +1 = S, Rk +1 = R.
Hk +1 = H , P̃+1 = 4. If the (34) is satisfied, then return to Step 2 after
decreasing γ0 to some extent. If the (34) is not
satisfied within a specified number of iterations, then
exit. Otherwise, set k = k + 1 and go to Step 3.
5. Output suboptimal guaranteed cost γ and corresponding guaranteed cost feedback gain K = YX −T .
The above algorithm gives the controller feedback
gain K = YX −T under the constraint of guaranteed cost
γ , and corresponding suboptimal guaranteed cost γ . In
Section 5, we will illustrate the effectiveness of the
above algorithm. In the following section, we extend the
obtained stabilization conditions for uncertain systems
(8).
Considering the effect of the parameter uncertainties
A, A1 , and B , the following theorem provides the
condition for existence of the guaranteed cost controller
for the uncertain time-delay systems (8).
Theorem 2 For given scalars η and λi (i = 2, 3, 4),
1 + λ2 + λ3 + λ4 > 0, if there exist scalars εi > 0 (i =
i (i =
1, 2, 3), matrices P, S and R > 0, X , Y and M
1, 2, 3, 4) with appropriate dimension, if
11 11
<0
(42)
∗ 22
where
11 =

11 + ∗


∗


∗
∗
12
13
14
22 + λ2 23
24
33 + λ3 34
∗
44 + λ4 ∗
∗
∗
∗
∗
1 
ηM
2 
ηM
3 
ηM

4 
ηM
η
R
Asia-Pac. J. Chem. Eng. 2007; 2: 650–658
DOI: 10.1002/apj
655
656
C. PENG

12


=

Asia-Pacific Journal of Chemical Engineering
λXE1T
0
λXE2T
0
0
0
0
0
0
0
0
0
X
0
0
0
0
λY T E3T
0
0
0
0
YT
0
0

The example system


,

From Tian and Gao (1999), when choosing the secondorder plus delay model parameters (K , , ξ, d) =
(0.858 m · s −1 × 103 , 128.5 rad/s, 0.953, 15 ms), we
can convert the second-order plus delay model of the
thermoplastic injection molding system described in
Tian and Gao (1999) into the following to resemble
the state-space model:
λ = 1 + λ2 + λ3 + λ4
22 = diag{−λε1 I , −λε2 I , −λε3 I , −Q1−1 , −R1−1 },
= (ε1 + ε2 + ε3 )DD T .
Then the system (8) with the feedback gain K = YX −T
is asymptotically stable and the cost function J in
(9) satisfies the following bound:
J ≤ ϕ (0)X
T
+
0
−η
0
−1 PX
−T
ϕ(0) +
0
T
ϕ (t)X
−1 −T
SX
ϕ(t)dt
−d
ϕ̇ T (v )X −1
RX −T ϕ̇(v )dvds
(43)
∗

∗
∗
∗
(45)
y(t) = Cx (t)
(46)
0.1
−0.2449 −0.0165
, A1 =
0
1.000
0
1
0
B=
,C =
0
0.0142
A=
0
,
0.1
(47)
s
Proof: Replace A, A1 , and B with A + DF (t)E1 , A1 +
DF (t)E2 and B + DF (t)E3 in (21). Then, following a
similar procedure as in the proof of theorem 1, we have

where
ẋ (t) = Ax (t) + A1 x (t − d) + Bu(t)
12
∗
∗
∗
13
23
∗
∗
14
24
34
44 + λ4 ∗
1 
ηM
2 
ηM
3 
ηM
<0
4 
ηM
η
R
Then, when considering the uncertainty in system
identification, we have the following model based on
(45)–(46)
ẋ (t) = (A + A)x (t) + (A1 + A1 )x (t − d)
+ (B + B )u(t)
(44)
y(t) = Cx (t)
22 + λ2 + λε−1 XE2T E2 X T ,
=
2
33 + λ3 +
=
λε3−1 Y T E3T E3 Y
D=
E2 =
+ Y T R1 Y .
Using Schur complement, we can obtain (42) from (44)
for λ > 0. This completes the proof.
Remark 4 To obtain the suboptimal guaranteed cost
γ of the system (8) with parameter uncertainties, we
can construct the same nonlinear minimization problem
as (30), and the corresponding solution is similar to
Algorithm 2; it is omitted here.
NUMERICAL EXAMPLES
In this section, we consider networked injection velocity
control for Thermoplastic Injection Molding taken from
(Tian and Gao, 1999). Our objective is to control
the injection velocity over Ethernet-based networks
to guarantee the given performance index γ defined
in (9).
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
(49)
where uncertain matrices A, A1 , and B with form
(3) and the following parameters:
where
11 + + λε−1 XE1T E1 X T + XQ1 X T ,
=
1
(48)
0.1 0
0.01
0
, E1 =
0 0
0
0.01
T
0.1 0
, E3 = [ 0 0 0 ]T .
0.1 0
0.01
0
0
0
0
0
T
,
(50)
Assume that the initial conditions are x1 (t) = 0.5e t+1
and x2 (t) = −0.5e t+1 for t ∈ [−d, 0], the system delay
is included in state delay x (t − d), the network-induced
delay will be considered in the controller design.
Controller design with stability
First, we consider the stabilization problem to derive
the SAEDB η. When the controller is implemented
through a network, the problem can be written in the
form of (8). Suppose that the full state variables are
available for measurement. Applying Algorithm 1 with
λ2 = −0.2, λ3 = 1.9, λ4 = 7.8, we can find that the
SAEDB that guarantees the stability of system (48)
is 3.8 ms and the corresponding feedback gain K =
[−0.0754 − 0.0261]. It means that, if h = 1 ms and the
data packet dropout can be neglected in the transmission, the maximum allowable network-induced delay
τik ≤ 2.8 ms. The designed controller can stabilize the
Asia-Pac. J. Chem. Eng. 2007; 2: 650–658
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
COST CONTROL FOR A CLASS OF INDUSTRIAL PROCESSES
system (6) as long as the upper bound of the networkinduced delay is less than 2.8 ms.
50
control point
set point r
Controller design with guaranteed cost
Now we consider the above system with guaranteed
cost under networked control. According to algorithm 2,
Table 1 shows the obtained suboptimal guaranteed cost
and corresponding controller feedback gain with η =
2.8, Q1 = 0.1I2×2 and R1 = 0.1. It shows that under
the constraint of guaranteed cost performance index γ ,
the designed controller can stabilize system (6).
The state responses of system (48) with controller
feedback gain K = [−0.2565 −0.0728] are shown
in Fig. 1. Figure 1 also shows that the system is
asymptotically stable with given controller.
Given injection velocity set-point as solid line in
Fig. 2, the tracing curve based on controller feedback
gain K = [−0.2565 −0.0728] is shown as dotted line
in Fig. 2. It can been seen that the method developed
in this paper can trace the varying injection velocity.
Furthermore, compared to the keeping time of injection
velocity(about 0.8 s), the adjust time of our method
(about 0.2 s) is less.
Table 1. Suboptimal guaranteed cost and
corresponding controller feedback gain.
γ
Number of
iterations
State feedback gain, K
50
45
40
34
36
46
[−0.2571 −0.0713]
[−0.2575 −0.0720]
[−0.2565 −0.0728]
2
x1
x2
1
Ram Velocity r* 103 (m/s)
40
30
20
10
0
10
0
0.5
1
1.5
2
Simulation Time t (s)
2.5
3
Figure 2.
Trace of injection velocity set-points with
K = [−0.2565 −0.0728].
CONCLUSION
When industrial processes with delay are controlled
over communication networks, the time-varying characteristics of the nonideal network conditions cannot be
neglected, which not only degrade control performance
but also introduce distortion of the controller signal. In
this paper, we have presented a solution to the delaydependent guaranteed cost control problem via a memoryless state feedback for a class of uncertain state-delay
industrial systems interconnected over data network.
First, an NCS model is presented to include nonideal
network conditions, such as data packet dropout. Then,
the guaranteed cost control for nominal and uncertain
systems with state delay has been introduced, and a cone
complementary linearization algorithm is presented to
solve the nonconvex problem of NCS controller design
and to obtain a suboptimal upper bound of the cost.
Numerical examples have been given to demonstrate
the effectiveness of the proposed method.
0
Acknowledgment
State
−1
This work is partially supported by Australian Research
Council (ARC) under the Discovery Projects Grant
Scheme (grant ID: DP0559111), Natural Science Foundation of China (grant ID: 60474079), and Natural Science Foundation of Jiangsu Province of China (grant
ID: BK2006573).
−2
−3
−4
−5
−6
0
0.1
0.2
0.3
0.4
Simulation Time t (s)
0.5
0.6
Figure 1. Simulation of system (9) with γ = 40, K =
[−0.2565 − 0.0728].
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
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