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Improved stabilization method for networked control systems with variable transmission delays and packet dropout
ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Research Article
Improved stabilization method for networked control systems with variable transmission delays and packet dropout Arash Farnam, Reza Mahboobi Esfanjani n Electrical Engineering Department, Sahand University of Technology, Tabriz, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 29 April 2012 Received in revised form 17 March 2014 Accepted 8 May 2014 This paper was recommended for publication by Prof. Y. Chen
This paper investigates the problem of stability analysis and stabilization for networked control systems with the network-induced delay and data dropout. In order to obtain less conservative results, a novel augmented Lyapunov–Krasovskii functional is introduced and new free-weighting matrices are employed to make some extra degrees of freedom in the sufficient stabilizability condition. The gain of the memoryless state-feedback controller is computed by solving a set of linear matrix inequalities (LMIs). Illustrative examples are given to verify the applicability and outperformance of the proposed method compared to the existing approaches in the literature. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Networked control systems Stabilization Lyapunov–Krasovskii theorem Linear matrix inequality (LMI)
1. Introduction Networked control system (NCS) is a feedback control structure wherein the control loop is closed via a communication network. NCS integrates communication network with control system to attain low cost, simple installation, easy maintenance and high flexibility. However, NCSs' challenges such as network-induced data transmission delay and dropout in communication between the loop components bring hard to solve problems in this scheme compared with traditional point-to-point control architectures. These issues motivated a lot of researches in this field during the recent years [1–7]. The Lyapunov–Krasovskii theorem is a common tool for stability analysis and stabilizing controller synthesis for time-delay systems [8–12] and consequently for NCSs. In [5], the basics of the statefeedback controller design methods using Lyapunov–Krasovskii functional (LKF) are presented. Jensen's inequality was employed in [8] to derive simple conditions for determination of controller gain. In spite of a straightforward design procedure, the obtained results are considerably conservative. In order to attain less conservative results, a free-weighting matrices technique was used in [6] and [7] for stabilization of NCSs. The method of [6] was improved in [13] by using the augmented LKF and further enhanced in [14] with new free-weighting matrices. The utilized LKF in the most of the papers n
Corresponding author. E-mail addresses: [email protected] (A. Farnam), [email protected] (R. Mahboobi Esfanjani).
involve only double-integral terms of state rate, [5–7] and [13–16]. In [11], a triple-integral term was included in LKF to obtain less conservative results in stability analysis of time-varying delay systems. The benefits of the existence of the triple-integral terms in LKF were discussed thoroughly in [12]. In this paper, an efficient approach is developed to design a static state-feedback stabilizing controller for NCSs. Variable delay is considered in the system model to represent both of the data packet latency and dropout in NCSs. New augmented LKF and novel free-weighting matrices are employed to derive less conservative delay-dependent sufficient condition in terms of linear matrix inequalities. Unlike all the existing methods for stabilization of NCSs which utilize the LKFs with double-integral terms, the introduced LKF includes triple-integral phrases. Moreover, for the first time, double-integral terms of the state rate are involved in the augmented vector of the energy functional to reduce the conservativeness of the design scheme. Illustrative examples are presented to demonstrate that the resulting maximum allowable delay to retain the stability of closed-loop system with the proposed controller is higher than the values obtained with the methods in the literature. This paper is organized as follows: in Section 2, NCSs are modeled as a continuous time-delay system. Sufficient conditions for the stability analysis and state feedback control design of NCSs are introduced in Section 3. In Section 4, two numerical benchmark examples and a practical system are resolved by the developed scheme to illustrate the effectiveness of the proposed approach. Section 5 concludes the paper.
http://dx.doi.org/10.1016/j.isatra.2014.05.006 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i
A. Farnam, R. Mahboobi Esfanjani / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
Notation: in this paper, n denotes symmetric block in the symmetric matrices. I is identity matrix of appropriate dimensions. The notation P 4 0 (P Z 0Þ means that P is real symmetric positive definite (positive semidefinite). The superscript T stands for matrix transposition. col ½ Represents the column vector of the elements in the bracket. diag½ symbolizes a diagonal matrix composed of the matrices listed in the bracket.
2. System description and preliminaries The controlled system is described as follows: x_ ðtÞ ¼ A xðtÞ þ B uðtÞ
ð1Þ
where xðtÞϵR and uðtÞϵR are the state vector and the control input vector, respectively; A and B are known system matrices with appropriate dimensions. It is assumed that the pair (A, B) is completely controllable. The considered NCS structure is shown in Fig. 1, where the controller and the actuator are event-driven and sampler is clockdriven. The sampling period is assumed to be h, where h is a positive constant. The transmission delays may not be necessarily integer multiplies of the sampling period; so, Zero order hold (ZOH) device's information may be updated between sampling instants. Since the controller is a constant gain, the feedback and forward delays are combined together at each sampling time. The updating instants of ZOH are t k and the successfully transmitted signals from the sampler to the controller and from the controller to ZOH at the instant t k experience delay ηk ¼ ηsck þ ηcak , where ηsck and ηcak are the time-delays from the sampler to the controller and from the controller to the ZOH at the updating instant t k , respectively. Therefore, the state feedback controller with considering the behavior of the ZOH takes the following form: n
m
uðt k Þ ¼ K xðt k ηk Þ;
tk r t o tk þ 1
ð2Þ
in which t k þ 1 is the next updating instant after t k . The networkinduced delay, ηk is bounded as the following inequality:
ηm r ηk r ηM
ð3Þ
where ηm and ηM are the lower and upper bounds of the networkinduced delay, respectively. Then, the closed-loop system in Fig. 1 is described by: x_ ðtÞ ¼ AxðtÞ þBKxðt k ηk Þ; t k rt o t k þ 1
ð4Þ
On the other hand, at the updating instant t k , the number of accumulated data packet dropouts since the last updating instant t k 1 is denoted by δk , with 0 r δk r δM , where δM stands for the maximum number of accumulated data packet dropouts. Combining the above-mentioned facts yields to: t k þ 1 t k ¼ ηk þ 1 ηk þ ðδk þ 1 þ1Þh
Z.O.H
Actuator and Plant
ð5Þ
If ηðtÞ ¼ t t k þ ηk is replaced in (4), then the following continuous time model is obtained for the closed-loop NCS in Fig. 1: x_ ðtÞ ¼ A xðtÞ þ BK xðt ηðtÞÞ which is in the form of a continuous time-delay system, with:
ηm r ηðtÞ r η
3. Main results In this section, a new delay-dependent stability criterion is derived in Theorem 1 to ensure the asymptotic stability of the closed-loop system (6) for all delays satisfying (7). Afterwards, a simple condition is extracted in Theorem 2, to determine the controller gain in terms of linear matrix inequalities. Theorem 1. For the given ηm , η and K, the closed-loop system (6) is asymptotically stable if there exist matrices N 0 , N 1 , N 2 , M, L0 , L1 , L2 , R, S, F and symmetric matrices P ¼ ½P ij 55 4 0, Q 1 ¼ ½Q 1ij 22 4 0, Q 2 ¼ ½Q 2ij 22 4 0, T 1 ¼ ½T 1ij 22 4 0, T 2 ¼ ½T 2ij 22 4 0, Z 1 4 0, Z 2 4 0,U 1 ; U 2 , V 0 ¼ ½V 0ij 77 , V 1 ¼ ½V 1ij 77 , V 2 ¼ ½V 2ij 77 , W 0 ¼ ½W 0ij 77 , W 1 ¼ ½W 1ij 77 , W 2 ¼ ½W 2ij 77 , X 0 , X 1 and X 2 with appropriate dimensions, satisfying the matrix inequalities (8)–(15): 2 3 P R S 6n U F 7 ð8Þ 4 540 1 n n U2
Ω1 ¼ π 1 þ π 2i þ π T2i þ π 3 þ π T3 þ ηm V 0 þ η^ V i þ for i ¼ 1; 2 2 V 0 L0 þ ψ 0 T 111 Ω2 ¼ 6 4 n n
2
V1 Ω3 ¼ 6 4 n n
2
V2 Ω4 ¼ 6 4 n "
Ω5 ¼ "
Ω6 ¼ "
Fig. 1. Considered networked control system.
ð7Þ
wherein η ¼ ηM þ ðδM þ 1Þh. It is obvious that η is specified by the maximum number of accumulated data packet dropouts, δM , the upper bound of network-induced delay, ηM and the sampling period, h. The aim of this paper is to develop an improved procedure to determine state-feedback gain K in (2), such that the closed-loop system described by (6) is asymptotically stable for all delay satisfying (7).
Sampler
Static Controller
ð6Þ
Ω7 ¼
N2
T 211
Z1
W1
L 1 þ φ1
n
Z2
ð11Þ
T 222
L2 þ ψ 1
n
ð10Þ
3
T 212 þ X 1 7 5 Z0
n
L 0 þ φ0
n
N1
T 211
W0
W 0 þ η2 W i o 0
T 122
L1 þ ψ 1
n
2
3
T 112 þ X 0 7 5Z0
n
n
W2
N0
η2m
3
T 212 þ X 2 7 5Z0 T 222
ð12Þ
Z0
ð13Þ
Z0
ð14Þ
#
#
# L 2 þ φ1 Z0 Z2
ð15Þ
where η^ ¼ η ηm , η ¼ 12ðη2 η2m Þ and
Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i
A. Farnam, R. Mahboobi Esfanjani / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
"
2
#
3
Z
Λ11 Λ12 Λ13 Λ14 Λ22 Λ23 Λ24 7 7 7 n Λ33 Λ34 5
V 2 ðxt Þ ¼
6 n Σ1 ¼ 6 6 4 n
Σ1 Σ2 π1 ¼ ; n Σ3
n
n
n
t
τT ðαÞQ 1 τðαÞ dα þ
t ηm
Z
44
V 3 ðxt Þ ¼
Z
0
3
ηm
t
Z
t ηm
τT ðαÞT 1 τðαÞ dα dβ þ
t þβ
τT ðαÞQ 2 τðαÞ dα
tη
Z
ηm
Z
η
t tþβ
τT ðαÞT 2 τðαÞ dα dβ ð19Þ
Λ11 ¼ P 14 R1 þ P T14 RT1 þ Q 111 þ ηm T 111 þ η^ T 211 þ X 0 Z
Λ12 ¼ P 14 þ P 15 þ R1 S1 þ P T24 RT2
V 4 ðxt Þ ¼
Z
0
þ
η
V 5 ðxt Þ ¼ 2ξ ðtÞR T
þ Q 211 X 0 þ X 1
η
2 R0 V 6 ðxt Þ ¼ 4 R
Λ24 ¼ Λ33 ¼
P 35 þ S3 P T35 þ ST3 Q 211
η^ S2
η2m
2
P 12 6 6 P 22 Q 112 þ Q 212 Σ2 ¼ 6 6 P T23 4
Σ3 ¼ 6 4
0
3
P 33 Q 212
0
0
n
Q 222
0
n
n
X1 þ X2
"
0
0
N0 þ N1
M
0
ηm
t tþβ
T x_ ðαÞ dα dβ þ 2ξ ðtÞS
ηm
ηm η
Rt
tþβ
x_ ðαÞ dα dβ
tþβ
x_ ðαÞ dα dβ
Rt
3T " 5
Z
0
ð20Þ
ηm η
Z
t
t þβ
x_ ðαÞ dα dβ
3 #2 R 0 R t _ η t þ β xðαÞ dα dβ 4R m R 5 ηm t U2 _ η t þ β xðαÞ dα dβ
U1
F
n
T V_ 1 ðxt Þ ¼ 2ξ ðtÞP ξ_ ðtÞ
V_ 2 ðxt Þ ¼ τT ðtÞ Q 1 τðtÞ τT ðt ηm ÞQ 1 τðt ηm Þ þ τT ðt ηm Þ Q 2 τðt ηm Þ τT ðt ηÞQ 2 τðt ηÞ
ð24Þ
V_ 3 ðxt Þ ¼ τT ðtÞðηm T 1 þ η^ T 2 Þ τðtÞ
7 5
N2
MBk
ð22Þ R t ηm
ð23Þ
3
Q 122 þ Q 222
π 3 ¼ MA
Z
0
x_ T ðαÞZ 2 x_ ðαÞ dα dθ dβ
where in, ξðtÞ ¼ col½ xðtÞ xðt ηm Þ xðt ηÞ t η xðαÞ dα t η m xðαÞ dα and τðαÞ ¼ col ½ xðαÞ x_ ðαÞ . The relation (8) guarantees that the LKF V in (16) is a positive definite functional. The time derivative of Vðxt Þ along the trajectories of (6) is obtained as follows:
7 07 7 07 5 0
P 23 0
π 2i ¼ N0 þ ηm L0 þ η^ Li
Z
t þθ
Rt
Z 1 þ ηZ 2
P 13
0 2
2
β
X2
Λ34 ¼ P T13 þ ηm R3 þ η^ S3 Λ44 ¼ Q 122 þ ηm T 122 þ η^ T 222 þ
x_ T ðαÞZ 1 x_ ðαÞdαdθdβ
ð21Þ
Λ23 ¼ P 25 þ S2 P T34 þ P T35 þRT3 ST3 P T12 þ m R2 þ
t
tþθ β Z ηm Z 0 Z t
Λ14 ¼ P 11 þ ηm R1 þ η^ S1 þ Q 112 þ ηm T 112 þ η^ T 212 Λ22 ¼
0Z
ηm
Λ13 ¼ P 15 þ S1 þ P T34 RT3
P 24 þ P 25 þ R2 S2 P T24 þ P T25 þ RT2 ST2 Q 111
ð18Þ
Z 0
0
0
N1 þ N2
#
t ηm t ηðtÞ
Z
t
t ηm
τT ðαÞT 2 τðαÞ dα
τT ðαÞT 1 τðαÞ dα Z
t ηðtÞ t η
τT ðαÞT 2 τðαÞ dα
ð25Þ
2 Z 0 Z t η x_ T ðαÞZ 1 x_ ðαÞ dα dβ V_ 4 ðxt Þ ¼ x_ T ðtÞ m Z 1 þ ηZ 2 x_ ðtÞ 2 ηm t þ β Z ηm Z t Z ηðtÞ Z t x_ T ðαÞZ 2 x_ ðαÞ dα dβ x_ T ðαÞZ 2 x_ ðαÞ dα dβ
ηðtÞ
tþβ
η
tþβ
ð26Þ
T T ψ 0 ¼ P 44 þR4 P 44 P 45 R4 þ S4
T T T ψ 1 ¼ P 45 þR5 P 45 P 55 R5 þ S5
T T T T T φ0 ¼ R4 þ U 1 R4 R5 U 1 þF
T T T T T T φ1 ¼ S4 þ F S4 S5 F þU 2
P 45 S4
P 55 T S5
P T44 ηm R4 η^ S4 P T15 ηm R5 η^ S5
RT5 F
RT1 ηm U T1 η^ F
ST5 U T2
ST1 ηm F T η^ U T2
RT2 ST2
Proof. Define the LKF as follows: Vðxt Þ ¼ V 1 ðxt Þ þ V 2 ðxt Þ þ V 3 ðxt Þ þV 4 ðxt Þ þ V 5 ðxt Þ þ V 6 ðxt Þ
ð16Þ
P T24
P T25
P T34
P T35
RT3
0
ST3
0
T
0
0
T
T
T
T V_ 5 ðxt Þ ¼ 2ξ_ ðtÞR
Z
0 ηm
Z
t
t þβ
T x_ ðαÞ dα dβ þ 2ξ ðtÞRðηm x_ ðtÞ
xðtÞ þ xðt ηm ÞÞ þ 2ξ_ ðtÞSð T
V 1 ðxt Þ ¼ ξ ðtÞP ξðtÞ T
ð17Þ
Z
ηm ηðt Þ
Z
t
tþβ
x_ ðαÞ dα dβ
Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i
A. Farnam, R. Mahboobi Esfanjani / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
Z þ
ηðt Þ
Z
η
t tþβ
ε16 ðtÞ ¼ xT ðt ηðtÞÞX 2 xðt ηðtÞÞ xT ðt ηÞX 2 xðt ηÞ
T x_ ðαÞ dα dβÞ þ 2ξ ðtÞSðη^ x_ ðtÞ
Z
xðt ηm Þ þ xðt ηÞÞ " V_ 6 ðxt Þ ¼
ð27Þ
ηm x_ ðtÞ xðtÞ þ xðt ηm Þ η^ x_ ðtÞ xðt ηm Þ þ xðt ηÞ
#T "
U1
R0 Rt
ε2 ðtÞ ¼ 2ζ ðtÞN 1 xðt ηm Þ xðt ηðtÞÞ ε3 ðtÞ ¼ 2ζ T ðtÞN 2 xðt ηðtÞÞ xðt ηÞ
t ηm t ηðtÞ
Z
t ηðtÞ tη
! x_ ðαÞ dα
¼0
¼0
ε4 ðtÞ ¼ 2ζ ðtÞMðx_ ðtÞ AxðtÞ Bkxðt ηðtÞÞÞ ¼ 0 Z
t
t ηm
Z
xðαÞ dα
ð30Þ
ð31Þ ð32Þ
Z
0 ηm
t
x_ ðαÞ dα dβ Þ ¼ 0
t þβ
ð33Þ Z
ε6 ðtÞ ¼ 2ζ T ðtÞL1 ððηðtÞ ηm ÞxðtÞ
t ηm
t ηðtÞ
xðαÞ dα
Z
ηm
Z
ηðtÞ
t
t þβ
ε7 ðtÞ ¼ 2ζ T ðtÞL2 ððη ηðtÞÞxðtÞ
t ηðtÞ
tη
xðαÞ dα
ηðtÞ Z
Z
η
t
tþβ
Z
t t ηm
ζ T ðtÞV 0 ζ ðtÞ dα ¼ 0
ε9 ðtÞ ¼ ðηðtÞ ηm Þζ T ðtÞV 1 ζ ðtÞ ε10 ðtÞ ¼ ðη ηðtÞÞζ ðtÞV 2 ζ ðtÞ T
ε11 ðtÞ ¼
η2m 2
ζ ðtÞW 0 ζ ðtÞ T
Z
0 ηm
Z
t ηm t ηðtÞ
Z
t ηðtÞ t η
Z
ζ T ðtÞV 1 ζ ðtÞ dα ¼ 0 ζ ðtÞV 2 ζ ðtÞ dα ¼ 0 T
ζ ðtÞW 0 ζ ðtÞ dα dβ ¼ 0 T
tþβ
Z
ð36Þ
ð37Þ
ð38Þ
Z
ηm ηðtÞ
t t þβ
ð39Þ
ζ ðtÞW 1 ζ ðtÞ dα dβ ¼ 0 T
ð40Þ
ε13 ðtÞ ¼
ðη2 η2 ðtÞÞ T ζ ðtÞW 2 ζ ðtÞ 2
Z
ηðtÞ
Z
η
t
tþβ
ζ T ðtÞW 2 ζ ðtÞ dα dβ ¼ 0 ð41Þ
ε14 ðtÞ ¼ xT ðtÞX 0 xðtÞ xT ðt ηm ÞX 0 xðt ηm Þ 2
Z
t
t ηm
x_ T ðαÞX 0 xðαÞ dα ¼ 0
ð42Þ
ε15 ðtÞ ¼ xT ðt ηm ÞX 1 xðt ηm Þ xT ðt ηðtÞÞX 1 xðt ηðtÞÞ Z
2
t ηm
t ηðtÞ
ð46Þ
i¼2
where,
Ω1 ðtÞ ¼ π 1 þ π 2 ðtÞ þ π T2 ðtÞ þ π 3 þ π T3 þ ηm V 0 þ ðηðtÞ ηm ÞV 1 þ ðη ηðtÞÞV 2 η2m ðη2 ðtÞ η2m Þ ðη2 η2 ðtÞÞ 2
W0 þ
W1 þ
2
3 2 ζ ðtÞ T V 0 6 xðαÞ 7 6 n Ω2 ðtÞ ¼ 4 5 4 t ηm n x_ ðαÞ Z
Ω3 ðtÞ ¼
Ω4 ðtÞ ¼
Ω5 ðtÞ ¼ Ω6 ðtÞ ¼ Ω7 ðtÞ ¼
Z
2
ηm
32
3
2
3 2
32
3
"
t
Z
ηðtÞ η
N2 ζ ðtÞ T V 2 L2 þ ψ 1 ζ ðtÞ 6 xðαÞ 7 6 n 76 xðαÞ 7 T T þ X 211 212 2 54 4 5 4 5 dα n n T 222 x_ ðαÞ x_ ðαÞ
t þβ
ηðtÞ
Z
n
3 2
Z
ηm
Z
T 111
2
t ηðtÞ
0
32
N0
N1 ζ ðtÞ T V 1 L1 þ ψ 1 ζ ðtÞ 6 xðαÞ 7 6 n 6 7 T 211 T 212 þ X 1 7 4 5 4 54 xðαÞ 5 dα _xðαÞ _ n n T 222 x ðα Þ
t ηm
t η
Z
W2
3 ζ ðtÞ 6 7 T 112 þ X 0 54 xðαÞ 7 5 dα T 122 x_ ðαÞ
L0 þ ψ 0
t
t ηðtÞ
Z
2
ζ ðtÞ x_ ðαÞ
"
t t þβ
Z
t
ζ ðtÞ x_ ðαÞ
"
tþβ
#T "
W0
L0 þ φ0
n
Z1
#T "
ζ ðtÞ x_ ðαÞ
#"
#"
W1
L1 þ φ1
n
Z2
W2
L 2 þ φ1
n
Z2
#T "
#
ζ ðtÞ dα dβ x_ ðαÞ #
ζ ðtÞ dα dβ x_ ðαÞ
#"
#
ζ ðtÞ dα dβ x_ ðαÞ
and
π 2 ðtÞ ¼ ½N 0 þ ηm L0 þ ðηðtÞ ηm ÞL1 þ ðη ηðtÞÞL2
t
ðη2 ðtÞ η2m Þ T ε12 ðtÞ ¼ ζ ðtÞW 1 ζ ðtÞ 2
i¼7
_ t Þ ¼ ζ T ðtÞΩ1 ðtÞ ζ ðtÞ þ ∑ Ωi ðtÞ Vðx
x_ ðαÞ dα dβ Þ ¼ 0
ð35Þ
ε8 ðtÞ ¼ ηm ζ T ðtÞV 0 ζ ðtÞ
The V_ in (45) can be rewritten as the following:
x_ ðαÞ dα dβ Þ ¼ 0
ð34Þ Z
i¼1
þ
T
ε5 ðtÞ ¼ 2ζ T ðtÞL0 ðηm xðtÞ
_ t Þ ¼ V_ 1 ðxt Þ þ V_ 2 ðxt Þ þ V_ 3 ðxt Þ þ V_ 4 ðxt Þ þ V_ 5 ðxt Þ þ V_ 6 ðxt Þ þ ∑ εi ðtÞ Vðx
! x_ ðαÞ dα
ð44Þ
ð45Þ
t ηm
Z
x_ T ðαÞX 2 xðαÞ dα ¼ 0
tη
i ¼ 16
ð28Þ
On the other hand, for any matrices N 0 , N 1 , N 2 , M, L0 , L1 and L2 and symmetric matrices V 0 , V 1 , V 2 , W 0 , W 1 , W 2 , X 0 , X 1 and X 2 with appropriate dimensions, the following equalities hold: Z t x_ ðαÞ dαÞ ¼ 0 ε1 ðtÞ ¼ 2ζ T ðtÞN 0 ðxðtÞ xðt ηm Þ ð29Þ
T
t ηðtÞ
where εðtÞ ¼ col½ xðtÞ xðt ηm Þ xðt ηÞ x_ ðtÞ x_ ðt ηm Þ x_ ðt ηÞ xðt ηðtÞÞ. Now, regarding (23)–(28) and (29)–(44), V_ can be stated as follows:
#
F
n U2 3 _ – η m t þ β x ðα Þ dα dβ 4 5 Rη R ηðtÞ R t Rt m _ _ ηðt Þ t þ β xðαÞ dα dβ þ η t þ β x ðα Þ dα dβ
2
2
x_ T ðαÞX 1 xðαÞ dα ¼ 0
ð43Þ
N0 þ N1 N2
0
0
0
N 1 þ N 2 :
Provided Ω1 ðtÞ o 0 and Ωi Z 0; ði ¼ 2; …; 7Þ, the Lyapunov–Krasovskii theorem ensures the asymptotic stability of the system (6). Note that replacing ηm and η by the ηðtÞ in the condition Ω1 ðtÞ o 0 yields to the conditions of Eq. (9). It should be emphasized that regarding the (17), (21) and (22), the condition (8) is needed to assure the positive-definiteness of the introduced LKF in (16). □ Theorem 1 presents sufficient conditions for the stability analysis of the closed-loop control system (6) in the form of matrix inequalities. For the given controller gain K, this theorem can be used to determine the maximum value of allowable delay η in (7) which retain the stability of the system (6). In Theorem 2, utilizing the variable changing technique, the bilinear conditions in Theorem 1 is modified to obtain equivalent linear matrix inequalities (LMIs) which are computationally more tractable to determine the controller gain K. The auxiliary scalar parameters appeared in the design conditions are tuned by trial and error to attain a feasible solution for the LMIs.
Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i
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Theorem 2. For the given constants ηm , η and scalars ρi 4 0 ði ¼ 2; …; 7Þ, the closed-loop system (6) is asymptotically
Λ~ 11 ¼ P~ 14 R~ 1 þ P~ 14 R~ 1 þ Q~ 111 þ ηm T~ 111 þ η^ T~ 211 þ X~ 0
stable with the controller gain K ¼ YX T if there exist a nonsingular ~ F~ ; Y and symmetric ~ S, matrix X, matrices N~ 0 , N~ 1 , N~ 2 , L~ 0 , L~ 1 , L~ 2 , R, ~ ~ ~ ~ ~ 40, Q ¼ ½Q~ 4 0, T~ 1 ¼ matrices P ¼ ½P ij , Q ¼ ½Q
Λ~ 12 ¼ P~ 14 þ P~ 15 þ R~ 1 S~ 1 þ P~ 24 R~ 2
½T~ 1ij 22 4 0, T~ 2 ¼ ½T~ 2ij 22 40,Z~ 1 4 0,Z~ 2 4 0, U~ 1 , U~ 2 , V~ 0 ¼ ½V~ 0ij 77 , ~ 0 ¼ ½W ~ 0 ,W ~ 1 ¼ ½W ~ 1ij , W ~ 2¼ V~ 1 ¼ ½V~ 1 , V~ 2 ¼ ½V~ 2 , W
Λ~ 13 ¼ P~ 15 þ S~ 1 þ P~ 34 R~ 3
~ 2ij , X~ 0 ,X~ 1 and X~ 2 with appropriate dimensions such that the ½W 77 following LMIs are satisfied (47–54): 2~ 3 P R~ S~ 6 7 ð47Þ 4 n U~ 1 F~ 5 4 0 ~ n n U2
Λ~ 14 ¼ P~ 11 þ ηm R~ 1 þ η^ S~ 1 þ Q~ 112 þ ηm T~ 112 þ η^ T~ 212
1
55
ij
77
ij
1ij 22
2
77
ij
6
~ ¼6 Ω 3 4
L~ 0 þ ψ~ 0 T~ 1
n
n
V~ 1 n
L~ 1 þ ψ~ 1 T~ 2
n
n
2
n
~ 0 W n
"
~ 1 W n
" ~ ¼ Ω 7
~ 2 W n
π~ 1 ¼
"
ψ~ 0 ¼ "
ψ~ 1 ¼
ð49Þ
T
T
Λ~ 34 ¼ P~ 13 þ ηm R~ 3 þ η^ S~ 3 ;
Λ~ 44 ¼ Q~ 122 þ ηm T~ 122 þ η^ T~ 222 þ P~ 13 P~ 23
P~ 12 6~ 6 P 22 Q~ 112 þ Q~ 212 Σ~ 2 ¼ 6 6 T 6 P~ 23 4 0
ð50Þ
2 6 Σ~ 3 ¼ 6 4
ð51Þ
Q~ 122 þ Q~ 222
T
T
2
P~ 33 Q~ 212 0
n
0 Q~
n
n
2
Z 1 þ ηZ~ 2
3 0 7 07 7 7 07 5 0 3
0 0
222
η2m ~
X~ 1 þ X~ 2
7 7 5
#
~0 L~ 0 þ φ Z0 Z~ 1
ð52Þ
# ~1 L~ 1 þ φ Z0 Z~ 2
ð53Þ
# ~1 L~ 2 þ φ Z0 Z~ 2
Σ~ 1 Σ~ 2 ; n Σ~ 3
P~ 44 þ R~ 4
P~ 45 þ R~ 5 T
T
φ~ 1 ¼ S~ 4 þ F~
T
T
2
AX T 6 6 ρ2 AX T 6 6 T 6 ρ3 AX 6 6 π~ 3 ¼ 6 ρ4 AX T 6 6 ρ AX T 5 6 6 6 ρ6 AX T 4 ρ7 AX T
3
Λ~ 11 Λ~ 12 Λ~ 13 Λ~ 14 6 7 6 n Λ~ 22 Λ~ 23 Λ~ 24 7 7 Σ~ 1 ¼ 6 6 n ~ ~ n Λ 33 Λ 34 7 4 5 n n n Λ~ 44
#
T
"
T P~ 44 P~ 45 R~ 4 þ S~ 4
T T P~ 45 P~ 55 R~ 5 þ S~ 5
T
T
T
T T T T S~ 4 S~ 5 F~ þ U~ 2
P~ 45 S~ 4
T P~ 55 S~ 5
T R~ 5 F~
T T S~ 5 U~ 2
P~ 24 T
T P~ 14 ηm R~ 4 η^ S~ 4
P~ 25 T
T P~ 15 ηm R~ 5 η^ S~ 5
T T R~ 1 ηm U~ 1 η^ F~
T T T S~ 1 ηm F~ η^ U~ 2
N~ 0 þ N~ 1
π~ 2i ¼ N~ 0 þ ηm L~ 0 þ η^ L~ i
ð54Þ
2
T
T
Λ~ 33 ¼ P~ 35 þ S~ 3 P~ 35 þ S~ 3 Q~ 211 X~ 2
T
3
φ~ 0 ¼ R~ 4 þ U~ 1 R~ 4 R~ 5 U~ 1 þ F~
T
Λ~ 24 ¼ P~ 12 þ ηm R~ 2 þ η^ S~ 2 ;
22
where, "
T
T
7 T~ 212 þ X~ 2 7 5Z0 T~ 2
11
n
~ 6¼ Ω
T
3
N~ 2
T
Λ~ 23 ¼ P~ 25 þ S~ 2 P~ 34 þ P~ 35 þ R~ 3 S~ 3
22
L~ 2 þ ψ~ 1 T~ 2
V~ 2 6 ~ 6 Ω4 ¼ 4 n
~ ¼ Ω 5
T
η ~ ~ i o0; W 0 þ η2 W
7 T~ 212 þ X~ 1 7 5Z0 T~ 2
11
T
2 m
7 T~ 112 þ X~ 0 7 5Z0 T~ 122 N~ 1
T
Λ~ 22 ¼ P~ 24 þ P~ 25 þ R~ 2 S~ 2 P~ 24 þ P~ 25 þ R~ 2 S~ 2 Q~ 111 þ Q~ 211 X~ 0 þ X~ 1
3
N~ 0
11
2
"
T
ð48Þ
V~ 0 6 ~ 6 Ω2 ¼ 4 n 2
T
77
8 i ¼ 1; 2 2
T
2ij 22
77
~ ¼ π~ þ π~ þ π~ T þ π~ þ π~ T þ η V~ þ η^ V~ þ Ω 1 1 3 2i i m 0 2i 3
T
R~ 3
R~ 2 T
T
S~ 2 T
0 0
0 0
XT ρ2 X T
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
ρ3 X T ρ4 X T ρ5 X T ρ6 X T ρ7 X T
0
0
T
T
0
0
0
N~ 1 þ N~ 2
#
3 BY 7 ρ2 BY 7 7 7 ρ3 BY 7 7 ρ4 BY 7 7 7 ρ5 BY 7 7 7 ρ6 BY 7 5 ρ7 BY
#T
P~ 34
P~ 35
N~ 2
0 #T 0
T 0
S~ 3 T
T 0
Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i
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6
Table 1 MADBs corresponding to the different design methods for Example 1 (with ηm ¼ 0). Method
[4]
[6]
[13]
[5]
[9]
[7]
[10]
[14]
[16]
[11]
Proposed method
MADB (in second)
0.7805
0.8695
0.8871
0.9410
0.9412
1.0081
1.0081
1.0213
1.0432
1.0629
1.0772
Fig. 2. Time response of systems' states and control effort for the Example 1.
and M 7 ¼ ρ7 M 0 . Feasibility of inequality (48) implies that M 0 is nonsingular. Let X ¼ M 0 1 , then pre- and neously the two sides of (9) with diag X (10)–(12) with diag X X X X X X with diag X X X X X X X X h i and its diag X X X X X X X inequalities Y ¼ KX T .
Fig. 3. Delay profile in the simulation of Example 1.
Table 2 MADBs resulting from different methods for Example 2. Method
MADB (in second)
[6] [7] [16]
0.5635 0.8597 0.9028
Proposed method
0.9685
(8)–(15)
leads
to
post-multiply simultaX X X , X X X X X X, (13)–(15) and (8) with transpose. Therefore
inequalities
(47)–(54),
with
□
Remark 1. Differently from the existing controller synthesis methods for NCSs which rely on the LKFs with double-integral terms, energy functional including triple-integral terms, V 4 ðxt Þ; is used to develop a more efficient stabilization scheme for the systems controlled over a network. Moreover, for the R0 Rt first time, new augmented variables η t þ β x_ ðαÞ dα dβ and R ηm R t m _ η t þ β xðαÞ dα dβ are involved in the introduced LKF which appear as the phrases V 5 ðxt Þ and V 6 ðxt Þ in (16). Remark 2. In order to increase the degree of freedom in the sufficient design condition, new free-weighting matrices are employed in (42)–(44). Unlike the common approach for injection of free-weighting matrices which are based on the following relation: Z t ηm T ðη ηm Þζ ðtÞV ζ ðtÞ ζ T ðtÞV ζ ðtÞ dα ¼ 0 t η
with any matrix V with appropriate dimension, we utilize the below relations to obtain less conservative results: ðηðtÞ ηm Þζ ðtÞV 1 ζ ðtÞ T
ðη ηðtÞÞζ ðtÞV 2 ζ ðtÞ T
Fig. 4. Networked control of the mobile robot for reference tracking in Example 3.
Proof. Let M ¼ ½ M 1 M 2 M 3 M 4 M 5 M 6 M 7 T . Replace M 1 ¼ M 0 ; M 2 ¼ ρ2 M 0 ; M 3 ¼ ρ3 M 0 ; M 4 ¼ ρ4 M 0 ; M 5 ¼ ρ5 M 0 ; M 6 ¼ ρ6 M 0
Z
Z
t ηm t ηðtÞ
t ηðtÞ tη
ζ T ðtÞV 1 ζ ðtÞ dα ¼ 0
ζ T ðtÞV 2 ζ ðtÞ dα ¼ 0
wherein V 1 and V 2 are arbitrary matrices with appropriate dimensions. It should be noted that the similar technique is applied to handle double-integral terms.
Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i
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Fig. 5. Simulation results for the tracking problem of Example 3.
4. Illustrative examples Two numerical benchmark examples and a practical system are presented to compare the results of the proposed method with existing approaches in the literature. The YALMIP Toolbox is utilized to solve the LMI feasibility problems [17]. Example 1. Consider the following system from [6]: 0 1 0 x_ ðtÞ ¼ xðtÞ þ uðtÞ 0 0:1 0:1
ð67Þ
which is controlled over a network. By the suggested method in Theorem 2, the controller gain is obtained as K ¼ 3:75 11:5 which leads to the Maximum Allowable Delay Bound (MADB) equals to 1.0772 s with ηm ¼ 0. In Table 1, the MADBs corresponding to the rival design methods are compared. Table 1 verifies the superiority of the proposed method compared to some existing methods in the literature. Time-delayed system (6) which describes considered NCS with the designed controller is simulated by Simulinks of Matlabs. Fig. 2 depicts the time response of the systems' state and control effort starting form initial state [0.1, 0.1] where varying delay profile is shown in Fig. 3. Example 2. Consider the following system as in (6), [16]: 0:5 1 0:5 2 x_ ðtÞ ¼ xðtÞ þ xðt ηðtÞÞ 0 0:6 1 1 To validate the less conservativeness of the proposed approach, the Theorem 1 which is the cornerstone of the proposed design procedure, is utilized for stability analysis of the closed-loop system. Table 2 shows the maximum allowable delay bound η, in the case ηm ¼ 0, obtained from the developed and some rival methods. As seen, the admissible upper bound of delay obtained by the method of this paper is larger than the existing ones; so the superiority of the suggested scheme is confirmed. Example 3. Dynamical model of a practical wheeled mobile robot which is controlled over a network is considered as follows [18]: 2 3 2 3 0 0 0 1 0 0 60 6 0:381 7 0 0:0175 0:01 7 0 6 7 6 7 x_ ðtÞ ¼ 6 7xðtÞ þ 6 7 uðtÞ 4 0 0:0175 4 0 0:319 5 0 0 5 0
0
0
0
0
3:542
The overall time-delay ηðtÞ in the network varies from 0.10 s to 0.24 s. Utilizing Theorem 2, by choosing ρ2 ¼ 0:01, ρ3 ¼ 0:01, ρ4 ¼ 146, ρ5 ¼ 0:01; ρ6 ¼ 0:01 and ρ7 ¼ 0:01, the stabilizing controller gain is obtained as follows: " # 0:0231 5:5152 0:0796 0:0032 K¼ 0:0016 2:5655 6:7624 0:0386 In a real scenario, the mobile robot tracks the desired reference. So, the closed-loop system is configured as shown in Fig. 4,
Fig. 6. Delay profile in the simulation of Example 3.
wherein Hð0Þ is defined as the following: Hð0Þ ¼
Pð0Þ 1 þ Pð0ÞK
with PðsÞ ¼ ðsI AÞ 1 B. Fig. 5 shows the control effort and robot states signals which track satisfactorily the reference trajectory xd ¼ ½3; 3; 0; 0T , starting from the initial condition xð0Þ ¼ ½4; 1; 0:1; 0:1 with the delay profile shown in Fig. 6.
5. Conclusion This paper has proposed an approach to synthesise stabilizing state feedback controller for the linear time invariant system which is controlled via communication network. Data packet loss and latency arising in NCS has been incorporated in the system model as a time-varying delay. In the suggested method, stabilizing state feedback controller is computed via the feasible solution of a set of linear matrix inequalities. New augmented LKF and novel free-weighting matrices has been introduced to make extra degrees of freedom in the design condition which leads to the less conservative results. The advantages of the developed method were demonstrated by illustrative examples. References [1] Hespanha JP, Naghshtabrizi P, Xu Y. A survey of recent results in networked control systems. Proc IEEE 2007;95:138–62. [2] Yu M, Wang L, Chu T, Xie G. Modeling and control of networked systems via jump systems approach. IET Control Theory Appl 2008;2:535–41.
Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i
8
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[3] Yu M, Wang L, Chu T, Hao F. Stabilization of networked control systems with data packet dropout and transmission delay: continues-time case. Eur J Control 2005;11:40–9. [4] Kim DS, Lee YS, Kwan WH, Park HS. Maximum allowable delay bound of networked control systems. Control Eng Pract 2003;11:1301–13. [5] Peng C, Tian YC, Tade M. State feedback controller design of networked control systems with interval time-varying delay and nonlinearity. Int J Robust Nonlinear Control 2008;18:1285–301. [6] Yue D, Han QL, Peng C. State feedback controller design of networked control systems. IEEE Trans Circuits Syst-II: Express Briefs 2004;11:640–4. [7] Tang B, Liu GP, Gui WH. Improvement of state feedback controller design for networked control systems. IEEE Trans Circuits Syst-II: Express Briefs 2008;55:464–8. [8] Gu L, Kharytonov VL, Chen J. Stability of time-delay systems. Brikhauser 2003. [9] Jiang X, Han QL. Delay-dependent robust stability for uncertain linear systems with interval time-varying delay. Automatica 2006;42:1059–65. [10] He Y, Wang QG, Xie L, Lin C. Further improvement of free weighting matrices technique for systems with time-varying delay. IEEE Trans Autom Control 2007;52:293–9.
[11] Sun J, Liu GP, Chen J, Rees D. Improved stability criteria for linear systems with time-varying delay. IET Control Theory Appl 2010;4:683–9. [12] Ariba Y, Gouaisbaut F, Delay-dependent stability analysis of linear systems with time-varying delay, In: Proceedings of the 46th IEEE conference on decision and control (CDC), New Orleans, USA; 2007. p. 2053–2058. [13] Yue D, Han QL, Lam J. Networked-based robust H1 control of systems with uncertainty. Autmatica 2005;41:999–1007. [14] Kim SH, Park P, Jeong C. Robust H 1 stabilization of networked control systems with packet analyser. IET Control Theory Appl 2010;4:1828–38. [15] Gao H, Chen T. Networked-based H1 output tracking control. IEEE Trans Autom Control 2008;53:655–66. [16] Yue D, Tian E, Wang Z, Lam J. Stabilization of systems with probabilistic interval input delays and its applications to networked control systems. IEEE Trans Syst Man Cybern-Part: Syst Human 2009;39:939–45. [17] Lofberg J, YALMIP: a toolbox for modeling and optimization in MATLAB, Proceedings of the CACSD conference. Taipei, Taiwan; 2004. [18] Mary Dolly, Mathew AT, Jacob J. Robust H-infinity ðH1 Þ stabilization of uncertain wheeled mobile robots. Glob J Res Eng Electr Electron Eng 2012;12:23–31.
Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i
Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Research Article
Improved stabilization method for networked control systems with variable transmission delays and packet dropout Arash Farnam, Reza Mahboobi Esfanjani n Electrical Engineering Department, Sahand University of Technology, Tabriz, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 29 April 2012 Received in revised form 17 March 2014 Accepted 8 May 2014 This paper was recommended for publication by Prof. Y. Chen
This paper investigates the problem of stability analysis and stabilization for networked control systems with the network-induced delay and data dropout. In order to obtain less conservative results, a novel augmented Lyapunov–Krasovskii functional is introduced and new free-weighting matrices are employed to make some extra degrees of freedom in the sufficient stabilizability condition. The gain of the memoryless state-feedback controller is computed by solving a set of linear matrix inequalities (LMIs). Illustrative examples are given to verify the applicability and outperformance of the proposed method compared to the existing approaches in the literature. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Networked control systems Stabilization Lyapunov–Krasovskii theorem Linear matrix inequality (LMI)
1. Introduction Networked control system (NCS) is a feedback control structure wherein the control loop is closed via a communication network. NCS integrates communication network with control system to attain low cost, simple installation, easy maintenance and high flexibility. However, NCSs' challenges such as network-induced data transmission delay and dropout in communication between the loop components bring hard to solve problems in this scheme compared with traditional point-to-point control architectures. These issues motivated a lot of researches in this field during the recent years [1–7]. The Lyapunov–Krasovskii theorem is a common tool for stability analysis and stabilizing controller synthesis for time-delay systems [8–12] and consequently for NCSs. In [5], the basics of the statefeedback controller design methods using Lyapunov–Krasovskii functional (LKF) are presented. Jensen's inequality was employed in [8] to derive simple conditions for determination of controller gain. In spite of a straightforward design procedure, the obtained results are considerably conservative. In order to attain less conservative results, a free-weighting matrices technique was used in [6] and [7] for stabilization of NCSs. The method of [6] was improved in [13] by using the augmented LKF and further enhanced in [14] with new free-weighting matrices. The utilized LKF in the most of the papers n
Corresponding author. E-mail addresses: [email protected] (A. Farnam), [email protected] (R. Mahboobi Esfanjani).
involve only double-integral terms of state rate, [5–7] and [13–16]. In [11], a triple-integral term was included in LKF to obtain less conservative results in stability analysis of time-varying delay systems. The benefits of the existence of the triple-integral terms in LKF were discussed thoroughly in [12]. In this paper, an efficient approach is developed to design a static state-feedback stabilizing controller for NCSs. Variable delay is considered in the system model to represent both of the data packet latency and dropout in NCSs. New augmented LKF and novel free-weighting matrices are employed to derive less conservative delay-dependent sufficient condition in terms of linear matrix inequalities. Unlike all the existing methods for stabilization of NCSs which utilize the LKFs with double-integral terms, the introduced LKF includes triple-integral phrases. Moreover, for the first time, double-integral terms of the state rate are involved in the augmented vector of the energy functional to reduce the conservativeness of the design scheme. Illustrative examples are presented to demonstrate that the resulting maximum allowable delay to retain the stability of closed-loop system with the proposed controller is higher than the values obtained with the methods in the literature. This paper is organized as follows: in Section 2, NCSs are modeled as a continuous time-delay system. Sufficient conditions for the stability analysis and state feedback control design of NCSs are introduced in Section 3. In Section 4, two numerical benchmark examples and a practical system are resolved by the developed scheme to illustrate the effectiveness of the proposed approach. Section 5 concludes the paper.
http://dx.doi.org/10.1016/j.isatra.2014.05.006 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i
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2
Notation: in this paper, n denotes symmetric block in the symmetric matrices. I is identity matrix of appropriate dimensions. The notation P 4 0 (P Z 0Þ means that P is real symmetric positive definite (positive semidefinite). The superscript T stands for matrix transposition. col ½ Represents the column vector of the elements in the bracket. diag½ symbolizes a diagonal matrix composed of the matrices listed in the bracket.
2. System description and preliminaries The controlled system is described as follows: x_ ðtÞ ¼ A xðtÞ þ B uðtÞ
ð1Þ
where xðtÞϵR and uðtÞϵR are the state vector and the control input vector, respectively; A and B are known system matrices with appropriate dimensions. It is assumed that the pair (A, B) is completely controllable. The considered NCS structure is shown in Fig. 1, where the controller and the actuator are event-driven and sampler is clockdriven. The sampling period is assumed to be h, where h is a positive constant. The transmission delays may not be necessarily integer multiplies of the sampling period; so, Zero order hold (ZOH) device's information may be updated between sampling instants. Since the controller is a constant gain, the feedback and forward delays are combined together at each sampling time. The updating instants of ZOH are t k and the successfully transmitted signals from the sampler to the controller and from the controller to ZOH at the instant t k experience delay ηk ¼ ηsck þ ηcak , where ηsck and ηcak are the time-delays from the sampler to the controller and from the controller to the ZOH at the updating instant t k , respectively. Therefore, the state feedback controller with considering the behavior of the ZOH takes the following form: n
m
uðt k Þ ¼ K xðt k ηk Þ;
tk r t o tk þ 1
ð2Þ
in which t k þ 1 is the next updating instant after t k . The networkinduced delay, ηk is bounded as the following inequality:
ηm r ηk r ηM
ð3Þ
where ηm and ηM are the lower and upper bounds of the networkinduced delay, respectively. Then, the closed-loop system in Fig. 1 is described by: x_ ðtÞ ¼ AxðtÞ þBKxðt k ηk Þ; t k rt o t k þ 1
ð4Þ
On the other hand, at the updating instant t k , the number of accumulated data packet dropouts since the last updating instant t k 1 is denoted by δk , with 0 r δk r δM , where δM stands for the maximum number of accumulated data packet dropouts. Combining the above-mentioned facts yields to: t k þ 1 t k ¼ ηk þ 1 ηk þ ðδk þ 1 þ1Þh
Z.O.H
Actuator and Plant
ð5Þ
If ηðtÞ ¼ t t k þ ηk is replaced in (4), then the following continuous time model is obtained for the closed-loop NCS in Fig. 1: x_ ðtÞ ¼ A xðtÞ þ BK xðt ηðtÞÞ which is in the form of a continuous time-delay system, with:
ηm r ηðtÞ r η
3. Main results In this section, a new delay-dependent stability criterion is derived in Theorem 1 to ensure the asymptotic stability of the closed-loop system (6) for all delays satisfying (7). Afterwards, a simple condition is extracted in Theorem 2, to determine the controller gain in terms of linear matrix inequalities. Theorem 1. For the given ηm , η and K, the closed-loop system (6) is asymptotically stable if there exist matrices N 0 , N 1 , N 2 , M, L0 , L1 , L2 , R, S, F and symmetric matrices P ¼ ½P ij 55 4 0, Q 1 ¼ ½Q 1ij 22 4 0, Q 2 ¼ ½Q 2ij 22 4 0, T 1 ¼ ½T 1ij 22 4 0, T 2 ¼ ½T 2ij 22 4 0, Z 1 4 0, Z 2 4 0,U 1 ; U 2 , V 0 ¼ ½V 0ij 77 , V 1 ¼ ½V 1ij 77 , V 2 ¼ ½V 2ij 77 , W 0 ¼ ½W 0ij 77 , W 1 ¼ ½W 1ij 77 , W 2 ¼ ½W 2ij 77 , X 0 , X 1 and X 2 with appropriate dimensions, satisfying the matrix inequalities (8)–(15): 2 3 P R S 6n U F 7 ð8Þ 4 540 1 n n U2
Ω1 ¼ π 1 þ π 2i þ π T2i þ π 3 þ π T3 þ ηm V 0 þ η^ V i þ for i ¼ 1; 2 2 V 0 L0 þ ψ 0 T 111 Ω2 ¼ 6 4 n n
2
V1 Ω3 ¼ 6 4 n n
2
V2 Ω4 ¼ 6 4 n "
Ω5 ¼ "
Ω6 ¼ "
Fig. 1. Considered networked control system.
ð7Þ
wherein η ¼ ηM þ ðδM þ 1Þh. It is obvious that η is specified by the maximum number of accumulated data packet dropouts, δM , the upper bound of network-induced delay, ηM and the sampling period, h. The aim of this paper is to develop an improved procedure to determine state-feedback gain K in (2), such that the closed-loop system described by (6) is asymptotically stable for all delay satisfying (7).
Sampler
Static Controller
ð6Þ
Ω7 ¼
N2
T 211
Z1
W1
L 1 þ φ1
n
Z2
ð11Þ
T 222
L2 þ ψ 1
n
ð10Þ
3
T 212 þ X 1 7 5 Z0
n
L 0 þ φ0
n
N1
T 211
W0
W 0 þ η2 W i o 0
T 122
L1 þ ψ 1
n
2
3
T 112 þ X 0 7 5Z0
n
n
W2
N0
η2m
3
T 212 þ X 2 7 5Z0 T 222
ð12Þ
Z0
ð13Þ
Z0
ð14Þ
#
#
# L 2 þ φ1 Z0 Z2
ð15Þ
where η^ ¼ η ηm , η ¼ 12ðη2 η2m Þ and
Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i
A. Farnam, R. Mahboobi Esfanjani / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
"
2
#
3
Z
Λ11 Λ12 Λ13 Λ14 Λ22 Λ23 Λ24 7 7 7 n Λ33 Λ34 5
V 2 ðxt Þ ¼
6 n Σ1 ¼ 6 6 4 n
Σ1 Σ2 π1 ¼ ; n Σ3
n
n
n
t
τT ðαÞQ 1 τðαÞ dα þ
t ηm
Z
44
V 3 ðxt Þ ¼
Z
0
3
ηm
t
Z
t ηm
τT ðαÞT 1 τðαÞ dα dβ þ
t þβ
τT ðαÞQ 2 τðαÞ dα
tη
Z
ηm
Z
η
t tþβ
τT ðαÞT 2 τðαÞ dα dβ ð19Þ
Λ11 ¼ P 14 R1 þ P T14 RT1 þ Q 111 þ ηm T 111 þ η^ T 211 þ X 0 Z
Λ12 ¼ P 14 þ P 15 þ R1 S1 þ P T24 RT2
V 4 ðxt Þ ¼
Z
0
þ
η
V 5 ðxt Þ ¼ 2ξ ðtÞR T
þ Q 211 X 0 þ X 1
η
2 R0 V 6 ðxt Þ ¼ 4 R
Λ24 ¼ Λ33 ¼
P 35 þ S3 P T35 þ ST3 Q 211
η^ S2
η2m
2
P 12 6 6 P 22 Q 112 þ Q 212 Σ2 ¼ 6 6 P T23 4
Σ3 ¼ 6 4
0
3
P 33 Q 212
0
0
n
Q 222
0
n
n
X1 þ X2
"
0
0
N0 þ N1
M
0
ηm
t tþβ
T x_ ðαÞ dα dβ þ 2ξ ðtÞS
ηm
ηm η
Rt
tþβ
x_ ðαÞ dα dβ
tþβ
x_ ðαÞ dα dβ
Rt
3T " 5
Z
0
ð20Þ
ηm η
Z
t
t þβ
x_ ðαÞ dα dβ
3 #2 R 0 R t _ η t þ β xðαÞ dα dβ 4R m R 5 ηm t U2 _ η t þ β xðαÞ dα dβ
U1
F
n
T V_ 1 ðxt Þ ¼ 2ξ ðtÞP ξ_ ðtÞ
V_ 2 ðxt Þ ¼ τT ðtÞ Q 1 τðtÞ τT ðt ηm ÞQ 1 τðt ηm Þ þ τT ðt ηm Þ Q 2 τðt ηm Þ τT ðt ηÞQ 2 τðt ηÞ
ð24Þ
V_ 3 ðxt Þ ¼ τT ðtÞðηm T 1 þ η^ T 2 Þ τðtÞ
7 5
N2
MBk
ð22Þ R t ηm
ð23Þ
3
Q 122 þ Q 222
π 3 ¼ MA
Z
0
x_ T ðαÞZ 2 x_ ðαÞ dα dθ dβ
where in, ξðtÞ ¼ col½ xðtÞ xðt ηm Þ xðt ηÞ t η xðαÞ dα t η m xðαÞ dα and τðαÞ ¼ col ½ xðαÞ x_ ðαÞ . The relation (8) guarantees that the LKF V in (16) is a positive definite functional. The time derivative of Vðxt Þ along the trajectories of (6) is obtained as follows:
7 07 7 07 5 0
P 23 0
π 2i ¼ N0 þ ηm L0 þ η^ Li
Z
t þθ
Rt
Z 1 þ ηZ 2
P 13
0 2
2
β
X2
Λ34 ¼ P T13 þ ηm R3 þ η^ S3 Λ44 ¼ Q 122 þ ηm T 122 þ η^ T 222 þ
x_ T ðαÞZ 1 x_ ðαÞdαdθdβ
ð21Þ
Λ23 ¼ P 25 þ S2 P T34 þ P T35 þRT3 ST3 P T12 þ m R2 þ
t
tþθ β Z ηm Z 0 Z t
Λ14 ¼ P 11 þ ηm R1 þ η^ S1 þ Q 112 þ ηm T 112 þ η^ T 212 Λ22 ¼
0Z
ηm
Λ13 ¼ P 15 þ S1 þ P T34 RT3
P 24 þ P 25 þ R2 S2 P T24 þ P T25 þ RT2 ST2 Q 111
ð18Þ
Z 0
0
0
N1 þ N2
#
t ηm t ηðtÞ
Z
t
t ηm
τT ðαÞT 2 τðαÞ dα
τT ðαÞT 1 τðαÞ dα Z
t ηðtÞ t η
τT ðαÞT 2 τðαÞ dα
ð25Þ
2 Z 0 Z t η x_ T ðαÞZ 1 x_ ðαÞ dα dβ V_ 4 ðxt Þ ¼ x_ T ðtÞ m Z 1 þ ηZ 2 x_ ðtÞ 2 ηm t þ β Z ηm Z t Z ηðtÞ Z t x_ T ðαÞZ 2 x_ ðαÞ dα dβ x_ T ðαÞZ 2 x_ ðαÞ dα dβ
ηðtÞ
tþβ
η
tþβ
ð26Þ
T T ψ 0 ¼ P 44 þR4 P 44 P 45 R4 þ S4
T T T ψ 1 ¼ P 45 þR5 P 45 P 55 R5 þ S5
T T T T T φ0 ¼ R4 þ U 1 R4 R5 U 1 þF
T T T T T T φ1 ¼ S4 þ F S4 S5 F þU 2
P 45 S4
P 55 T S5
P T44 ηm R4 η^ S4 P T15 ηm R5 η^ S5
RT5 F
RT1 ηm U T1 η^ F
ST5 U T2
ST1 ηm F T η^ U T2
RT2 ST2
Proof. Define the LKF as follows: Vðxt Þ ¼ V 1 ðxt Þ þ V 2 ðxt Þ þ V 3 ðxt Þ þV 4 ðxt Þ þ V 5 ðxt Þ þ V 6 ðxt Þ
ð16Þ
P T24
P T25
P T34
P T35
RT3
0
ST3
0
T
0
0
T
T
T
T V_ 5 ðxt Þ ¼ 2ξ_ ðtÞR
Z
0 ηm
Z
t
t þβ
T x_ ðαÞ dα dβ þ 2ξ ðtÞRðηm x_ ðtÞ
xðtÞ þ xðt ηm ÞÞ þ 2ξ_ ðtÞSð T
V 1 ðxt Þ ¼ ξ ðtÞP ξðtÞ T
ð17Þ
Z
ηm ηðt Þ
Z
t
tþβ
x_ ðαÞ dα dβ
Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i
A. Farnam, R. Mahboobi Esfanjani / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Z þ
ηðt Þ
Z
η
t tþβ
ε16 ðtÞ ¼ xT ðt ηðtÞÞX 2 xðt ηðtÞÞ xT ðt ηÞX 2 xðt ηÞ
T x_ ðαÞ dα dβÞ þ 2ξ ðtÞSðη^ x_ ðtÞ
Z
xðt ηm Þ þ xðt ηÞÞ " V_ 6 ðxt Þ ¼
ð27Þ
ηm x_ ðtÞ xðtÞ þ xðt ηm Þ η^ x_ ðtÞ xðt ηm Þ þ xðt ηÞ
#T "
U1
R0 Rt
ε2 ðtÞ ¼ 2ζ ðtÞN 1 xðt ηm Þ xðt ηðtÞÞ ε3 ðtÞ ¼ 2ζ T ðtÞN 2 xðt ηðtÞÞ xðt ηÞ
t ηm t ηðtÞ
Z
t ηðtÞ tη
! x_ ðαÞ dα
¼0
¼0
ε4 ðtÞ ¼ 2ζ ðtÞMðx_ ðtÞ AxðtÞ Bkxðt ηðtÞÞÞ ¼ 0 Z
t
t ηm
Z
xðαÞ dα
ð30Þ
ð31Þ ð32Þ
Z
0 ηm
t
x_ ðαÞ dα dβ Þ ¼ 0
t þβ
ð33Þ Z
ε6 ðtÞ ¼ 2ζ T ðtÞL1 ððηðtÞ ηm ÞxðtÞ
t ηm
t ηðtÞ
xðαÞ dα
Z
ηm
Z
ηðtÞ
t
t þβ
ε7 ðtÞ ¼ 2ζ T ðtÞL2 ððη ηðtÞÞxðtÞ
t ηðtÞ
tη
xðαÞ dα
ηðtÞ Z
Z
η
t
tþβ
Z
t t ηm
ζ T ðtÞV 0 ζ ðtÞ dα ¼ 0
ε9 ðtÞ ¼ ðηðtÞ ηm Þζ T ðtÞV 1 ζ ðtÞ ε10 ðtÞ ¼ ðη ηðtÞÞζ ðtÞV 2 ζ ðtÞ T
ε11 ðtÞ ¼
η2m 2
ζ ðtÞW 0 ζ ðtÞ T
Z
0 ηm
Z
t ηm t ηðtÞ
Z
t ηðtÞ t η
Z
ζ T ðtÞV 1 ζ ðtÞ dα ¼ 0 ζ ðtÞV 2 ζ ðtÞ dα ¼ 0 T
ζ ðtÞW 0 ζ ðtÞ dα dβ ¼ 0 T
tþβ
Z
ð36Þ
ð37Þ
ð38Þ
Z
ηm ηðtÞ
t t þβ
ð39Þ
ζ ðtÞW 1 ζ ðtÞ dα dβ ¼ 0 T
ð40Þ
ε13 ðtÞ ¼
ðη2 η2 ðtÞÞ T ζ ðtÞW 2 ζ ðtÞ 2
Z
ηðtÞ
Z
η
t
tþβ
ζ T ðtÞW 2 ζ ðtÞ dα dβ ¼ 0 ð41Þ
ε14 ðtÞ ¼ xT ðtÞX 0 xðtÞ xT ðt ηm ÞX 0 xðt ηm Þ 2
Z
t
t ηm
x_ T ðαÞX 0 xðαÞ dα ¼ 0
ð42Þ
ε15 ðtÞ ¼ xT ðt ηm ÞX 1 xðt ηm Þ xT ðt ηðtÞÞX 1 xðt ηðtÞÞ Z
2
t ηm
t ηðtÞ
ð46Þ
i¼2
where,
Ω1 ðtÞ ¼ π 1 þ π 2 ðtÞ þ π T2 ðtÞ þ π 3 þ π T3 þ ηm V 0 þ ðηðtÞ ηm ÞV 1 þ ðη ηðtÞÞV 2 η2m ðη2 ðtÞ η2m Þ ðη2 η2 ðtÞÞ 2
W0 þ
W1 þ
2
3 2 ζ ðtÞ T V 0 6 xðαÞ 7 6 n Ω2 ðtÞ ¼ 4 5 4 t ηm n x_ ðαÞ Z
Ω3 ðtÞ ¼
Ω4 ðtÞ ¼
Ω5 ðtÞ ¼ Ω6 ðtÞ ¼ Ω7 ðtÞ ¼
Z
2
ηm
32
3
2
3 2
32
3
"
t
Z
ηðtÞ η
N2 ζ ðtÞ T V 2 L2 þ ψ 1 ζ ðtÞ 6 xðαÞ 7 6 n 76 xðαÞ 7 T T þ X 211 212 2 54 4 5 4 5 dα n n T 222 x_ ðαÞ x_ ðαÞ
t þβ
ηðtÞ
Z
n
3 2
Z
ηm
Z
T 111
2
t ηðtÞ
0
32
N0
N1 ζ ðtÞ T V 1 L1 þ ψ 1 ζ ðtÞ 6 xðαÞ 7 6 n 6 7 T 211 T 212 þ X 1 7 4 5 4 54 xðαÞ 5 dα _xðαÞ _ n n T 222 x ðα Þ
t ηm
t η
Z
W2
3 ζ ðtÞ 6 7 T 112 þ X 0 54 xðαÞ 7 5 dα T 122 x_ ðαÞ
L0 þ ψ 0
t
t ηðtÞ
Z
2
ζ ðtÞ x_ ðαÞ
"
t t þβ
Z
t
ζ ðtÞ x_ ðαÞ
"
tþβ
#T "
W0
L0 þ φ0
n
Z1
#T "
ζ ðtÞ x_ ðαÞ
#"
#"
W1
L1 þ φ1
n
Z2
W2
L 2 þ φ1
n
Z2
#T "
#
ζ ðtÞ dα dβ x_ ðαÞ #
ζ ðtÞ dα dβ x_ ðαÞ
#"
#
ζ ðtÞ dα dβ x_ ðαÞ
and
π 2 ðtÞ ¼ ½N 0 þ ηm L0 þ ðηðtÞ ηm ÞL1 þ ðη ηðtÞÞL2
t
ðη2 ðtÞ η2m Þ T ε12 ðtÞ ¼ ζ ðtÞW 1 ζ ðtÞ 2
i¼7
_ t Þ ¼ ζ T ðtÞΩ1 ðtÞ ζ ðtÞ þ ∑ Ωi ðtÞ Vðx
x_ ðαÞ dα dβ Þ ¼ 0
ð35Þ
ε8 ðtÞ ¼ ηm ζ T ðtÞV 0 ζ ðtÞ
The V_ in (45) can be rewritten as the following:
x_ ðαÞ dα dβ Þ ¼ 0
ð34Þ Z
i¼1
þ
T
ε5 ðtÞ ¼ 2ζ T ðtÞL0 ðηm xðtÞ
_ t Þ ¼ V_ 1 ðxt Þ þ V_ 2 ðxt Þ þ V_ 3 ðxt Þ þ V_ 4 ðxt Þ þ V_ 5 ðxt Þ þ V_ 6 ðxt Þ þ ∑ εi ðtÞ Vðx
! x_ ðαÞ dα
ð44Þ
ð45Þ
t ηm
Z
x_ T ðαÞX 2 xðαÞ dα ¼ 0
tη
i ¼ 16
ð28Þ
On the other hand, for any matrices N 0 , N 1 , N 2 , M, L0 , L1 and L2 and symmetric matrices V 0 , V 1 , V 2 , W 0 , W 1 , W 2 , X 0 , X 1 and X 2 with appropriate dimensions, the following equalities hold: Z t x_ ðαÞ dαÞ ¼ 0 ε1 ðtÞ ¼ 2ζ T ðtÞN 0 ðxðtÞ xðt ηm Þ ð29Þ
T
t ηðtÞ
where εðtÞ ¼ col½ xðtÞ xðt ηm Þ xðt ηÞ x_ ðtÞ x_ ðt ηm Þ x_ ðt ηÞ xðt ηðtÞÞ. Now, regarding (23)–(28) and (29)–(44), V_ can be stated as follows:
#
F
n U2 3 _ – η m t þ β x ðα Þ dα dβ 4 5 Rη R ηðtÞ R t Rt m _ _ ηðt Þ t þ β xðαÞ dα dβ þ η t þ β x ðα Þ dα dβ
2
2
x_ T ðαÞX 1 xðαÞ dα ¼ 0
ð43Þ
N0 þ N1 N2
0
0
0
N 1 þ N 2 :
Provided Ω1 ðtÞ o 0 and Ωi Z 0; ði ¼ 2; …; 7Þ, the Lyapunov–Krasovskii theorem ensures the asymptotic stability of the system (6). Note that replacing ηm and η by the ηðtÞ in the condition Ω1 ðtÞ o 0 yields to the conditions of Eq. (9). It should be emphasized that regarding the (17), (21) and (22), the condition (8) is needed to assure the positive-definiteness of the introduced LKF in (16). □ Theorem 1 presents sufficient conditions for the stability analysis of the closed-loop control system (6) in the form of matrix inequalities. For the given controller gain K, this theorem can be used to determine the maximum value of allowable delay η in (7) which retain the stability of the system (6). In Theorem 2, utilizing the variable changing technique, the bilinear conditions in Theorem 1 is modified to obtain equivalent linear matrix inequalities (LMIs) which are computationally more tractable to determine the controller gain K. The auxiliary scalar parameters appeared in the design conditions are tuned by trial and error to attain a feasible solution for the LMIs.
Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i
A. Farnam, R. Mahboobi Esfanjani / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
5
Theorem 2. For the given constants ηm , η and scalars ρi 4 0 ði ¼ 2; …; 7Þ, the closed-loop system (6) is asymptotically
Λ~ 11 ¼ P~ 14 R~ 1 þ P~ 14 R~ 1 þ Q~ 111 þ ηm T~ 111 þ η^ T~ 211 þ X~ 0
stable with the controller gain K ¼ YX T if there exist a nonsingular ~ F~ ; Y and symmetric ~ S, matrix X, matrices N~ 0 , N~ 1 , N~ 2 , L~ 0 , L~ 1 , L~ 2 , R, ~ ~ ~ ~ ~ 40, Q ¼ ½Q~ 4 0, T~ 1 ¼ matrices P ¼ ½P ij , Q ¼ ½Q
Λ~ 12 ¼ P~ 14 þ P~ 15 þ R~ 1 S~ 1 þ P~ 24 R~ 2
½T~ 1ij 22 4 0, T~ 2 ¼ ½T~ 2ij 22 40,Z~ 1 4 0,Z~ 2 4 0, U~ 1 , U~ 2 , V~ 0 ¼ ½V~ 0ij 77 , ~ 0 ¼ ½W ~ 0 ,W ~ 1 ¼ ½W ~ 1ij , W ~ 2¼ V~ 1 ¼ ½V~ 1 , V~ 2 ¼ ½V~ 2 , W
Λ~ 13 ¼ P~ 15 þ S~ 1 þ P~ 34 R~ 3
~ 2ij , X~ 0 ,X~ 1 and X~ 2 with appropriate dimensions such that the ½W 77 following LMIs are satisfied (47–54): 2~ 3 P R~ S~ 6 7 ð47Þ 4 n U~ 1 F~ 5 4 0 ~ n n U2
Λ~ 14 ¼ P~ 11 þ ηm R~ 1 þ η^ S~ 1 þ Q~ 112 þ ηm T~ 112 þ η^ T~ 212
1
55
ij
77
ij
1ij 22
2
77
ij
6
~ ¼6 Ω 3 4
L~ 0 þ ψ~ 0 T~ 1
n
n
V~ 1 n
L~ 1 þ ψ~ 1 T~ 2
n
n
2
n
~ 0 W n
"
~ 1 W n
" ~ ¼ Ω 7
~ 2 W n
π~ 1 ¼
"
ψ~ 0 ¼ "
ψ~ 1 ¼
ð49Þ
T
T
Λ~ 34 ¼ P~ 13 þ ηm R~ 3 þ η^ S~ 3 ;
Λ~ 44 ¼ Q~ 122 þ ηm T~ 122 þ η^ T~ 222 þ P~ 13 P~ 23
P~ 12 6~ 6 P 22 Q~ 112 þ Q~ 212 Σ~ 2 ¼ 6 6 T 6 P~ 23 4 0
ð50Þ
2 6 Σ~ 3 ¼ 6 4
ð51Þ
Q~ 122 þ Q~ 222
T
T
2
P~ 33 Q~ 212 0
n
0 Q~
n
n
2
Z 1 þ ηZ~ 2
3 0 7 07 7 7 07 5 0 3
0 0
222
η2m ~
X~ 1 þ X~ 2
7 7 5
#
~0 L~ 0 þ φ Z0 Z~ 1
ð52Þ
# ~1 L~ 1 þ φ Z0 Z~ 2
ð53Þ
# ~1 L~ 2 þ φ Z0 Z~ 2
Σ~ 1 Σ~ 2 ; n Σ~ 3
P~ 44 þ R~ 4
P~ 45 þ R~ 5 T
T
φ~ 1 ¼ S~ 4 þ F~
T
T
2
AX T 6 6 ρ2 AX T 6 6 T 6 ρ3 AX 6 6 π~ 3 ¼ 6 ρ4 AX T 6 6 ρ AX T 5 6 6 6 ρ6 AX T 4 ρ7 AX T
3
Λ~ 11 Λ~ 12 Λ~ 13 Λ~ 14 6 7 6 n Λ~ 22 Λ~ 23 Λ~ 24 7 7 Σ~ 1 ¼ 6 6 n ~ ~ n Λ 33 Λ 34 7 4 5 n n n Λ~ 44
#
T
"
T P~ 44 P~ 45 R~ 4 þ S~ 4
T T P~ 45 P~ 55 R~ 5 þ S~ 5
T
T
T
T T T T S~ 4 S~ 5 F~ þ U~ 2
P~ 45 S~ 4
T P~ 55 S~ 5
T R~ 5 F~
T T S~ 5 U~ 2
P~ 24 T
T P~ 14 ηm R~ 4 η^ S~ 4
P~ 25 T
T P~ 15 ηm R~ 5 η^ S~ 5
T T R~ 1 ηm U~ 1 η^ F~
T T T S~ 1 ηm F~ η^ U~ 2
N~ 0 þ N~ 1
π~ 2i ¼ N~ 0 þ ηm L~ 0 þ η^ L~ i
ð54Þ
2
T
T
Λ~ 33 ¼ P~ 35 þ S~ 3 P~ 35 þ S~ 3 Q~ 211 X~ 2
T
3
φ~ 0 ¼ R~ 4 þ U~ 1 R~ 4 R~ 5 U~ 1 þ F~
T
Λ~ 24 ¼ P~ 12 þ ηm R~ 2 þ η^ S~ 2 ;
22
where, "
T
T
7 T~ 212 þ X~ 2 7 5Z0 T~ 2
11
n
~ 6¼ Ω
T
3
N~ 2
T
Λ~ 23 ¼ P~ 25 þ S~ 2 P~ 34 þ P~ 35 þ R~ 3 S~ 3
22
L~ 2 þ ψ~ 1 T~ 2
V~ 2 6 ~ 6 Ω4 ¼ 4 n
~ ¼ Ω 5
T
η ~ ~ i o0; W 0 þ η2 W
7 T~ 212 þ X~ 1 7 5Z0 T~ 2
11
T
2 m
7 T~ 112 þ X~ 0 7 5Z0 T~ 122 N~ 1
T
Λ~ 22 ¼ P~ 24 þ P~ 25 þ R~ 2 S~ 2 P~ 24 þ P~ 25 þ R~ 2 S~ 2 Q~ 111 þ Q~ 211 X~ 0 þ X~ 1
3
N~ 0
11
2
"
T
ð48Þ
V~ 0 6 ~ 6 Ω2 ¼ 4 n 2
T
77
8 i ¼ 1; 2 2
T
2ij 22
77
~ ¼ π~ þ π~ þ π~ T þ π~ þ π~ T þ η V~ þ η^ V~ þ Ω 1 1 3 2i i m 0 2i 3
T
R~ 3
R~ 2 T
T
S~ 2 T
0 0
0 0
XT ρ2 X T
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
ρ3 X T ρ4 X T ρ5 X T ρ6 X T ρ7 X T
0
0
T
T
0
0
0
N~ 1 þ N~ 2
#
3 BY 7 ρ2 BY 7 7 7 ρ3 BY 7 7 ρ4 BY 7 7 7 ρ5 BY 7 7 7 ρ6 BY 7 5 ρ7 BY
#T
P~ 34
P~ 35
N~ 2
0 #T 0
T 0
S~ 3 T
T 0
Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i
A. Farnam, R. Mahboobi Esfanjani / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
Table 1 MADBs corresponding to the different design methods for Example 1 (with ηm ¼ 0). Method
[4]
[6]
[13]
[5]
[9]
[7]
[10]
[14]
[16]
[11]
Proposed method
MADB (in second)
0.7805
0.8695
0.8871
0.9410
0.9412
1.0081
1.0081
1.0213
1.0432
1.0629
1.0772
Fig. 2. Time response of systems' states and control effort for the Example 1.
and M 7 ¼ ρ7 M 0 . Feasibility of inequality (48) implies that M 0 is nonsingular. Let X ¼ M 0 1 , then pre- and neously the two sides of (9) with diag X (10)–(12) with diag X X X X X X with diag X X X X X X X X h i and its diag X X X X X X X inequalities Y ¼ KX T .
Fig. 3. Delay profile in the simulation of Example 1.
Table 2 MADBs resulting from different methods for Example 2. Method
MADB (in second)
[6] [7] [16]
0.5635 0.8597 0.9028
Proposed method
0.9685
(8)–(15)
leads
to
post-multiply simultaX X X , X X X X X X, (13)–(15) and (8) with transpose. Therefore
inequalities
(47)–(54),
with
□
Remark 1. Differently from the existing controller synthesis methods for NCSs which rely on the LKFs with double-integral terms, energy functional including triple-integral terms, V 4 ðxt Þ; is used to develop a more efficient stabilization scheme for the systems controlled over a network. Moreover, for the R0 Rt first time, new augmented variables η t þ β x_ ðαÞ dα dβ and R ηm R t m _ η t þ β xðαÞ dα dβ are involved in the introduced LKF which appear as the phrases V 5 ðxt Þ and V 6 ðxt Þ in (16). Remark 2. In order to increase the degree of freedom in the sufficient design condition, new free-weighting matrices are employed in (42)–(44). Unlike the common approach for injection of free-weighting matrices which are based on the following relation: Z t ηm T ðη ηm Þζ ðtÞV ζ ðtÞ ζ T ðtÞV ζ ðtÞ dα ¼ 0 t η
with any matrix V with appropriate dimension, we utilize the below relations to obtain less conservative results: ðηðtÞ ηm Þζ ðtÞV 1 ζ ðtÞ T
ðη ηðtÞÞζ ðtÞV 2 ζ ðtÞ T
Fig. 4. Networked control of the mobile robot for reference tracking in Example 3.
Proof. Let M ¼ ½ M 1 M 2 M 3 M 4 M 5 M 6 M 7 T . Replace M 1 ¼ M 0 ; M 2 ¼ ρ2 M 0 ; M 3 ¼ ρ3 M 0 ; M 4 ¼ ρ4 M 0 ; M 5 ¼ ρ5 M 0 ; M 6 ¼ ρ6 M 0
Z
Z
t ηm t ηðtÞ
t ηðtÞ tη
ζ T ðtÞV 1 ζ ðtÞ dα ¼ 0
ζ T ðtÞV 2 ζ ðtÞ dα ¼ 0
wherein V 1 and V 2 are arbitrary matrices with appropriate dimensions. It should be noted that the similar technique is applied to handle double-integral terms.
Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i
A. Farnam, R. Mahboobi Esfanjani / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Fig. 5. Simulation results for the tracking problem of Example 3.
4. Illustrative examples Two numerical benchmark examples and a practical system are presented to compare the results of the proposed method with existing approaches in the literature. The YALMIP Toolbox is utilized to solve the LMI feasibility problems [17]. Example 1. Consider the following system from [6]: 0 1 0 x_ ðtÞ ¼ xðtÞ þ uðtÞ 0 0:1 0:1
ð67Þ
which is controlled over a network. By the suggested method in Theorem 2, the controller gain is obtained as K ¼ 3:75 11:5 which leads to the Maximum Allowable Delay Bound (MADB) equals to 1.0772 s with ηm ¼ 0. In Table 1, the MADBs corresponding to the rival design methods are compared. Table 1 verifies the superiority of the proposed method compared to some existing methods in the literature. Time-delayed system (6) which describes considered NCS with the designed controller is simulated by Simulinks of Matlabs. Fig. 2 depicts the time response of the systems' state and control effort starting form initial state [0.1, 0.1] where varying delay profile is shown in Fig. 3. Example 2. Consider the following system as in (6), [16]: 0:5 1 0:5 2 x_ ðtÞ ¼ xðtÞ þ xðt ηðtÞÞ 0 0:6 1 1 To validate the less conservativeness of the proposed approach, the Theorem 1 which is the cornerstone of the proposed design procedure, is utilized for stability analysis of the closed-loop system. Table 2 shows the maximum allowable delay bound η, in the case ηm ¼ 0, obtained from the developed and some rival methods. As seen, the admissible upper bound of delay obtained by the method of this paper is larger than the existing ones; so the superiority of the suggested scheme is confirmed. Example 3. Dynamical model of a practical wheeled mobile robot which is controlled over a network is considered as follows [18]: 2 3 2 3 0 0 0 1 0 0 60 6 0:381 7 0 0:0175 0:01 7 0 6 7 6 7 x_ ðtÞ ¼ 6 7xðtÞ þ 6 7 uðtÞ 4 0 0:0175 4 0 0:319 5 0 0 5 0
0
0
0
0
3:542
The overall time-delay ηðtÞ in the network varies from 0.10 s to 0.24 s. Utilizing Theorem 2, by choosing ρ2 ¼ 0:01, ρ3 ¼ 0:01, ρ4 ¼ 146, ρ5 ¼ 0:01; ρ6 ¼ 0:01 and ρ7 ¼ 0:01, the stabilizing controller gain is obtained as follows: " # 0:0231 5:5152 0:0796 0:0032 K¼ 0:0016 2:5655 6:7624 0:0386 In a real scenario, the mobile robot tracks the desired reference. So, the closed-loop system is configured as shown in Fig. 4,
Fig. 6. Delay profile in the simulation of Example 3.
wherein Hð0Þ is defined as the following: Hð0Þ ¼
Pð0Þ 1 þ Pð0ÞK
with PðsÞ ¼ ðsI AÞ 1 B. Fig. 5 shows the control effort and robot states signals which track satisfactorily the reference trajectory xd ¼ ½3; 3; 0; 0T , starting from the initial condition xð0Þ ¼ ½4; 1; 0:1; 0:1 with the delay profile shown in Fig. 6.
5. Conclusion This paper has proposed an approach to synthesise stabilizing state feedback controller for the linear time invariant system which is controlled via communication network. Data packet loss and latency arising in NCS has been incorporated in the system model as a time-varying delay. In the suggested method, stabilizing state feedback controller is computed via the feasible solution of a set of linear matrix inequalities. New augmented LKF and novel free-weighting matrices has been introduced to make extra degrees of freedom in the design condition which leads to the less conservative results. The advantages of the developed method were demonstrated by illustrative examples. References [1] Hespanha JP, Naghshtabrizi P, Xu Y. A survey of recent results in networked control systems. Proc IEEE 2007;95:138–62. [2] Yu M, Wang L, Chu T, Xie G. Modeling and control of networked systems via jump systems approach. IET Control Theory Appl 2008;2:535–41.
Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i
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Please cite this article as: Farnam A, Mahboobi Esfanjani R. Improved stabilization method for networked control systems with variable transmission delays and packet dropout. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.006i