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Delayed observer-based H∞ control for networked control systems
Author’s Accepted Manuscript Delayed Observer-based H∞ Control for Networked Control Systems Lijia Liu, Xianli Liu, Chuntao Man, Chengyang Xu
www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(15)01884-6 http://dx.doi.org/10.1016/j.neucom.2015.11.075 NEUCOM16475
To appear in: Neurocomputing Received date: 17 July 2015 Revised date: 28 October 2015 Accepted date: 24 November 2015 Cite this article as: Lijia Liu, Xianli Liu, Chuntao Man and Chengyang Xu, Delayed Observer-based H∞ Control for Networked Control Systems, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.11.075 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Delayed Observer-based H∞ Control for Networked Control Systems Lijia Liua,c,∗, Xianli Liua , Chuntao Man b , Chengyang Xu a a The
Lab of National and Local United Engineering for High-Efficiency Cutting & Tools, Harbin University of Science and Technology, China b Harbin University of Science and Technology, China c Institute of Electrical Engineering, Heilongjiang Polytechnic, China
Abstract In this paper, the observer-based H ∞ control problem for networked control systems is investigated. Moreover, network-induced time-varying delays are considered to model the real world applications. The Luenberger-type observer is first designed for state estimation, then the observer-based controller is further developed. By applying the Lyapunov-Krasovskii functional approach, delay-dependent conditions are established to ensure that the prescribed H ∞ performance can be achieved. Finally, the effectiveness and usefulness of our obtained theoretical results are demonstrated by two examples. Keywords: Networked-observer; Network-based H ∞ control; Networked control systems.
1. Introduction The past decade has witnessed rapid developments of network and communication technologies, which gives rise to the networked control approaches in the real world applications [1–6]. Generally speaking, networked control systems are composed of 5
the communication network, the sensors, the controllers and the actuators [ 7–10]. In contrast to classical control systems, significant advantages can be obtained by networked control systems including high reliability, ease of maintenance, low cost, etc ∗ Corresponding
author Email addresses: [email protected] (Lijia Liu), [email protected] (Xianli Liu), [email protected] (Chuntao Man), [email protected] (Chengyang Xu)
Preprint submitted to Neurocomputing
December 10, 2015
[11–13]. As a consequence, many remarkable theoretical and practical methodologies are developed and can be found in the literature; see e.g. [ 14–17]. 10
On another research front line, observers are often needed in some control engineering practice since certain states are difficult or even unavailable to measure [ 18– 22]. Under this context, observer-based control schemes are developed to solve this problem. Moreover, it is now well recognized that the observer and the controller can be determined independently, such that the effectiveness of the system design can be
15
increased [23, 24]. It should be pointed out that in the networked control environment, the system output measurements are in the discrete-time form according to the sampling periods of the zero-order holds (ZOH). Moreover, the network-induced timevarying delays should be taken into account due to communication constraints, and they may degrade or destroy the control system performance [ 25–30]. So far, to the
20
best of the authors’ knowledge, the observer-based H ∞ control problem for networked control systems with time-varying delays has not been fully addressed in the reported works and remains open. As a result, the aim of this paper is to shorten such a gap. Motivated by the discussion made by far, in this paper, we aim to solve the observerbased H∞ control problem for networked control systems. In particular, the Luenberger-
25
type observer is introduced to obtain the estimated state. By employing the LyapunovKrasovskii functional approach, delay-dependent conditions are established in terms of linear matrix inequalities (LMIs) to guarantee the pre-specified H ∞ disturbance attenuation level. Based on the developed results, the observer along with the state feedback controller are designed. Compared with the existing results, the main contributions of
30
our paper lie in: (i) a unified observer-based control framework is developed for the networked control systems. (ii) the phenomena of network-induced time-varying delays which can model the networked engineering practice are considered. The remainder of this paper is organized as follows. In Section 2, the model of networked control systems under consideration is introduced and the observer-based
35
H∞ problem is formulated. Section 3 gives the main results on the design procedure of networked observer and controller. In Section 4, two examples are presented to show the effectiveness of the proposed algorithm. Finally, concluding remarks are given in Section 5. 2
Notation: The notation in our paper is standard. R n denotes the n dimensional 40
Euclidean space, R m×n represents the set of all m × n real matrices. I and 0 denote identity matrix and zero matrix with appropriate dimensions, respectively. The notation P > 0 means that P is real symmetric and positive definite, and the superscript “T ” denotes matrix transposition. In symmetric block matrices, * is used as an ellipsis for the terms which are introduced by symmetry and diag{· · · } denotes a block-diagonal
45
matrix. Finally, if not explicitly stated, all matrices are supposed to have compatible dimensions.
2. Problem Formulation and Preliminaries Consider the following class of networked control systems: ⎧ ⎨ x(t) ˙ = Ax(t) + Bu(t) + D(t) , ⎩ y(t) = Cx(t ), t ≤ t < t k
k
(1)
k+1
where x(t) ∈ Rn denotes the state vector, u(t) ∈ R m is the networked control input to be designed, (t) ∈ L 2 [0, ∞) represents the external disturbance, y(t) ∈ R p de50
notes the output measurement. All A, B, C, D are constant matrices of appropriate dimensions which represent the linear dynamics. In some network environment, x(t) is difficult to measure, such that the state observer is required. Consequently, a networked Luenberger-like state observer can be constructed as follows: ⎧ ⎨ x ˆ˙ (t) = Aˆ x(t) + Bu(t) + D(t) + L(y(tk ) − yˆ(tk )), tk ≤ t < tk+1 , ⎩ yˆ(t) = C x ˆ(t ),
(2)
k
where x ˆ(tk ) denotes the estimate of x(t k ) in the networked environments and L ∈ 55
Rn×p is the gain matrix of the state observer. Moreover, the following assumptions are given for the networked state observer. Assumption 1. The sensor is clock-driven, the ZOH is event-driven. The updating instants of ZOH are at tk , k = 1, 2, . . . , ∞. The sampling period of the sensor is h(t) ¯ where ¯h is a positive constant. satisfying 0 < h(t) ≤ h, 3
60
Assumption 2. The data is transmitted by a single-packet. Assumption 3. The time-varying delays from the networked control system to the networked state observer at the updating instant t k can be considered as τ k . Moreover, it is assumed that 0 ≤ τk ≤ τ¯, k = 1, 2, . . . , ∞, where τ¯ is a positive constant. Assumption 4. The matrix B is of full-column rank, then there exist two orthogonal matrices U ∈ Rn×n and V ∈ Rm×m , such that ⎡ ⎤ B ⎦V T, B=U⎣ 0
(3)
where B = diag{b 1 , b2 , . . . , bm }, bi (i = 1, 2, . . . , m) are nonzero singular values of 65
B. As a result, the observer error system can be obtained as follows: e(t) ˙ = Ae(t) − LCe(tk − τk ), tk ≤ t < tk+1 , where e(t) := x(t) − x ˆ(t) denotes the observer error. Then, in order to solve the H ∞ control problem, the following controller is designed: ˆx u(t) = K ˆ(tk ), tk ≤ t < tk+1 ,
(4)
ˆ ∈ Rm×n is the controller gain to be determined later. where K Note that (4) can be further rewritten as ˆ k ). ˆ k ) − Ke(t u(t) = Kx(t In addition, the time-varying delay from the sensor to the controller at the updating instant tk is σk , k = 1, 2, . . . , ∞, where 0 ≤ σk ≤ σ ¯ , with σ ¯ being a positive constant. Based on the above analysis, the following augmented system can be formed: ⎧ ⎨ z(t) ¯ ˙ = Az(t) + A¯1 z(tk − σk ) + A¯2 z(tk − τk ) + A¯3 (t) , (5) ⎩ y(t) = Cz(t ¯ ), t ≤ t < t k
k
k+1
4
where z(t) =[xT (t), eT (t)]T , ⎡ ⎤ A 0 ⎦, A¯ = ⎣ 0 A ⎡ ⎤ ˆ −B K ˆ BK ⎦, A¯1 = ⎣ 0 0 ⎡ A¯2 = ⎣ ⎡ A¯3 = ⎣ C¯ =
⎤
0
0
0
−LC ⎤
D
⎦,
0
0
C
⎦,
.
By utilizing the input delay approach, (5) can be rewritten as ⎧ ⎨ z(t) ¯ ˙ = Az(t) + A¯1 z(t − d1 (t)) + A¯2 z(t − d2 (t)) + A¯3 (t) , ⎩ y(t) = Cz(t ¯ − d(t)), t ≤ t < t k
(6)
k+1
where d(t) := t − tk , d1 (t) := t − tk + σk and d2 (t) := t − tk + τk denote the virtual delays satisfying ¯ 0 ≤ d(t) ≤ h, ¯, 0 ≤ d1 (t) ≤ d¯1 := ¯h + σ 0 ≤ d2 (t) ≤ d¯2 := ¯h + τ¯. 70
Remark 1. It is worth mentioning that the sampling period is assumed to be nonuniform in this paper. The upper bound of all the possible sampling periods is given as ¯ which means that the sampling periods can very randomly within the range of (0, ¯h]. h, It can be found that the non-uniform sampling period is more applicable and practical than the uniform cases with fixed sampling period.
75
Before proceeding, the following definition and lemmas are essential in establishing the main results. 5
Definition 1. System (5) is said to satisfy the H∞ performance if under zero initial condition there exists a positive constant γ such that it holds that ∞ ∞ y T (t)y(t)dt < γ 2 T (t)(t)dt, 0
0
for all nonzero (t) ∈ L 2 [0, ∞), t > 0. Lemma 1. [31] For the matrix B ∈ R n×m that is of full-column rank, if matrix Υ is of the following structure:
⎡ Υ = UT ⎣
Υ11
0
0
Υ22
⎤ ⎦ U,
where Υ11 ∈ Rm×m and Υ22 ∈ R(n−m)×(n−m) , U is defined in (3), then there exists a nonsingular matrix such that ΥB = B. ⎡ ⎤ M S ⎦ ≥ 0, scalars τ > 0, τ (t) satisfying 0 ≤ Lemma 2. [32] For any matrix ⎣ ∗ M τ (t) ≤ τ , vector function x(t) ˙ : [−τ, 0] → R n such that the concerned integrations are well defined, then −τ
t
t−τ
x˙ T (s)M x(s)ds ˙ ≤ ζ T (t)Ωζ(t),
where ζ(t) =[xT (t), xT (t − τ (t)), xT (t − τ )]T , ⎡ −M M −S S ⎢ ⎢ Ω =⎢ ∗ −2M + S + S T −S + M ⎣ ∗ ∗ −M 80
⎤ ⎥ ⎥ ⎥. ⎦
The purpose of this paper is to design a networked state feedback controller ( 4) and a networked state observer (2) to ensure that the system (5) is asymptotically stable while achieving the H ∞ performance γ. 3. Main Results In this section, delay-dependent criteria are developed to solve the networked-
85
observer-based H ∞ problem by the LMI techniques. Both controller gain and state observer gain can be obtained by the following established conditions. 6
¯ Theorem 1. For given H ∞ performance γ, the upper bound of sampling period h, the upper bounds of network-induced delays σ ¯ and τ¯, the networked-observer-based ˆ H∞ problem of system (5) can be solved with the given controller ⎡ gain K and ⎤ the P1 0 ⎦ > 0, given state observer gain L, if there exist symmetric matrices P = ⎣ 0 P2 ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 0 Q1 0 R1 0 M1 ⎦ > 0, Q = ⎣ ⎦ > 0, R = ⎣ ⎦ > 0, M = ⎣ 0 M2 0 Q2 0 R2 ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 0 0 N W S1 0 ⎦,N =⎣ 1 ⎦ and W = ⎣ 1 ⎦ satisfying matrices S = ⎣ 0 S2 0 N2 0 W2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Q S M N R W ⎣ ⎦ ≥ 0, ⎣ ⎦ ≥ 0, ⎣ ⎦ ≥ 0 and a constant γ > 0 such that ∗ Q ∗ M ∗ R the following inequalities hold: ⎤ ⎡ Π1 Π2 ⎦ < 0, Π := ⎣ ∗ Π3 where
⎡ Π1 = ⎣ ⎡ Π11 = ⎣
Π11
Π12
∗
Π13
⎤ ⎦,
Π111
Π112
∗
Π113
⎡ Π111 = ⎣
⎤ ⎦,
2P1 A − M1 − Q1 − R1
⎡ ⎡ Π112 = ⎣ ⎣
0 M 1 − N1 0
0
2P2 A − M2 − Q2 − R2 ⎤ ⎡ ⎤ ⎤ 0 N1 0 ⎦ ⎣ ⎦ ⎦, M 2 − N2 0 N2
⎤ ⎦,
Π113 = [Π1131 , Π1132 ] , ⎤ ⎤ ⎡ ⎡ −2M1 + N1 + N1T + C T C 0 ⎦ ⎥ ⎢ ⎣ ⎥ ⎢ 0 −2M2 + N2 + N2T ⎥ ⎢ ⎥, ⎢ ⎤ ⎡ Π1131 = ⎢ T ⎥ ⎥ ⎢ + M 0 −N 1 1 ⎦ ⎣ ⎦ ⎣ 0 −N2 + M2
7
⎡ ⎡
Π1132
⎢ ⎣ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣
−N1 + M1
⎤ ⎤
0
−N2 + M2 ⎤ M1 0 ⎦ −⎣ 0 M2 0 ⎡
⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
Π12 = [Π121 , Π122 ] , ⎤ ⎡ ⎤ ⎤ ⎡ ⎡ ˆ + Q1 − S1 −P1 B K ˆ P1 B K S1 0 ⎢ ⎣ ⎦ ⎣ ⎦ ⎥ ⎥ ⎢ ⎥ ⎢ 0 Q 2 − S2 0 S2 ⎥, ⎢ Π121 = ⎢ ⎥ ⎥ ⎢ 0 0 ⎦ ⎣ 0 0 ⎤ ⎡ ⎤ ⎡ ⎡ ⎡ ⎤ ⎤ R1 − W1 0 W1 0 P1 D ⎢ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎥ ⎢ ⎥ ⎢ ⎥ 0 −P2 LC + R2 − W2 0 W2 0 ⎢ ⎥, Π122 = ⎢ ⎥ ⎢ ⎥ 0 0 0 ⎣ ⎦ 0 0 0 ⎡ ⎤ Π131 Π132 ⎦, Π13 = ⎣ ∗ Π133 ⎤ ⎡ ⎤ ⎤ ⎡ ⎡ −2Q1 + S1 + S1T 0 0 −S1 + Q1 ⎢ ⎣ ⎦ ⎣ ⎦ ⎥ ⎢ ⎥ T ⎢ ⎥ 0 −2Q2 + S2 + S2 0 −S2 + Q2 ⎢ ⎥, ⎡ ⎤ Π131 = ⎢ ⎥ ⎢ ⎥ Q1 0 ⎣ ⎦ ⎣ ⎦ ∗ − 0 Q2 Π132 =0, ⎡
Π133
Π1331
⎡
R1 − W1
⎤
0
⎤
⎢ Π1331 ⎣ ⎦ 0 ⎥ ⎢ ⎥ ⎢ ⎥ 0 R2 − W2 ⎢ ⎥ ⎤ ⎡ ⎢ ⎥ =⎢ ⎥, R1 0 ⎢ ∗ ⎥ ⎦ ⎣ 0 − ⎢ ⎥ ⎢ ⎥ 0 R2 ⎣ ⎦ ∗ ∗ −γ 2 I ⎡ ⎤ −2R1 + W1 + W1T 0 ⎦, =⎣ 0 −2R2 + W2 + W2T
Π2 = [Π21 , Π22 , Π23 ] ,
8
⎡
⎡
AT P1
⎤ ⎤
0
⎦ ⎥ ⎢ h⎣ ⎢ ⎥ ⎢ 0 AT P2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎡ ⎤ ⎥ ⎢ ⎢ ⎥ ˆ T BT P T 0 ⎢ ⎥ K 1 ⎦ ⎥ ⎢ h⎣ ⎢ ⎥ T T T ˆ B P1 0 Π21 =Π22 = Π23 = ⎢ ⎥, K ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎤ ⎥ ⎢ ⎡ ⎢ ⎥ ⎢ ⎥ 0 0 ⎦ ⎥ ⎢ h⎣ ⎢ ⎥ ⎢ ⎥ 0 −C T LT P2T ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎣ ⎦
h DT P1 0 ⎡ ⎤ ⎡ ⎤ ⎡ M1 0 P1 0 Q ⎦−2⎣ ⎦,⎣ 1 Π3 =diag{⎣ 0 M2 0 P2 0 ⎤ ⎡ ⎤ ⎡ 0 0 P R ⎦−2⎣ 1 ⎦}. ⎣ 1 0 R2 0 P2
0 Q2
⎤
⎡
⎦−2⎣
P1
0
0
P2
⎤ ⎦
Proof. Choose the following Lyapunov-Krasovskii function V (t) =
4
Vi (t),
(7)
i=1
where V1 (t) = z T (t)P z(t), 0 t ¯ V2 (t) = h z˙ T (η)M z(η)dηdϕ, ˙ V3 (t) = d¯1 V4 (t) = d¯2
¯ t+ϕ −h 0 t
−d¯1 0
t+ϕ t
−d¯2
t+ϕ
z˙ T (η)Qz(η)dηdϕ, ˙ z˙ T (η)Rz(η)dηdϕ. ˙
By taking the derivative of (7) along the solution of system (6), one has V˙ 1 (t) = 2z T (t)P z(t), ˙ ⎡ ⎡ ⎤ P P1 0 1 ¯ ⎦ Az(t) = 2z T (t) ⎣ + 2z T (t) ⎣ 0 P2 0 9
0 P2
⎤ ⎦ A¯1 z(t − d1 (t))
⎡ + 2z T (t) ⎣ ⎡ = 2z T (t) ⎣
0
0
P2
P1
0
0
P2
⎦ A¯2 z(t − d2 (t)) + 2z T (t) ⎣ ⎤⎡ ⎦⎣ ⎡
z(t − d1 (t)) + 2z T (t) ⎣ ⎡ = 2z T (t) ⎣
⎡
⎤
P1
P1 A
0
0
P2 A ⎡
+ 2z T (t) ⎣
A
0
0
A
0
0
P2
⎦ z(t) + 2z T (t) ⎣
⎤ ⎦ A¯3 (t)
ˆ P1 B K
ˆ −P1 B K
0
0
⎤
0
⎡
⎡
ˆ P1 B K
ˆ −P1 B K
0
0
⎦ z(t − d2 (t)) + 2z T (t) ⎣ 0 −P2 LC t ¯ 2 z˙ T (t)M z(t) ¯ ˙ −h z˙ T (ϕ)M z(ϕ)dϕ, ˙ V˙ 2 (t) = h ˙ − d¯1 V˙ 3 (t) = d¯21 z˙ T (t)Qz(t) V˙ 4 (t) = d¯22 z˙ T (t)Rz(t) ˙ − d¯2
t−d¯1 t t−d¯2
⎦×
0
⎦ (t)
⎦ z(t − d1 (t))
⎡
⎤
¯ t−h t
⎤
⎤
⎤
P1 D
⎦ z(t − d2 (t)) + 2z T (t) ⎣
⎦ z(t) + 2z T (t) ⎣
0
0
⎡
⎤
0 −P2 LC ⎤
0
P1
⎤ P1 D 0
⎦ (t),
(8)
(9)
z˙ T (ϕ)Qz(ϕ)dϕ, ˙
(10)
z˙ T (ϕ)Rz(ϕ)dϕ. ˙
(11)
Moreover, it can be obtained by Lemma 2 that t z˙ T (ϕ)M z(ϕ)dϕ ˙ −¯ h ⎡
¯ t−h
z(t)
⎤T ⎡
−M
M −N
⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ≤ ⎢ z(t − d(t)) ⎥ ⎢ ∗ −2M + N + N T ⎣ ⎦ ⎣ ¯ z(t − h) ∗ ∗ t − d¯1 z˙ T (ϕ)Qz(ϕ)dϕ ˙ ⎡
t−d¯1
⎤T ⎡
z(t) −Q Q−S ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ≤ ⎢ z(t − d1 (t)) ⎥ ⎢ ∗ −2Q + S + S T ⎣ ⎦ ⎣ z(t − d¯1 ) ∗ ∗ t − d¯2 z˙ T (ϕ)Rz(ϕ)dϕ ˙ t−d¯2
10
⎤⎡
⎤ z(t)
N
⎥⎢ ⎥⎢ −N + M ⎥ ⎢ z(t − d(t)) ⎦⎣ ¯ z(t − h) −M
⎤⎡ z(t) ⎥⎢ ⎥⎢ −S + Q ⎥ ⎢ z(t − d1 (t)) ⎦⎣ −Q z(t − d¯1 ) S
⎥ ⎥ ⎥, ⎦
⎤ ⎥ ⎥ ⎥, ⎦
⎡
⎤T ⎡
z(t)
⎢ ⎢ ≤ ⎢ z(t − d2 (t)) ⎣ z(t − d¯2 )
⎥ ⎥ ⎥ ⎦
⎢ ⎢ ⎢ ⎣
⎤
⎤⎡
−R
R−W
∗
−2R + W + W T
∗
∗
W
z(t)
⎥⎢ ⎥⎢ −W + R ⎥ ⎢ z(t − d2 (t)) ⎦⎣ −R z(t − d¯2 )
Then, it follows that V˙ (t) + y T (t)y(t) − γ 2 T (t)(t) ⎡ ⎤ T C C 0 ⎦ z(t − d(t)) − γ 2 T (t)(t) =V˙ (t) + z T (t − d(t)) ⎣ 0 0 ⎡ ⎤ ˆ ˆ P A + P B K −P B K 1 1 1 ⎦ z(t) − 2z T (t)× ≤2z T (t) ⎣ 0 P2 A − P2 LC ⎡ ⎣
ˆ P1 B K
ˆ −P1 B K
0
0
t
t−d2 (t)
⎤ ⎦
t−d1 (t)
z(ϕ)dϕ ˙ − 2z T (t) ⎣
⎡
z(ϕ)dϕ ˙ + 2z T (t) ⎣
˙ − + d¯21 z˙ T (t)Qz(t)
⎡
t
t
t−d¯1 t
⎤
P1 D 0
0
0
0 −P2 LC
⎤ ⎦×
⎦ (t)
z˙ T (ϕ)dϕQ
t
t−d¯1 t
z(ϕ)dϕ ˙
+ d¯22 z˙ T (t)Rz(t) ˙ − z˙ T (ϕ)dϕR z(ϕ)dϕ ˙ t−d¯2 t−d¯2 ⎡ ⎤ CT C 0 ⎦ z(t − d(t)) − γ 2 T (t)(t) + z T (t − d(t)) ⎣ 0 0 ¯ 2 z˙ T (t)M z(t) ¯ =ς T (t)Πς(t) + d¯21 z˙ T (t)Qz(t) ˙ +h ˙ + d¯22 z˙ T (t)Rz(t), ˙ where ¯ z T (t − d1 (t)), z T (t − d¯1 ), ς(t) =[z T (t), z T (t − d(t)), z T (t − h), z T (t − d2 (t)), z T (t − d¯2 ), T (t)]T , ⎡ ⎤ ¯1 Π ¯2 Π ¯ =⎣ ⎦, Π ¯ ∗ Π3
11
⎥ ⎥ ⎥. ⎦
⎡ ¯1 = ⎣ Π
¯ 11 Π
¯ 12 Π
∗ ⎡
¯ 13 Π
¯ 11 = ⎣ Π
¯ 12 = ⎣ ⎣ Π
¯ 131 Π
¯ 132 Π
¯2 Π
¯ 21 Π
¯ 22 Π
¯3 Π
⎦,
2P1 A − M1 − Q1 − R1
⎡ ⎡
¯ 13 Π
⎤
0 M 1 − N1
0
2P2 A − M2 − Q2 − R2 ⎤ ⎡ ⎤ ⎤ 0 N1 0 ⎦ ⎣ ⎦ ⎦, M 2 − N2 0 N2
⎤ ⎦,
0 ¯ 131 , Π ¯ 132 , = Π ⎤ ⎡ ⎡ −2M1 + N1 + N1T + C T C 0 ⎦ ⎢ ⎣ ⎢ 0 −2M2 + N2 + N2T ⎢ ⎤T ⎡ =⎢ ⎢ ⎢ + M 0 −N 1 1 ⎣ ⎦ ⎣ 0 −N2 + M2 ⎤ ⎤ ⎡ ⎡ −N1 + M1 0 ⎢ ⎣ ⎦ ⎥ ⎢ ⎥ ⎢ ⎥ 0 −N2 + M2 ⎥, ⎡ ⎤ =⎢ ⎢ ⎥ ⎢ ⎥ M1 0 ⎣ ⎦ ⎣ ⎦ − 0 M2 ¯ 21 , Π ¯ 22 , = Π ⎤ ⎡ ⎤ ⎤ ⎡ ⎡ ˆ + Q1 − S1 −P1 B K ˆ P1 B K S1 0 ⎢ ⎣ ⎦ ⎣ ⎦ ⎥ ⎥ ⎢ ⎥ ⎢ 0 Q 2 − S2 0 S2 ⎥, ⎢ =⎢ ⎥ ⎥ ⎢ 0 0 ⎦ ⎣ 0 0 ⎤ ⎡ ⎤ ⎡ ⎡ R1 − W1 0 0 W1 ⎢ ⎣ ⎦ ⎣ ⎦ ⎢ ⎢ 0 −P2 LC + R2 − W2 0 W2 =⎢ ⎢ ⎢ 0 0 ⎣ 0 0 ⎡ ⎤ ¯ 31 Π ¯ 32 Π ⎦, =⎣ ¯ 33 ∗ Π
12
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
⎡ ⎣
⎤ ⎤ P1 D 0 0 0
⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
⎡ ⎡ ¯ 31 Π
⎢ ⎣ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣
¯ 32 =0, Π ⎡
¯ 33 Π
¯ 331 Π
−2Q1 + S1 + S1T
0
0
−2Q2 + S2 + S2T
⎤
⎡
⎦
⎣
∗
⎡
R1 − W1
0
⎤
¯ ⎢ Π ⎣ ⎦ 0 ⎢ 331 ⎢ 0 R2 − W2 ⎢ ⎤ ⎡ ⎢ =⎢ R1 0 ⎢ ∗ ⎦ 0 −⎣ ⎢ ⎢ 0 R2 ⎣ ∗ ∗ −γ 2 I ⎡ −2R1 + W1 + W1T 0 =⎣ 0 −2R2 + W2 + W2T
−S1 + Q1
0
−S2 + Q2 ⎤ Q1 0 ⎦ −⎣ 0 Q2 0 ⎡
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦ ⎤ ⎦.
Note that ¯ 2 M + d¯2 Q + d¯2 R)z(t) ˙ z˙ T (t)(h 1 2 ⎡ ⎡ ⎤ A¯T ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ ¯T ⎥ ⎢ ⎢ ⎢ ⎥ A 1 ⎥ ¯2 T 2 2 ¯ ¯ ⎢ ⎢ =ς (t) ⎢ ⎥ (h M + d1 Q + d2 R) ⎢ ⎢ 0 ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ ¯T ⎥ ⎢ ⎢ A2 ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎣ ⎣ ⎦ A¯T3
A¯T
⎤T
⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ A¯T1 ⎥ ⎥ ς(t). ⎥ 0 ⎥ ⎥ ⎥ A¯T2 ⎥ ⎥ ⎥ 0 ⎥ ⎦ A¯T3
Then, noting that −P M −1 P ≤ M − 2P , −P Q−1 P ≤ Q − 2P and −P R−1 P ≤ R − 2P , it can be verified by Schur complement lemma [ 33] that V˙ (t) + y T (t)y(t) −
90
γ 2 T (t)(t) < 0 is equivalent to Π < 0. Therefore, under zero initial condi∞ tion, one has V (0) = 0 and V (∞) ≥ 0, which implies that 0 y T (t)y(t)dt ≤ ∞ γ 2 0 T (t)(t)dt and completes the proof. Remark 2. In some cases where the state variables of the mechanical systems are difficult to obtain, the observer-based control scheme is needed to solve the H ∞ control 13
⎤ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
problem. ¯ Theorem 2. For given H ∞ performance γ, the upper bound of sampling period h, the upper bounds of network-induced delays σ ¯ and τ¯, the networked-observer-based can be solved,⎤ if there exist symmetric matrices P = H ⎡ ∞ problem ⎤ of system (5) ⎡ 0 0 P P ⎣ 1 ⎦ > 0, P1 = U T ⎣ 11 ⎦ U , P11 ∈ Rm×m , P22 ∈ R(n−m)×(n−m) , 0 P2 0 P22 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 M1 Q1 0 R1 0 ⎦ > 0, Q = ⎣ ⎦ > 0, R = ⎣ ⎦ > 0, M = ⎣ 0 M2 0 Q2 0 R2 ⎡ ⎤ ⎡ ⎤ S1 0 N1 0 ⎦ , N = ⎣ ⎦ and W = matrices G1 , G2 , matrices S = ⎣ 0 S2 0 N2 ⎤ ⎡ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Q S W1 0 M N R W ⎦ satisfying ⎣ ⎣ ⎦ ≥ 0, ⎣ ⎦ ≥ 0, ⎣ ⎦ ≥ 0 0 W2 ∗ Q ∗ M ∗ R and a constant γ > 0 such that the following inequalities hold: ⎤ ⎡ Ω 1 Ω2 ⎦ < 0, Ω := ⎣ ∗ Ω3
where
⎡ Ω1 = ⎣ ⎡ Ω11 = ⎣ ⎡ Ω111 = ⎣
Ω11
Ω12
∗
Ω13
⎤ ⎦,
Ω111
Ω112
∗
Ω113
⎤ ⎦,
2P1 A − M1 − Q1 − R1
⎡ ⎡ Ω112 = ⎣ ⎣
0 M 1 − N1 0
0
2P2 A − M2 − Q2 − R2 ⎤ ⎡ ⎤ ⎤ 0 N1 0 ⎦ ⎣ ⎦ ⎦, M 2 − N2 0 N2
Ω113 = [Ω1131 , Ω1132 ] ,
14
⎤ ⎦,
⎡ ⎡
Ω1131
⎢ ⎣ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣ ⎡ ⎡
Ω1132
⎢ ⎣ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣
−2M1 + N1 + N1T + C T C −N1 + M1
⎣
0
−N1 + M1
0
−N2 + M2 ⎤ M1 0 ⎦ −⎣ 0 M2 0 ⎡
⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
Ω12 = [Ω121 , Ω122 ] , ⎤ ⎡ ⎡ ⎡ BG1 + Q1 − S1 −BG1 S ⎢ ⎣ ⎦ ⎣ 1 ⎢ ⎢ 0 Q 2 − S2 0 Ω121 = ⎢ ⎢ ⎢ 0 ⎣ 0 ⎤ ⎡ ⎡ ⎡ R1 − W1 0 ⎢ ⎣ ⎦ ⎣ ⎢ ⎢ 0 −G2 C + R2 − W2 Ω122 = ⎢ ⎢ ⎢ 0 ⎣ 0 ⎡ ⎤ Ω131 Ω132 ⎦, Ω13 = ⎣ ∗ Ω133 ⎡ ⎡ −2Q1 + S1 + S1T 0 ⎢ ⎣ ⎢ ⎢ 0 −2Q2 + S2 + S2T Ω131 = ⎢ ⎢ ⎢ ⎣ ∗ Ω132 =0, ⎡
Ω1331
⎢ Ω1331 ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ∗ ⎢ ⎢ ⎣ ∗
⎡ ⎣
R1 − W1
⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
−2M2 + N2 + N2T ⎤T 0 ⎦ −N2 + M2 ⎤ ⎤
0
⎡
⎤ ⎤
0
S2 0 0
R2 − W2 ⎤ R1 0 ⎦ −⎣ 0 R2 0 ⎡
⎦
0
0
W2
15
0
0
0
⎤
0
⎦ ⎣
⎤ ⎤ P1 D 0
⎦ ⎣
0
⎤ ⎡
0
⎤ ⎡
−γ 2 I
∗
⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
W1
⎤
0
⎤ ⎤
0
⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦
−S1 + Q1
⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
0
−S2 + Q2 ⎤ Q1 0 ⎦ −⎣ 0 Q2 0 ⎡
⎤ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
⎡ Ω1331 = ⎣
⎤
−2R1 + W1 + W1T
0
0
−2R2 + W2 + W2T
Ω2 = [Ω21 , Ω22 , Ω23 ] , ⎡ ⎡
AT P1
0
⎦,
⎤ ⎤
⎦ ⎥ ⎢ h⎣ ⎢ ⎥ ⎢ ⎥ 0 AT P2 ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎡ ⎤ ⎥ ⎢ ⎢ ⎥ T T ⎢ ⎥ 0 G1 B ⎦ ⎥ ⎢ h⎣ ⎢ ⎥ Ω21 =Ω22 = Ω23 = ⎢ ⎥, −GT1 B T 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎤ ⎥ ⎡ ⎢ ⎢ ⎥ ⎢ ⎥ 0 0 ⎦ ⎥ ⎢ h⎣ ⎢ ⎥ T T ⎢ ⎥ 0 −C G2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎣ ⎦
T h D P1 0 ⎡ ⎤ ⎡ ⎤ ⎡ M1 0 P1 0 Q ⎦−2⎣ ⎦,⎣ 1 Ω3 =diag{⎣ 0 M2 0 P2 0 ⎤ ⎡ ⎤ ⎡ 0 0 P R ⎦−2⎣ 1 ⎦}. ⎣ 1 0 R2 0 P2
0 Q2
⎤
⎡
⎦−2⎣
P1
0
0
P2
⎤ ⎦
ˆ and the state observer gain L can be obtained by Moreover, the controller gain K solving the following equations: ˆ =V B −1 P −1 BV T G1 , K 11 L =P2−1 G2 . 95
Proof. Under the condition of Assumption 4 and based on Lemma 1, the proof follows directly from Theorem 1. Remark 3. It is worth mentioning that the established results impose no constrains on the derivatives of the network-induced time-varying delays, which is applicable for the real world applications.
16
100
Remark 4. It should be pointed out that by utilizing the separation principle, the network-based controller and observer can be both designed by the LMI approach, which can be solved by MATLAB LMI toolbox.
4. Numerical Examples In this section, the following two illustrative examples are presented to verify our 105
theoretical results. Example 1. Consider the following system described in ( 1) as: ⎧ ⎨ x(t) ˙ = Ax(t) + Bu(t) + D(t) , ⎩ y(t) = Cx(tk ), tk ≤ t < tk+1 where
⎡ A =⎣
C=
0
0.1
−0.2 0.6 0 1
⎤
⎡
⎦,B = ⎣ ⎡
,D = ⎣
1 1
⎤
0 2
(12)
⎤ ⎦,
⎦.
In this example, the sampling period is assumed to be 0.02s and the networkinduced time-varying delays are given as σ(t) = 0.01 + 0.01 sin t and τ (t) = 0.01 + 0.01 cos t with σ ¯ = 0.02 and τ¯ = 0.02. The external disturbance is set as (t) = 0.2 sin 10t. Table 1 shows the minimum allowable γ for different upper bounds of 110
network-induced delays. Table 1: Minimum allowable γ
τ¯
0
0.02
0.04
0
0.18
0.34
0.35
0.02
0.34
0.36
0.36
0.04
0.36
0.37
0.38
σ ¯
17
By choosing γ = 0.4 and solving the LMIs in Theorem 2 with the above paˆ = rameters, the desired controller gain and state observer gain are obtained as K [−0.2286, −0.4470] and L = [0.0067, 0.8973]T , respectively. 0.5
x 10
29
0
State response of x(t)
-0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 0
20
40
60 Time (s)
80
100
120
Figure 1: State response of x(t) without control input
2.5 2 1.5
State response of x(t)
1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 0
20
40
60 Time (s)
80
100
Figure 2: State response of x(t) with control input
18
120
0.2
Networked-observer error e(tk )
0
-0.2
-0.4
-0.6
-0.8
-1 0
20
40
60 Time (s)
80
100
120
Figure 3: Networked-observer error e(tk )
Figure 1 shows the state response of the open-loop system and Figures 2-3 depict 115
the state response of the closed-loop system and the networked-observer error, which implies that our designed state observer and controller are applicable to the H ∞ control problem. Example 2. Consider the network-based boring bar system with equivalent model depicted in Figure 4. The mathematical model is written as follows: m
k2
x2(t)
P c2 T
M
x1(t)
k1 c1
Figure 4: An equivalent model of boring bar system
19
⎧ ⎨ Mx ¨1 (t) + (c1 + c2 )x˙ 1 (t) − c2 x˙ 2 (t) + (k1 + k2 )x1 (t) − kx2 (t) = F (t) + f (t), ⎩ m¨ x (t) + c x˙ (t) − c x˙ (t) + k x (t) − k x (t) = −F (t), 2
2 2
2 1
2 2
2 1
(13) 120
where x1 (t) and x2 (t) are the equivalent displacements, F (t) = −ικV is the control force, V denotes the piezoelectric voltage, f (t) is the unknown cutting force which can be considered as external disturbance, M and m denote the equivalent masses, k 1 and k2 represent the equivalent stiffness, c 1 and c2 denote the equivalent damping. Let ke = k2 + ι, x˙ 1 (t) = x3 (t) and x˙ 2 (t) = x4 (t). Then, system (13) can be ⎧ ⎨ x(t) ˙ = Ax(t) + Bu(t) + D(t) , ⎩ y(t) = Cx(t ), t ≤ t < t
rewritten as
k
where
⎡
0
k
0
k+1
I
0
⎤
⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 I ⎥, A =⎢ ⎥ ⎢ ⎢ −(k1 + ke )/M ke /M −(c1 + c2 )/M c2 /M ⎥ ⎦ ⎣ −ke /m c2 /m −c2 /m ke /m ⎤ ⎡ ⎤ ⎡ 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢
⎢ 0 ⎥ ⎥ ⎢ 0 ⎥, ⎢ ⎥ ⎢ B =⎢ ⎥ ⎥,C = 0 1 0 0 ,D = ⎢ ⎢ 1/M ⎥ ⎢ −ικ/M ⎥ ⎦ ⎣ ⎦ ⎣ 0 ικ/m x(t) =[xT1 (t), xT2 (t), xT3 (t), xT4 (t)]T , u(t) = V, (t) = f (t). Therefore, the H ∞ control problem of the above network-based boring bar system can 125
be solved by our designed methodology. In this example, the parameters are set as M =1.97kg, m=0.197kg, ι = 1/5k 1 , ke = ι, κ =2×10−8m/V, k1 =15×105N/m, c1 = 2k1 ε/wn , c2 = 2ke ε/wn , ε = 0.01, wn = 873.3r/s. Moreover, the external disturbance is chosen by (t) = 0.2 sin 10t. Set the sampling period as 0.01s and the upper bound of network-induced delays is assumed
130
to be 0.01s. The H∞ performance parameter γ is given as 1. The controller gain ˆ = [199740, −180960, 5162, −15726] and and state observer gain are obtained as K 20
L = [820, 24080, −7715, −10344]T , respectively. Figure 5 shows the state response of x1 (t), x2 (t) without control input while Figures 6-7 depict the state response of x 1 (t), x2 (t), and observer error e(t k ) with control input and observer, which demonstrates the effectiveness of the obtained results. 14
x 10
237
12
State response of x(t)
10 8 6 4 2 0 -2 0
0.2
0.4
0.6
0.8
1 1.2 Time (s)
1.4
1.6
1.8
2
Figure 5: State response of x1 (t) and x2 (t) without control input
6 5 4 State response of x(t)
135
3 2 1 0 -1 -2 -3 -4 0
0.5
1
1.5
2
2.5 3 Time (s)
3.5
4
4.5
Figure 6: State response of x1 (t) and x2 (t) with control input
21
5
Networked-observer error e(tk )
1 0.8 0.6 0.4 0.2 0 -0.2 0
1
2
3
4
5
Time (s)
Figure 7: Networked-observer error e(tk )
5. Conclusion This paper deals with the observer-based H ∞ control problem for a class of networked control systems with time-varying network-induced delays. A novel observerbased control scheme is given based on the network-based Luenberger-type observer. 140
By utilizing the Lyapunov-Krasovskii functional method, delay-dependent criteria are first developed to guarantee the prescribed H ∞ performance. Based on the obtained results, the controller and the observer are further designed. In the end, two illustrative examples are provided to verify the effectiveness of the designed approaches. Our future study will focus on extending the obtained results in this paper to the cases with
145
randomly occurring parameters, which is more complicated in the networked environment.
Acknowledgements This work was partially supported by the State Key Program of National Natural Science of China(Grant No. 51235003), National Natural Science Foundation
22
150
of China (51275139,51375127,51205095) and Natural Science Foundation of Heilongjiang Province of China (F201216).
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PII: DOI: Reference:
S0925-2312(15)01884-6 http://dx.doi.org/10.1016/j.neucom.2015.11.075 NEUCOM16475
To appear in: Neurocomputing Received date: 17 July 2015 Revised date: 28 October 2015 Accepted date: 24 November 2015 Cite this article as: Lijia Liu, Xianli Liu, Chuntao Man and Chengyang Xu, Delayed Observer-based H∞ Control for Networked Control Systems, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.11.075 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Delayed Observer-based H∞ Control for Networked Control Systems Lijia Liua,c,∗, Xianli Liua , Chuntao Man b , Chengyang Xu a a The
Lab of National and Local United Engineering for High-Efficiency Cutting & Tools, Harbin University of Science and Technology, China b Harbin University of Science and Technology, China c Institute of Electrical Engineering, Heilongjiang Polytechnic, China
Abstract In this paper, the observer-based H ∞ control problem for networked control systems is investigated. Moreover, network-induced time-varying delays are considered to model the real world applications. The Luenberger-type observer is first designed for state estimation, then the observer-based controller is further developed. By applying the Lyapunov-Krasovskii functional approach, delay-dependent conditions are established to ensure that the prescribed H ∞ performance can be achieved. Finally, the effectiveness and usefulness of our obtained theoretical results are demonstrated by two examples. Keywords: Networked-observer; Network-based H ∞ control; Networked control systems.
1. Introduction The past decade has witnessed rapid developments of network and communication technologies, which gives rise to the networked control approaches in the real world applications [1–6]. Generally speaking, networked control systems are composed of 5
the communication network, the sensors, the controllers and the actuators [ 7–10]. In contrast to classical control systems, significant advantages can be obtained by networked control systems including high reliability, ease of maintenance, low cost, etc ∗ Corresponding
author Email addresses: [email protected] (Lijia Liu), [email protected] (Xianli Liu), [email protected] (Chuntao Man), [email protected] (Chengyang Xu)
Preprint submitted to Neurocomputing
December 10, 2015
[11–13]. As a consequence, many remarkable theoretical and practical methodologies are developed and can be found in the literature; see e.g. [ 14–17]. 10
On another research front line, observers are often needed in some control engineering practice since certain states are difficult or even unavailable to measure [ 18– 22]. Under this context, observer-based control schemes are developed to solve this problem. Moreover, it is now well recognized that the observer and the controller can be determined independently, such that the effectiveness of the system design can be
15
increased [23, 24]. It should be pointed out that in the networked control environment, the system output measurements are in the discrete-time form according to the sampling periods of the zero-order holds (ZOH). Moreover, the network-induced timevarying delays should be taken into account due to communication constraints, and they may degrade or destroy the control system performance [ 25–30]. So far, to the
20
best of the authors’ knowledge, the observer-based H ∞ control problem for networked control systems with time-varying delays has not been fully addressed in the reported works and remains open. As a result, the aim of this paper is to shorten such a gap. Motivated by the discussion made by far, in this paper, we aim to solve the observerbased H∞ control problem for networked control systems. In particular, the Luenberger-
25
type observer is introduced to obtain the estimated state. By employing the LyapunovKrasovskii functional approach, delay-dependent conditions are established in terms of linear matrix inequalities (LMIs) to guarantee the pre-specified H ∞ disturbance attenuation level. Based on the developed results, the observer along with the state feedback controller are designed. Compared with the existing results, the main contributions of
30
our paper lie in: (i) a unified observer-based control framework is developed for the networked control systems. (ii) the phenomena of network-induced time-varying delays which can model the networked engineering practice are considered. The remainder of this paper is organized as follows. In Section 2, the model of networked control systems under consideration is introduced and the observer-based
35
H∞ problem is formulated. Section 3 gives the main results on the design procedure of networked observer and controller. In Section 4, two examples are presented to show the effectiveness of the proposed algorithm. Finally, concluding remarks are given in Section 5. 2
Notation: The notation in our paper is standard. R n denotes the n dimensional 40
Euclidean space, R m×n represents the set of all m × n real matrices. I and 0 denote identity matrix and zero matrix with appropriate dimensions, respectively. The notation P > 0 means that P is real symmetric and positive definite, and the superscript “T ” denotes matrix transposition. In symmetric block matrices, * is used as an ellipsis for the terms which are introduced by symmetry and diag{· · · } denotes a block-diagonal
45
matrix. Finally, if not explicitly stated, all matrices are supposed to have compatible dimensions.
2. Problem Formulation and Preliminaries Consider the following class of networked control systems: ⎧ ⎨ x(t) ˙ = Ax(t) + Bu(t) + D(t) , ⎩ y(t) = Cx(t ), t ≤ t < t k
k
(1)
k+1
where x(t) ∈ Rn denotes the state vector, u(t) ∈ R m is the networked control input to be designed, (t) ∈ L 2 [0, ∞) represents the external disturbance, y(t) ∈ R p de50
notes the output measurement. All A, B, C, D are constant matrices of appropriate dimensions which represent the linear dynamics. In some network environment, x(t) is difficult to measure, such that the state observer is required. Consequently, a networked Luenberger-like state observer can be constructed as follows: ⎧ ⎨ x ˆ˙ (t) = Aˆ x(t) + Bu(t) + D(t) + L(y(tk ) − yˆ(tk )), tk ≤ t < tk+1 , ⎩ yˆ(t) = C x ˆ(t ),
(2)
k
where x ˆ(tk ) denotes the estimate of x(t k ) in the networked environments and L ∈ 55
Rn×p is the gain matrix of the state observer. Moreover, the following assumptions are given for the networked state observer. Assumption 1. The sensor is clock-driven, the ZOH is event-driven. The updating instants of ZOH are at tk , k = 1, 2, . . . , ∞. The sampling period of the sensor is h(t) ¯ where ¯h is a positive constant. satisfying 0 < h(t) ≤ h, 3
60
Assumption 2. The data is transmitted by a single-packet. Assumption 3. The time-varying delays from the networked control system to the networked state observer at the updating instant t k can be considered as τ k . Moreover, it is assumed that 0 ≤ τk ≤ τ¯, k = 1, 2, . . . , ∞, where τ¯ is a positive constant. Assumption 4. The matrix B is of full-column rank, then there exist two orthogonal matrices U ∈ Rn×n and V ∈ Rm×m , such that ⎡ ⎤ B ⎦V T, B=U⎣ 0
(3)
where B = diag{b 1 , b2 , . . . , bm }, bi (i = 1, 2, . . . , m) are nonzero singular values of 65
B. As a result, the observer error system can be obtained as follows: e(t) ˙ = Ae(t) − LCe(tk − τk ), tk ≤ t < tk+1 , where e(t) := x(t) − x ˆ(t) denotes the observer error. Then, in order to solve the H ∞ control problem, the following controller is designed: ˆx u(t) = K ˆ(tk ), tk ≤ t < tk+1 ,
(4)
ˆ ∈ Rm×n is the controller gain to be determined later. where K Note that (4) can be further rewritten as ˆ k ). ˆ k ) − Ke(t u(t) = Kx(t In addition, the time-varying delay from the sensor to the controller at the updating instant tk is σk , k = 1, 2, . . . , ∞, where 0 ≤ σk ≤ σ ¯ , with σ ¯ being a positive constant. Based on the above analysis, the following augmented system can be formed: ⎧ ⎨ z(t) ¯ ˙ = Az(t) + A¯1 z(tk − σk ) + A¯2 z(tk − τk ) + A¯3 (t) , (5) ⎩ y(t) = Cz(t ¯ ), t ≤ t < t k
k
k+1
4
where z(t) =[xT (t), eT (t)]T , ⎡ ⎤ A 0 ⎦, A¯ = ⎣ 0 A ⎡ ⎤ ˆ −B K ˆ BK ⎦, A¯1 = ⎣ 0 0 ⎡ A¯2 = ⎣ ⎡ A¯3 = ⎣ C¯ =
⎤
0
0
0
−LC ⎤
D
⎦,
0
0
C
⎦,
.
By utilizing the input delay approach, (5) can be rewritten as ⎧ ⎨ z(t) ¯ ˙ = Az(t) + A¯1 z(t − d1 (t)) + A¯2 z(t − d2 (t)) + A¯3 (t) , ⎩ y(t) = Cz(t ¯ − d(t)), t ≤ t < t k
(6)
k+1
where d(t) := t − tk , d1 (t) := t − tk + σk and d2 (t) := t − tk + τk denote the virtual delays satisfying ¯ 0 ≤ d(t) ≤ h, ¯, 0 ≤ d1 (t) ≤ d¯1 := ¯h + σ 0 ≤ d2 (t) ≤ d¯2 := ¯h + τ¯. 70
Remark 1. It is worth mentioning that the sampling period is assumed to be nonuniform in this paper. The upper bound of all the possible sampling periods is given as ¯ which means that the sampling periods can very randomly within the range of (0, ¯h]. h, It can be found that the non-uniform sampling period is more applicable and practical than the uniform cases with fixed sampling period.
75
Before proceeding, the following definition and lemmas are essential in establishing the main results. 5
Definition 1. System (5) is said to satisfy the H∞ performance if under zero initial condition there exists a positive constant γ such that it holds that ∞ ∞ y T (t)y(t)dt < γ 2 T (t)(t)dt, 0
0
for all nonzero (t) ∈ L 2 [0, ∞), t > 0. Lemma 1. [31] For the matrix B ∈ R n×m that is of full-column rank, if matrix Υ is of the following structure:
⎡ Υ = UT ⎣
Υ11
0
0
Υ22
⎤ ⎦ U,
where Υ11 ∈ Rm×m and Υ22 ∈ R(n−m)×(n−m) , U is defined in (3), then there exists a nonsingular matrix such that ΥB = B. ⎡ ⎤ M S ⎦ ≥ 0, scalars τ > 0, τ (t) satisfying 0 ≤ Lemma 2. [32] For any matrix ⎣ ∗ M τ (t) ≤ τ , vector function x(t) ˙ : [−τ, 0] → R n such that the concerned integrations are well defined, then −τ
t
t−τ
x˙ T (s)M x(s)ds ˙ ≤ ζ T (t)Ωζ(t),
where ζ(t) =[xT (t), xT (t − τ (t)), xT (t − τ )]T , ⎡ −M M −S S ⎢ ⎢ Ω =⎢ ∗ −2M + S + S T −S + M ⎣ ∗ ∗ −M 80
⎤ ⎥ ⎥ ⎥. ⎦
The purpose of this paper is to design a networked state feedback controller ( 4) and a networked state observer (2) to ensure that the system (5) is asymptotically stable while achieving the H ∞ performance γ. 3. Main Results In this section, delay-dependent criteria are developed to solve the networked-
85
observer-based H ∞ problem by the LMI techniques. Both controller gain and state observer gain can be obtained by the following established conditions. 6
¯ Theorem 1. For given H ∞ performance γ, the upper bound of sampling period h, the upper bounds of network-induced delays σ ¯ and τ¯, the networked-observer-based ˆ H∞ problem of system (5) can be solved with the given controller ⎡ gain K and ⎤ the P1 0 ⎦ > 0, given state observer gain L, if there exist symmetric matrices P = ⎣ 0 P2 ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 0 Q1 0 R1 0 M1 ⎦ > 0, Q = ⎣ ⎦ > 0, R = ⎣ ⎦ > 0, M = ⎣ 0 M2 0 Q2 0 R2 ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 0 0 N W S1 0 ⎦,N =⎣ 1 ⎦ and W = ⎣ 1 ⎦ satisfying matrices S = ⎣ 0 S2 0 N2 0 W2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Q S M N R W ⎣ ⎦ ≥ 0, ⎣ ⎦ ≥ 0, ⎣ ⎦ ≥ 0 and a constant γ > 0 such that ∗ Q ∗ M ∗ R the following inequalities hold: ⎤ ⎡ Π1 Π2 ⎦ < 0, Π := ⎣ ∗ Π3 where
⎡ Π1 = ⎣ ⎡ Π11 = ⎣
Π11
Π12
∗
Π13
⎤ ⎦,
Π111
Π112
∗
Π113
⎡ Π111 = ⎣
⎤ ⎦,
2P1 A − M1 − Q1 − R1
⎡ ⎡ Π112 = ⎣ ⎣
0 M 1 − N1 0
0
2P2 A − M2 − Q2 − R2 ⎤ ⎡ ⎤ ⎤ 0 N1 0 ⎦ ⎣ ⎦ ⎦, M 2 − N2 0 N2
⎤ ⎦,
Π113 = [Π1131 , Π1132 ] , ⎤ ⎤ ⎡ ⎡ −2M1 + N1 + N1T + C T C 0 ⎦ ⎥ ⎢ ⎣ ⎥ ⎢ 0 −2M2 + N2 + N2T ⎥ ⎢ ⎥, ⎢ ⎤ ⎡ Π1131 = ⎢ T ⎥ ⎥ ⎢ + M 0 −N 1 1 ⎦ ⎣ ⎦ ⎣ 0 −N2 + M2
7
⎡ ⎡
Π1132
⎢ ⎣ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣
−N1 + M1
⎤ ⎤
0
−N2 + M2 ⎤ M1 0 ⎦ −⎣ 0 M2 0 ⎡
⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
Π12 = [Π121 , Π122 ] , ⎤ ⎡ ⎤ ⎤ ⎡ ⎡ ˆ + Q1 − S1 −P1 B K ˆ P1 B K S1 0 ⎢ ⎣ ⎦ ⎣ ⎦ ⎥ ⎥ ⎢ ⎥ ⎢ 0 Q 2 − S2 0 S2 ⎥, ⎢ Π121 = ⎢ ⎥ ⎥ ⎢ 0 0 ⎦ ⎣ 0 0 ⎤ ⎡ ⎤ ⎡ ⎡ ⎡ ⎤ ⎤ R1 − W1 0 W1 0 P1 D ⎢ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎥ ⎢ ⎥ ⎢ ⎥ 0 −P2 LC + R2 − W2 0 W2 0 ⎢ ⎥, Π122 = ⎢ ⎥ ⎢ ⎥ 0 0 0 ⎣ ⎦ 0 0 0 ⎡ ⎤ Π131 Π132 ⎦, Π13 = ⎣ ∗ Π133 ⎤ ⎡ ⎤ ⎤ ⎡ ⎡ −2Q1 + S1 + S1T 0 0 −S1 + Q1 ⎢ ⎣ ⎦ ⎣ ⎦ ⎥ ⎢ ⎥ T ⎢ ⎥ 0 −2Q2 + S2 + S2 0 −S2 + Q2 ⎢ ⎥, ⎡ ⎤ Π131 = ⎢ ⎥ ⎢ ⎥ Q1 0 ⎣ ⎦ ⎣ ⎦ ∗ − 0 Q2 Π132 =0, ⎡
Π133
Π1331
⎡
R1 − W1
⎤
0
⎤
⎢ Π1331 ⎣ ⎦ 0 ⎥ ⎢ ⎥ ⎢ ⎥ 0 R2 − W2 ⎢ ⎥ ⎤ ⎡ ⎢ ⎥ =⎢ ⎥, R1 0 ⎢ ∗ ⎥ ⎦ ⎣ 0 − ⎢ ⎥ ⎢ ⎥ 0 R2 ⎣ ⎦ ∗ ∗ −γ 2 I ⎡ ⎤ −2R1 + W1 + W1T 0 ⎦, =⎣ 0 −2R2 + W2 + W2T
Π2 = [Π21 , Π22 , Π23 ] ,
8
⎡
⎡
AT P1
⎤ ⎤
0
⎦ ⎥ ⎢ h⎣ ⎢ ⎥ ⎢ 0 AT P2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎡ ⎤ ⎥ ⎢ ⎢ ⎥ ˆ T BT P T 0 ⎢ ⎥ K 1 ⎦ ⎥ ⎢ h⎣ ⎢ ⎥ T T T ˆ B P1 0 Π21 =Π22 = Π23 = ⎢ ⎥, K ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎤ ⎥ ⎢ ⎡ ⎢ ⎥ ⎢ ⎥ 0 0 ⎦ ⎥ ⎢ h⎣ ⎢ ⎥ ⎢ ⎥ 0 −C T LT P2T ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎣ ⎦
h DT P1 0 ⎡ ⎤ ⎡ ⎤ ⎡ M1 0 P1 0 Q ⎦−2⎣ ⎦,⎣ 1 Π3 =diag{⎣ 0 M2 0 P2 0 ⎤ ⎡ ⎤ ⎡ 0 0 P R ⎦−2⎣ 1 ⎦}. ⎣ 1 0 R2 0 P2
0 Q2
⎤
⎡
⎦−2⎣
P1
0
0
P2
⎤ ⎦
Proof. Choose the following Lyapunov-Krasovskii function V (t) =
4
Vi (t),
(7)
i=1
where V1 (t) = z T (t)P z(t), 0 t ¯ V2 (t) = h z˙ T (η)M z(η)dηdϕ, ˙ V3 (t) = d¯1 V4 (t) = d¯2
¯ t+ϕ −h 0 t
−d¯1 0
t+ϕ t
−d¯2
t+ϕ
z˙ T (η)Qz(η)dηdϕ, ˙ z˙ T (η)Rz(η)dηdϕ. ˙
By taking the derivative of (7) along the solution of system (6), one has V˙ 1 (t) = 2z T (t)P z(t), ˙ ⎡ ⎡ ⎤ P P1 0 1 ¯ ⎦ Az(t) = 2z T (t) ⎣ + 2z T (t) ⎣ 0 P2 0 9
0 P2
⎤ ⎦ A¯1 z(t − d1 (t))
⎡ + 2z T (t) ⎣ ⎡ = 2z T (t) ⎣
0
0
P2
P1
0
0
P2
⎦ A¯2 z(t − d2 (t)) + 2z T (t) ⎣ ⎤⎡ ⎦⎣ ⎡
z(t − d1 (t)) + 2z T (t) ⎣ ⎡ = 2z T (t) ⎣
⎡
⎤
P1
P1 A
0
0
P2 A ⎡
+ 2z T (t) ⎣
A
0
0
A
0
0
P2
⎦ z(t) + 2z T (t) ⎣
⎤ ⎦ A¯3 (t)
ˆ P1 B K
ˆ −P1 B K
0
0
⎤
0
⎡
⎡
ˆ P1 B K
ˆ −P1 B K
0
0
⎦ z(t − d2 (t)) + 2z T (t) ⎣ 0 −P2 LC t ¯ 2 z˙ T (t)M z(t) ¯ ˙ −h z˙ T (ϕ)M z(ϕ)dϕ, ˙ V˙ 2 (t) = h ˙ − d¯1 V˙ 3 (t) = d¯21 z˙ T (t)Qz(t) V˙ 4 (t) = d¯22 z˙ T (t)Rz(t) ˙ − d¯2
t−d¯1 t t−d¯2
⎦×
0
⎦ (t)
⎦ z(t − d1 (t))
⎡
⎤
¯ t−h t
⎤
⎤
⎤
P1 D
⎦ z(t − d2 (t)) + 2z T (t) ⎣
⎦ z(t) + 2z T (t) ⎣
0
0
⎡
⎤
0 −P2 LC ⎤
0
P1
⎤ P1 D 0
⎦ (t),
(8)
(9)
z˙ T (ϕ)Qz(ϕ)dϕ, ˙
(10)
z˙ T (ϕ)Rz(ϕ)dϕ. ˙
(11)
Moreover, it can be obtained by Lemma 2 that t z˙ T (ϕ)M z(ϕ)dϕ ˙ −¯ h ⎡
¯ t−h
z(t)
⎤T ⎡
−M
M −N
⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ≤ ⎢ z(t − d(t)) ⎥ ⎢ ∗ −2M + N + N T ⎣ ⎦ ⎣ ¯ z(t − h) ∗ ∗ t − d¯1 z˙ T (ϕ)Qz(ϕ)dϕ ˙ ⎡
t−d¯1
⎤T ⎡
z(t) −Q Q−S ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ≤ ⎢ z(t − d1 (t)) ⎥ ⎢ ∗ −2Q + S + S T ⎣ ⎦ ⎣ z(t − d¯1 ) ∗ ∗ t − d¯2 z˙ T (ϕ)Rz(ϕ)dϕ ˙ t−d¯2
10
⎤⎡
⎤ z(t)
N
⎥⎢ ⎥⎢ −N + M ⎥ ⎢ z(t − d(t)) ⎦⎣ ¯ z(t − h) −M
⎤⎡ z(t) ⎥⎢ ⎥⎢ −S + Q ⎥ ⎢ z(t − d1 (t)) ⎦⎣ −Q z(t − d¯1 ) S
⎥ ⎥ ⎥, ⎦
⎤ ⎥ ⎥ ⎥, ⎦
⎡
⎤T ⎡
z(t)
⎢ ⎢ ≤ ⎢ z(t − d2 (t)) ⎣ z(t − d¯2 )
⎥ ⎥ ⎥ ⎦
⎢ ⎢ ⎢ ⎣
⎤
⎤⎡
−R
R−W
∗
−2R + W + W T
∗
∗
W
z(t)
⎥⎢ ⎥⎢ −W + R ⎥ ⎢ z(t − d2 (t)) ⎦⎣ −R z(t − d¯2 )
Then, it follows that V˙ (t) + y T (t)y(t) − γ 2 T (t)(t) ⎡ ⎤ T C C 0 ⎦ z(t − d(t)) − γ 2 T (t)(t) =V˙ (t) + z T (t − d(t)) ⎣ 0 0 ⎡ ⎤ ˆ ˆ P A + P B K −P B K 1 1 1 ⎦ z(t) − 2z T (t)× ≤2z T (t) ⎣ 0 P2 A − P2 LC ⎡ ⎣
ˆ P1 B K
ˆ −P1 B K
0
0
t
t−d2 (t)
⎤ ⎦
t−d1 (t)
z(ϕ)dϕ ˙ − 2z T (t) ⎣
⎡
z(ϕ)dϕ ˙ + 2z T (t) ⎣
˙ − + d¯21 z˙ T (t)Qz(t)
⎡
t
t
t−d¯1 t
⎤
P1 D 0
0
0
0 −P2 LC
⎤ ⎦×
⎦ (t)
z˙ T (ϕ)dϕQ
t
t−d¯1 t
z(ϕ)dϕ ˙
+ d¯22 z˙ T (t)Rz(t) ˙ − z˙ T (ϕ)dϕR z(ϕ)dϕ ˙ t−d¯2 t−d¯2 ⎡ ⎤ CT C 0 ⎦ z(t − d(t)) − γ 2 T (t)(t) + z T (t − d(t)) ⎣ 0 0 ¯ 2 z˙ T (t)M z(t) ¯ =ς T (t)Πς(t) + d¯21 z˙ T (t)Qz(t) ˙ +h ˙ + d¯22 z˙ T (t)Rz(t), ˙ where ¯ z T (t − d1 (t)), z T (t − d¯1 ), ς(t) =[z T (t), z T (t − d(t)), z T (t − h), z T (t − d2 (t)), z T (t − d¯2 ), T (t)]T , ⎡ ⎤ ¯1 Π ¯2 Π ¯ =⎣ ⎦, Π ¯ ∗ Π3
11
⎥ ⎥ ⎥. ⎦
⎡ ¯1 = ⎣ Π
¯ 11 Π
¯ 12 Π
∗ ⎡
¯ 13 Π
¯ 11 = ⎣ Π
¯ 12 = ⎣ ⎣ Π
¯ 131 Π
¯ 132 Π
¯2 Π
¯ 21 Π
¯ 22 Π
¯3 Π
⎦,
2P1 A − M1 − Q1 − R1
⎡ ⎡
¯ 13 Π
⎤
0 M 1 − N1
0
2P2 A − M2 − Q2 − R2 ⎤ ⎡ ⎤ ⎤ 0 N1 0 ⎦ ⎣ ⎦ ⎦, M 2 − N2 0 N2
⎤ ⎦,
0 ¯ 131 , Π ¯ 132 , = Π ⎤ ⎡ ⎡ −2M1 + N1 + N1T + C T C 0 ⎦ ⎢ ⎣ ⎢ 0 −2M2 + N2 + N2T ⎢ ⎤T ⎡ =⎢ ⎢ ⎢ + M 0 −N 1 1 ⎣ ⎦ ⎣ 0 −N2 + M2 ⎤ ⎤ ⎡ ⎡ −N1 + M1 0 ⎢ ⎣ ⎦ ⎥ ⎢ ⎥ ⎢ ⎥ 0 −N2 + M2 ⎥, ⎡ ⎤ =⎢ ⎢ ⎥ ⎢ ⎥ M1 0 ⎣ ⎦ ⎣ ⎦ − 0 M2 ¯ 21 , Π ¯ 22 , = Π ⎤ ⎡ ⎤ ⎤ ⎡ ⎡ ˆ + Q1 − S1 −P1 B K ˆ P1 B K S1 0 ⎢ ⎣ ⎦ ⎣ ⎦ ⎥ ⎥ ⎢ ⎥ ⎢ 0 Q 2 − S2 0 S2 ⎥, ⎢ =⎢ ⎥ ⎥ ⎢ 0 0 ⎦ ⎣ 0 0 ⎤ ⎡ ⎤ ⎡ ⎡ R1 − W1 0 0 W1 ⎢ ⎣ ⎦ ⎣ ⎦ ⎢ ⎢ 0 −P2 LC + R2 − W2 0 W2 =⎢ ⎢ ⎢ 0 0 ⎣ 0 0 ⎡ ⎤ ¯ 31 Π ¯ 32 Π ⎦, =⎣ ¯ 33 ∗ Π
12
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
⎡ ⎣
⎤ ⎤ P1 D 0 0 0
⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
⎡ ⎡ ¯ 31 Π
⎢ ⎣ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣
¯ 32 =0, Π ⎡
¯ 33 Π
¯ 331 Π
−2Q1 + S1 + S1T
0
0
−2Q2 + S2 + S2T
⎤
⎡
⎦
⎣
∗
⎡
R1 − W1
0
⎤
¯ ⎢ Π ⎣ ⎦ 0 ⎢ 331 ⎢ 0 R2 − W2 ⎢ ⎤ ⎡ ⎢ =⎢ R1 0 ⎢ ∗ ⎦ 0 −⎣ ⎢ ⎢ 0 R2 ⎣ ∗ ∗ −γ 2 I ⎡ −2R1 + W1 + W1T 0 =⎣ 0 −2R2 + W2 + W2T
−S1 + Q1
0
−S2 + Q2 ⎤ Q1 0 ⎦ −⎣ 0 Q2 0 ⎡
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦ ⎤ ⎦.
Note that ¯ 2 M + d¯2 Q + d¯2 R)z(t) ˙ z˙ T (t)(h 1 2 ⎡ ⎡ ⎤ A¯T ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ ¯T ⎥ ⎢ ⎢ ⎢ ⎥ A 1 ⎥ ¯2 T 2 2 ¯ ¯ ⎢ ⎢ =ς (t) ⎢ ⎥ (h M + d1 Q + d2 R) ⎢ ⎢ 0 ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ ¯T ⎥ ⎢ ⎢ A2 ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎣ ⎣ ⎦ A¯T3
A¯T
⎤T
⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ A¯T1 ⎥ ⎥ ς(t). ⎥ 0 ⎥ ⎥ ⎥ A¯T2 ⎥ ⎥ ⎥ 0 ⎥ ⎦ A¯T3
Then, noting that −P M −1 P ≤ M − 2P , −P Q−1 P ≤ Q − 2P and −P R−1 P ≤ R − 2P , it can be verified by Schur complement lemma [ 33] that V˙ (t) + y T (t)y(t) −
90
γ 2 T (t)(t) < 0 is equivalent to Π < 0. Therefore, under zero initial condi∞ tion, one has V (0) = 0 and V (∞) ≥ 0, which implies that 0 y T (t)y(t)dt ≤ ∞ γ 2 0 T (t)(t)dt and completes the proof. Remark 2. In some cases where the state variables of the mechanical systems are difficult to obtain, the observer-based control scheme is needed to solve the H ∞ control 13
⎤ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
problem. ¯ Theorem 2. For given H ∞ performance γ, the upper bound of sampling period h, the upper bounds of network-induced delays σ ¯ and τ¯, the networked-observer-based can be solved,⎤ if there exist symmetric matrices P = H ⎡ ∞ problem ⎤ of system (5) ⎡ 0 0 P P ⎣ 1 ⎦ > 0, P1 = U T ⎣ 11 ⎦ U , P11 ∈ Rm×m , P22 ∈ R(n−m)×(n−m) , 0 P2 0 P22 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 M1 Q1 0 R1 0 ⎦ > 0, Q = ⎣ ⎦ > 0, R = ⎣ ⎦ > 0, M = ⎣ 0 M2 0 Q2 0 R2 ⎡ ⎤ ⎡ ⎤ S1 0 N1 0 ⎦ , N = ⎣ ⎦ and W = matrices G1 , G2 , matrices S = ⎣ 0 S2 0 N2 ⎤ ⎡ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Q S W1 0 M N R W ⎦ satisfying ⎣ ⎣ ⎦ ≥ 0, ⎣ ⎦ ≥ 0, ⎣ ⎦ ≥ 0 0 W2 ∗ Q ∗ M ∗ R and a constant γ > 0 such that the following inequalities hold: ⎤ ⎡ Ω 1 Ω2 ⎦ < 0, Ω := ⎣ ∗ Ω3
where
⎡ Ω1 = ⎣ ⎡ Ω11 = ⎣ ⎡ Ω111 = ⎣
Ω11
Ω12
∗
Ω13
⎤ ⎦,
Ω111
Ω112
∗
Ω113
⎤ ⎦,
2P1 A − M1 − Q1 − R1
⎡ ⎡ Ω112 = ⎣ ⎣
0 M 1 − N1 0
0
2P2 A − M2 − Q2 − R2 ⎤ ⎡ ⎤ ⎤ 0 N1 0 ⎦ ⎣ ⎦ ⎦, M 2 − N2 0 N2
Ω113 = [Ω1131 , Ω1132 ] ,
14
⎤ ⎦,
⎡ ⎡
Ω1131
⎢ ⎣ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣ ⎡ ⎡
Ω1132
⎢ ⎣ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣
−2M1 + N1 + N1T + C T C −N1 + M1
⎣
0
−N1 + M1
0
−N2 + M2 ⎤ M1 0 ⎦ −⎣ 0 M2 0 ⎡
⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
Ω12 = [Ω121 , Ω122 ] , ⎤ ⎡ ⎡ ⎡ BG1 + Q1 − S1 −BG1 S ⎢ ⎣ ⎦ ⎣ 1 ⎢ ⎢ 0 Q 2 − S2 0 Ω121 = ⎢ ⎢ ⎢ 0 ⎣ 0 ⎤ ⎡ ⎡ ⎡ R1 − W1 0 ⎢ ⎣ ⎦ ⎣ ⎢ ⎢ 0 −G2 C + R2 − W2 Ω122 = ⎢ ⎢ ⎢ 0 ⎣ 0 ⎡ ⎤ Ω131 Ω132 ⎦, Ω13 = ⎣ ∗ Ω133 ⎡ ⎡ −2Q1 + S1 + S1T 0 ⎢ ⎣ ⎢ ⎢ 0 −2Q2 + S2 + S2T Ω131 = ⎢ ⎢ ⎢ ⎣ ∗ Ω132 =0, ⎡
Ω1331
⎢ Ω1331 ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ∗ ⎢ ⎢ ⎣ ∗
⎡ ⎣
R1 − W1
⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
−2M2 + N2 + N2T ⎤T 0 ⎦ −N2 + M2 ⎤ ⎤
0
⎡
⎤ ⎤
0
S2 0 0
R2 − W2 ⎤ R1 0 ⎦ −⎣ 0 R2 0 ⎡
⎦
0
0
W2
15
0
0
0
⎤
0
⎦ ⎣
⎤ ⎤ P1 D 0
⎦ ⎣
0
⎤ ⎡
0
⎤ ⎡
−γ 2 I
∗
⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
W1
⎤
0
⎤ ⎤
0
⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦
−S1 + Q1
⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
0
−S2 + Q2 ⎤ Q1 0 ⎦ −⎣ 0 Q2 0 ⎡
⎤ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
⎡ Ω1331 = ⎣
⎤
−2R1 + W1 + W1T
0
0
−2R2 + W2 + W2T
Ω2 = [Ω21 , Ω22 , Ω23 ] , ⎡ ⎡
AT P1
0
⎦,
⎤ ⎤
⎦ ⎥ ⎢ h⎣ ⎢ ⎥ ⎢ ⎥ 0 AT P2 ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎡ ⎤ ⎥ ⎢ ⎢ ⎥ T T ⎢ ⎥ 0 G1 B ⎦ ⎥ ⎢ h⎣ ⎢ ⎥ Ω21 =Ω22 = Ω23 = ⎢ ⎥, −GT1 B T 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎤ ⎥ ⎡ ⎢ ⎢ ⎥ ⎢ ⎥ 0 0 ⎦ ⎥ ⎢ h⎣ ⎢ ⎥ T T ⎢ ⎥ 0 −C G2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎣ ⎦
T h D P1 0 ⎡ ⎤ ⎡ ⎤ ⎡ M1 0 P1 0 Q ⎦−2⎣ ⎦,⎣ 1 Ω3 =diag{⎣ 0 M2 0 P2 0 ⎤ ⎡ ⎤ ⎡ 0 0 P R ⎦−2⎣ 1 ⎦}. ⎣ 1 0 R2 0 P2
0 Q2
⎤
⎡
⎦−2⎣
P1
0
0
P2
⎤ ⎦
ˆ and the state observer gain L can be obtained by Moreover, the controller gain K solving the following equations: ˆ =V B −1 P −1 BV T G1 , K 11 L =P2−1 G2 . 95
Proof. Under the condition of Assumption 4 and based on Lemma 1, the proof follows directly from Theorem 1. Remark 3. It is worth mentioning that the established results impose no constrains on the derivatives of the network-induced time-varying delays, which is applicable for the real world applications.
16
100
Remark 4. It should be pointed out that by utilizing the separation principle, the network-based controller and observer can be both designed by the LMI approach, which can be solved by MATLAB LMI toolbox.
4. Numerical Examples In this section, the following two illustrative examples are presented to verify our 105
theoretical results. Example 1. Consider the following system described in ( 1) as: ⎧ ⎨ x(t) ˙ = Ax(t) + Bu(t) + D(t) , ⎩ y(t) = Cx(tk ), tk ≤ t < tk+1 where
⎡ A =⎣
C=
0
0.1
−0.2 0.6 0 1
⎤
⎡
⎦,B = ⎣ ⎡
,D = ⎣
1 1
⎤
0 2
(12)
⎤ ⎦,
⎦.
In this example, the sampling period is assumed to be 0.02s and the networkinduced time-varying delays are given as σ(t) = 0.01 + 0.01 sin t and τ (t) = 0.01 + 0.01 cos t with σ ¯ = 0.02 and τ¯ = 0.02. The external disturbance is set as (t) = 0.2 sin 10t. Table 1 shows the minimum allowable γ for different upper bounds of 110
network-induced delays. Table 1: Minimum allowable γ
τ¯
0
0.02
0.04
0
0.18
0.34
0.35
0.02
0.34
0.36
0.36
0.04
0.36
0.37
0.38
σ ¯
17
By choosing γ = 0.4 and solving the LMIs in Theorem 2 with the above paˆ = rameters, the desired controller gain and state observer gain are obtained as K [−0.2286, −0.4470] and L = [0.0067, 0.8973]T , respectively. 0.5
x 10
29
0
State response of x(t)
-0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 0
20
40
60 Time (s)
80
100
120
Figure 1: State response of x(t) without control input
2.5 2 1.5
State response of x(t)
1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 0
20
40
60 Time (s)
80
100
Figure 2: State response of x(t) with control input
18
120
0.2
Networked-observer error e(tk )
0
-0.2
-0.4
-0.6
-0.8
-1 0
20
40
60 Time (s)
80
100
120
Figure 3: Networked-observer error e(tk )
Figure 1 shows the state response of the open-loop system and Figures 2-3 depict 115
the state response of the closed-loop system and the networked-observer error, which implies that our designed state observer and controller are applicable to the H ∞ control problem. Example 2. Consider the network-based boring bar system with equivalent model depicted in Figure 4. The mathematical model is written as follows: m
k2
x2(t)
P c2 T
M
x1(t)
k1 c1
Figure 4: An equivalent model of boring bar system
19
⎧ ⎨ Mx ¨1 (t) + (c1 + c2 )x˙ 1 (t) − c2 x˙ 2 (t) + (k1 + k2 )x1 (t) − kx2 (t) = F (t) + f (t), ⎩ m¨ x (t) + c x˙ (t) − c x˙ (t) + k x (t) − k x (t) = −F (t), 2
2 2
2 1
2 2
2 1
(13) 120
where x1 (t) and x2 (t) are the equivalent displacements, F (t) = −ικV is the control force, V denotes the piezoelectric voltage, f (t) is the unknown cutting force which can be considered as external disturbance, M and m denote the equivalent masses, k 1 and k2 represent the equivalent stiffness, c 1 and c2 denote the equivalent damping. Let ke = k2 + ι, x˙ 1 (t) = x3 (t) and x˙ 2 (t) = x4 (t). Then, system (13) can be ⎧ ⎨ x(t) ˙ = Ax(t) + Bu(t) + D(t) , ⎩ y(t) = Cx(t ), t ≤ t < t
rewritten as
k
where
⎡
0
k
0
k+1
I
0
⎤
⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 I ⎥, A =⎢ ⎥ ⎢ ⎢ −(k1 + ke )/M ke /M −(c1 + c2 )/M c2 /M ⎥ ⎦ ⎣ −ke /m c2 /m −c2 /m ke /m ⎤ ⎡ ⎤ ⎡ 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢
⎢ 0 ⎥ ⎥ ⎢ 0 ⎥, ⎢ ⎥ ⎢ B =⎢ ⎥ ⎥,C = 0 1 0 0 ,D = ⎢ ⎢ 1/M ⎥ ⎢ −ικ/M ⎥ ⎦ ⎣ ⎦ ⎣ 0 ικ/m x(t) =[xT1 (t), xT2 (t), xT3 (t), xT4 (t)]T , u(t) = V, (t) = f (t). Therefore, the H ∞ control problem of the above network-based boring bar system can 125
be solved by our designed methodology. In this example, the parameters are set as M =1.97kg, m=0.197kg, ι = 1/5k 1 , ke = ι, κ =2×10−8m/V, k1 =15×105N/m, c1 = 2k1 ε/wn , c2 = 2ke ε/wn , ε = 0.01, wn = 873.3r/s. Moreover, the external disturbance is chosen by (t) = 0.2 sin 10t. Set the sampling period as 0.01s and the upper bound of network-induced delays is assumed
130
to be 0.01s. The H∞ performance parameter γ is given as 1. The controller gain ˆ = [199740, −180960, 5162, −15726] and and state observer gain are obtained as K 20
L = [820, 24080, −7715, −10344]T , respectively. Figure 5 shows the state response of x1 (t), x2 (t) without control input while Figures 6-7 depict the state response of x 1 (t), x2 (t), and observer error e(t k ) with control input and observer, which demonstrates the effectiveness of the obtained results. 14
x 10
237
12
State response of x(t)
10 8 6 4 2 0 -2 0
0.2
0.4
0.6
0.8
1 1.2 Time (s)
1.4
1.6
1.8
2
Figure 5: State response of x1 (t) and x2 (t) without control input
6 5 4 State response of x(t)
135
3 2 1 0 -1 -2 -3 -4 0
0.5
1
1.5
2
2.5 3 Time (s)
3.5
4
4.5
Figure 6: State response of x1 (t) and x2 (t) with control input
21
5
Networked-observer error e(tk )
1 0.8 0.6 0.4 0.2 0 -0.2 0
1
2
3
4
5
Time (s)
Figure 7: Networked-observer error e(tk )
5. Conclusion This paper deals with the observer-based H ∞ control problem for a class of networked control systems with time-varying network-induced delays. A novel observerbased control scheme is given based on the network-based Luenberger-type observer. 140
By utilizing the Lyapunov-Krasovskii functional method, delay-dependent criteria are first developed to guarantee the prescribed H ∞ performance. Based on the obtained results, the controller and the observer are further designed. In the end, two illustrative examples are provided to verify the effectiveness of the designed approaches. Our future study will focus on extending the obtained results in this paper to the cases with
145
randomly occurring parameters, which is more complicated in the networked environment.
Acknowledgements This work was partially supported by the State Key Program of National Natural Science of China(Grant No. 51235003), National Natural Science Foundation
22
150
of China (51275139,51375127,51205095) and Natural Science Foundation of Heilongjiang Province of China (F201216).
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