On the Model-Based Networked Control for ... - Semantic Scholar
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
WeAIn1.8
On the Model-Based Networked Control for Singularly Perturbed Systems with Nonlinear Uncertainties Hongwang Yu, Xiaomei Zhang, Guoping Lu and Yufan Zheng Abstract— The model-based networked control for a class of singularly perturbed control systems with nonlinear uncertainties is addressed in this paper. The approximate slow and fast systems of the plant, which are obtained by omitting the nonlinear uncertainties, are used as a model to estimate the state behavior of the plant between transmission times. The stability of model-based networked control systems is investigated under the assumption that the controller/actuator is updated with the sensor information at constant time intervals. It is shown that there exists the allowable upper bound of the singular perturbation parameter such that the model-based networked control system is global exponentially stable.
I. INTRODUCTION Networked control systems (NCSs) are currently receiving considerable attention in the literature ([1] ∼ [8]). The insertion of the real time network in the feedback control loop has many advantages and meanwhile can also make the analysis and design of the systems more complicated. Recently, several specific control techniques have been developed. One of them is model-based control proposed by Luis A. Montestruque and Panos J. Antsaklis in [5] ∼ [8]. A model-based NCS provides an explicit model of the plant in the actuator side to estimate the plant’s state between transmission times and generate the appropriate control signals when sensor data are not available (see Fig.1). At each transmission time the model is updated with the measured state of the plant. A model-based NCS has as its main objective the reduction of the data packets transmitted over the network. There are several works are valid for linear or nonlinear plant when the controller/actuator is updated with the sensor information at constant or nonconstant time intervals. The study of singularly perturbed systems (SPS) has received much attention during the past decades, and the analysis and synthesis of uncertain SPS have been investigated by many researchers; see, [11] ∼ [13]. The modelbased networked control for linear SPS was first discussed
in [9]. The slow and fast systems of the plant are used to produce an estimation of the plant’s state between transmission instants. When the slow and fast systems are both stable, the model-based networked control system is shown to be asymptotically stable by means of deriving their approximate solutions. In [10], general linear SPS are used as a model to estimate the plant’s state. To the authors’ best knowledge, there is no literature about networked control for nonlinear SPS. Thus, we consider the plant governed by SPS with uncertainties. By omitting the uncertain terms, we obtain their approximate slow and fast systems and take them as a model in the actuator side of the plant in this paper. It is assumed that the controller/actuator is updated with the sensor information at constant time intervals. A sufficient condition on searching for the allowable upper bound of the singular perturbation parameter is proposed, under which the model-based networked control system is global exponentially stable. This paper is organized as follows. In Section II some notations are reviewed and the state feedback networked control architecture shown in Fig. 1 is stated. In Section III sufficient conditions of robust stability are derived. A numerical example is presented in Section IV. Conclusions are given in the last section. Plant
Sensor
Model
Controller
Fig. 1. Networked control for SPS. This work was supported by National Natural Science Foundation of China under Grants No. 60674046 and 60874021, Natural Science Foundation Grant BK2007061 from Jiangsu Province, Qing Lan Project from the Jiangsu Provincial Department for Education, the Fund for Doctorial Program at Nantong University No. 07B14 and the Nantong University Fund for Natural Science No. 08Z009. H. Yu is with School of Mathematics and Statistics, Nanjing Audit University, Nanjing 211815, China [email protected] X. Zhang is with School of Sciences, Nantong University, Nantong 226007, Jiangsu, China [email protected] G. Lu is with School of Electrical Engineering, Nantong University, Nantong 226019, Jiangsu, China [email protected] Y. Zheng is with Department of Mathematics, Shanghai University, Shanghai 200436, China [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
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II. PRELIMINARIES AND PROBLEM STATEMENT The following notations will be employed in this paper. • ||x|| denotes the standard Euclidean norm of the vector x ∈ Rn . • λmax (A) denotes the largest eigenvalue of matrix A. • λmin (A) denotes the smallest eigenvalue of matrix A. • det(A) denotes the determinant of matrix A. • p ||A|| denotes the norm of the matrix A with ||A|| = λmax (AT A).
WeAIn1.8 •
A vector function f (t, ε) is said to be O(ε) over an interval [t1 ,t2 ] if there exist positive constants k and ε ∗ such that || f (t, ε)|| ≤ kε,
∀ε ∈ [0, ε ∗ ], ∀t ∈ [t1 ,t2 ].
Consider networked singularly perturbed control systems shown in Fig. 1, where the plant behavior is described by x(t) ˙ =A11 x(t) + A12 z(t) + B1 u(t) + f1 (x(t), z(t)), (1) ε z˙(t) =A21 x(t) + A22 z(t) + B2 u(t) + f2 (x(t), z(t)),
with t ∈ [tk ,tk+1 ), where x ∈ Rn are z ∈ Rm are state vectors of the slow and fast systems, respectively, u ∈ R p is the control input, Ai j , Bi (i, j = 1, 2) are constant real matrices with appropriate dimensions, the scalar ε > 0 is a singular perturbation parameter. The functions fi (x(t), z(t)), i = 1, 2 may be nonlinear uncertainties and are bounded by i = 1, 2, k fi (x(t), z(t)) k≤ ai k x(t) k +bi k z(t) k with ai and bi are positive constants. Define [xT (t), zT (t)]T = T · ξ T (t), η T (t) with In εH , T= −L Im − εLH
(2)
where ξˆ ∈ Rn are ηˆ ∈ Rm are state vectors of the slow and fast systems, respectively, u ∈ R p is the control input, Aˆ 0 , Aˆ 22 , Bˆ 0 , Bˆ 2 are constant real matrices with appropriate dimensions. Consider the following state feedback control law: ˆ u(t) = K1 · ξˆ (t) + K2 · η(t),
t ∈ [tk ,tk+1 ).
(6)
where K1 and K2 are control gains to be designed. Since T is non-singular, system (1) is stable if and only if system (4) is stable. In this paper, our purpose is to find a appropriate controller in the form of (6) such that the plant (4) with the model (5) is robustly stabilized under the condition of network transmission. In order to achieve it, we assume that the fast system of the plant with control-free is asymptotically stable. That is to say, A22 is supposed to be Hurwitz stable in this work. III. ROBUST STABILITY FOR UNCERTAIN SPS WITH NETWORKED CONTROL
(3)
−1 where H = A12 A−1 22 and L = A22 A21 . Here the transformation −1 matrix T is non-singular if A22 exists. One may also obtain T that ξ T (t), η T (t) = T −1 · [xT (t), zT (t)] with −εH In − εHL −1 T = L Im
and the equivalent form of system (1) as follows. ˙ ξ =[A0 + α11 (ε)]ξ + α12 (ε)η + (B0 + β1 (ε))u + (In − εHL) f¯1 (ξ , η) − H f¯2 (ξ , η) ε η˙ =α21 (ε)ξ + [A22 + α22 (ε)]η + (B2 + β2 (ε))u + εL f¯1 (ξ , η) + f¯2 (ξ , η)
in real control system. One may take their approximate slow and fast control systems as a model. ( ˙ ξˆ (t) = Aˆ 0 · ξˆ (t) + Bˆ 0 u t ∈ [tk ,tk+1 ), (5) ˙ˆ = Aˆ 22 · η(t) ˆ + Bˆ 2 u ε η(t)
Define the state error between the plant (4) and the model (5) as e1 (t) = ξ (t) − ξˆ (t),
ˆ e2 (t) = η(t) − η(t),
with t ∈ [tk ,tk+1 ). Assume that the sensor sends the state information through the network every time h, the update times are at tk second where tk+1 − tk = h for all k = 0, 1, 2, · · · . Since the model’s state is updated at tk , it holds that e1 (tk ) = e2 (tk ) = 0,
(4)
with t ∈ [tk ,tk+1 ), where
k = 0, 1, 2, · · ·
B0 = B1 − A12 A−1 22 B2 , α11 (ε) = −εHLA0 , α21 (ε) = εLA0 , α12 (ε) = ε(A0 H − HLA12 ) − ε 2 HLA0 H, α22 (ε) = εLA12 + ε 2 LA0 H, β1 (ε) = −εHLB1 , β2 (ε) = εLB1 , ¯f1 (ξ , η) = f1 (ξ + εHη, −Lξ + (Im − εLH)η), f¯2 (ξ , η) = f2 (ξ + εHη, −Lξ + (Im − εLH)η). Omitting the nonlinear uncertainties of the plant, we would get the slow and fast control systems of the plant if the accurate dynamic of the plant is given. But it is impossible
(8)
This resetting of the state error at every update time is a key for model-based NCSs. Let # # " " ξ (t) η(t) , ζ2 (t) = . ζ1 (t) = e1 (t) e2 (t) From (7), one may get " # ξ (t) ζ1 (t) = Pn · ˆ , ξ (t)
A0 = A11 − A12 A−1 22 A21 ,
(7)
"
# η(t) ζ2 (t) = Pm · , ˆ η(t) Im 0m In 0n . , Pm = with Pn = Im − Im In − In By using the Kronecker product, we can obtain the following dynamics of ζ1 (t) and ζ2 (t) for t ∈ [tk ,tk+1 ). ζ˙1 (t) =Pn (Λ11 + α¯ 11 (ε))Pn ζ1 (t) + Pn (Λ12 + α¯ 12 (ε))Pm ζ2 (t) + 12 ⊗ F1 (ζ1 , ζ2 ) (9) ˙ ε ζ2 (t) =Pm (Λ21 + α¯ 21 (ε))Pn ζ1 (t) + Pm (Λ22 + α¯ (ε))P ζ (t) + 1 ⊗ F (ζ , ζ )
685
22
m 2
2
2
1
2
WeAIn1.8 with t ∈ [tk ,tk+1 ), where A0 A¯ 12 Λ0 = , 0n×n A∗0 ¯ + α¯ 11 (ε) ¯ 0 + α¯ 11 (ε) − α¯ 12 (ε)L) γ11 (ε) = − ε H¯ L(Λ ¯ ¯ α¯ 22 (ε)L¯ − α¯ 21 (ε)) − α¯ 12 (ε)L, + H( 2 ¯ 12 ) − ε H¯ LΛ ¯ 0 H¯ − H¯ α¯ 22 (ε) γ12 (ε) =ε(Λ0 H¯ − H¯ LΛ ¯ α¯ 22 (ε)L¯ − α¯ 21 (ε))H¯ − ε H¯ L¯ α¯ 12 (ε) + ε H( ¯ α¯ 11 (ε) − α¯ 12 (ε)L) ¯ H¯ + ε(1 − ε H¯ L)(
where Λ11 =
Λ22 =
A0 0
B0 K1 Aˆ 0 + Bˆ 0 K1
A22 0
B2 K2 Aˆ 22 + Bˆ 2 K2
,
Λ12 =
, Λ21 =
0 B2 K2 0 Bˆ 0 K2
,
0 B2 K1 0 Bˆ 0 K1
,
α¯ 11 (ε) =
α11 (ε) β1 (ε)K1 0 0
,
α¯ 12 (ε) =
α12 (ε) β1 (ε)K2 0 0
,
α¯ 21 (ε) =
α21 (ε) β2 (ε)K1 0 0
,
α¯ 22 (ε) =
α22 (ε) β2 (ε)K2 0 0
,
+ α¯ 12 (ε), ¯ 0 + α¯ 11 (ε) − α¯ 12 (ε)L) ¯ γ21 (ε) =ε L(Λ ¯ + α¯ 21 (ε) − α¯ 22 (ε)L, 2 ¯ 12 + ε LΛ ¯ 0 H¯ + ε L¯ α¯ 12 (ε) γ22 (ε) =ε LΛ ¯ H, ¯ ¯ α¯ 11 (ε) − α¯ 12 (ε)L) + ε 2 L( ¯ H¯ + α¯ 22 (ε), + ε(α¯ 21 (ε) − α¯ 22 (ε)L) ¯ F¯1 (X, Z) − H¯ F¯2 (X, Z), g1 (X, Z) = (I2n − ε H¯ L) g2 (X, Z) = ε L¯ F¯1 (X, Z) + F¯2 (X, Z),
and F1 = (In − εHL) f¯1 (E1 ζ1 , E2 ζ2 ) − H f¯2 (E1 ζ1 , E2 ζ2 ) F2 = εL f¯1 (E1 ζ1 , E2 ζ2 ) + f¯2 (E1 ζ1 , E2 ζ2 )
with
with 12 = [1 1]T , E1 = [In 0], E2 = [Im 0]. The nonlinear functions Fi , i = 1, 2 represent the uncertainties of system (9). Under the conditions of the bounded functions fi , i = 1, 2, one can get the bounds of their norms as follows. k Fi (ζ1 (t), ζ2 (t)) k≤ a¯i k ζ1 (t) k +b¯ i k ζ2 (t) k,
a¯2 =εa1 k L k +a2 + εb1 k L k2 +b2 k L k, b¯ 1 = k H k (εa1 λ1 + εa2 k H k +b2 λ2 ) + b1 λ1 λ2 , b¯ 2 =ε k H k (a2 + εa1 k L k) + λ2 (b2 + εb1 k L k)
with
ε H¯ I2m − ε L¯ H¯
,
k gi (X, Z) k≤ a˜i k X k +b˜ i k Z k,
(13)
b˜ 2 =ε k H¯ k (a¯2 + ε a¯1 k L¯ k) + λ¯ 2 (b¯ 2 + ε b¯ 1 k L¯ k)
with λ1 =k In − εHL k and λ2 =k Im − εLH k. If the controller gain K2 is designed such that (Aˆ 22 + ˆ B2 K2 )−1 exists, it is obvious that Λ−1 22 exists and one may define [ζ1T (t), ζ2T (t)]T = T¯ · X T (t), Z T (t) I2n −L¯
The nonlinear functions gi , i = 1, 2 may be the uncertainties of the equivalent system (12) and are bounded by
where a˜1 =λ¯ 1 (a¯1 + b¯ 1 k L¯ k)+ k H¯ k (a¯2 + b¯ 2 k L¯ k), a˜2 =ε a¯1 k L¯ k +a¯2 + ε b¯ 1 k L¯ k2 +b¯ 2 k L¯ k, b˜ 1 = k H¯ k (ε a¯1 λ¯ 1 + ε a¯2 k H¯ k +b¯ 2 λ2 ) + b¯ 1 λ¯ 1 λ¯ 2 ,
a¯1 =λ1 (a1 + b1 k L k)+ k H k (a2 + b2 k L k),
¯ −LX ¯ + (I2m − ε L¯ H)Z), ¯ F¯1 = 12 ⊗ F1 (X + ε HZ, ¯ −LX ¯ + (I2m − ε L¯ H)Z). ¯ F¯2 = 12 ⊗ F2 (X + ε HZ,
(10)
where
T¯ =
A∗0 = Aˆ 0 + Bˆ 0 K1 − Bˆ 0 K2 (Aˆ 22 + Bˆ 2 K2 )−1 B2 K1 , A¯ 12 = B0 K1 − B2 K2 (Aˆ 22 + Bˆ 2 K2 )−1 Bˆ 2 K1
(11)
−1 ¯ where H¯ = Λ12 Λ−1 22 and L = Λ22 Λ21 . Applying transformation (11) into system (9), we can get the following equivalent form of system (9). ˙ X =Pn [Λ0 + γ11 (ε)]Pn X + Pn γ12 (ε)Pm Z + g1 (X, Z) (12) ˙ ε Z =Pm γ21 (ε)Pn X + Pm [Λ22 + γ22 (ε)]Pm Z + g2 (X, Z)
with λ¯ 1 =k I2n − ε H¯ L¯ k and λ¯ 2 =k I2m − ε L¯ H¯ k. Suppose that Λ−1 22 exists, one may take the dynamic system (12) as a standard singularly perturbed system with nonlinear uncertainties at t ∈ [tk , tk+1 ). We will design a control law based on the slow and fast systems of system (12) such that system (12) is robustly stabilized. One can get their slow and fast systems of the linear part by omitting the nonlinear uncertainties. The slow system is derived as ¯ 0 Xs , t ∈ [tk , tk+1 ) X˙s = Λ
(14)
¯ 0 = Pn Λ0 Pn , and the fast system is described with Λ dZ f ¯ 22 Z f , τ ∈ [τk , τk+1 ) (15) =Λ dτ t ¯ 22 = Pm Λ22 Pm . with τ = and Λ ε For linear slow and fast systems, controller gains K1 and K2 may be designed such that they are both exponentially stable. Let In 0n Im 0m Rn = , Rm = , (16) 0n 0n 0m 0m
686
WeAIn1.8 we have the following result. Theorem 1: Consider systems (14) and (15) with t ∈ [tk , tk+1 ). Suppose that A22 is Hurwitz stable. If there exist ˆ 2 is Hurwitz controller gains K1 and K2 such that Aˆ + BK ¯ stable and Rn eΛ0 h Rn is Schur stable, then the solutions of systems (14) and (15) are globally exponentially stable. Proof Considering the fact that the model’s state is updated at tk , we can obtain the solution of system (14) ¯
i = 1, 2, one obtains k Gi (X, Z) k≤ µi (ε) k X k +νi (ε) k Z k, where i = 1, 2, µi (ε) = a˜i + k γi1 (ε) k, νi (ε) = b˜ i + k γi2 (ε) k . − w(tk+1 ) = H · w(tk− ) + Ω(tk− ),
and ¯
Xs (tk− ) = eΛ0 h Xs (tk−1 ).
with
H =
Then one may get (Rn e
Rn )k Xs (t0 ).
d1
¯
||(Rn eΛ0 h Rn )k || ≤ c1 e−d1 k ≤ c1 ed1 e− h t . Therefore, one gets that d1
¯
||Xs (t)|| ≤ e||Λ0 ||h+d1 c1 e− h t kXs (t0 )k, which also implies that the solution of system (14) is globally exponentially stable. Next, one may get the solution of system (15) ¯
Z f (τ) = eΛ22 (τ −τk ) (Rm e
¯ h Λ 22 ε
e
¯ h Λ 22 ε
Rm
,
Ω(tk− ) =
Z tk+1 ¯ 0 (tk+1 −s) Λ
e
tk
1 ε
Z tk+1
e
# Ω1 (tk− )
(21)
Ω2 (tk− )
G1 (X, Z)ds,
¯ Λ 22 ε (tk+1 −s)
tk
"
(22) G2 (X, Z)ds.
||Ω1 (tk− )|| ≤δ1 (ε)||w(tk− )||,
(23)
||Ω2 (tk− )|| ≤δ2 (ε)||w(tk− )||, where c
¯ 0 ||+µ1 (ε )+ν1 (ε ))h+ 0 (µ2 (ε )+ν2 (ε )) (||Λ d
ρ(ε) =(1 + c0 )e
¯
||eΛ22 (τ −τk ) || ≤ c2 e−d2 (τ −τk ) ≤ c2 . ¯ h Λ 22
We also obtain that Rm e ε Rm is Schur stable. So there exist positive constants c4 , c5 , d3 and d4 such that ¯ h Λ 22 ε
0
Considering system (20) with the nonlinear term Ω(tk− ), we will get the following result. Theorem 2: Consider system (12) and (20) with tk+1 − ˆ 2 are Hurwitz stable. There tk = h. Suppose A22 and Aˆ + BK exist positive constants c0 and d0 , independent of ε, such that ||w(t)|| ≤ρ(ε)||w(tk− )||,
Rm )k Z f (τ0 ).
¯ 22 is Hurwitz stable since A22 and A+ ˆ BK ˆ 2 It is obvious that Λ are Hurwitz stable. So there exist positive constants c2 and d2 , independent of ε, it holds that
0
,
¯ (ε))(e||Λ0 ||h − 1)
ρ(ε)(µ1 (ε) + ν1 , ¯ 0 || ||Λ ρ(ε)c0 (µ2 (ε) + ν2 (ε)) δ2 (ε) = d0 with µi (ε), νi (ε), i = 1, 2 are defined in (19). Proof From the first equation of (12), one gets δ1 (ε) =
X(t) =X(tk ) +
Rm )k || ≤ c3 e−d3 k ≤ c4 e−d4 τ .
Z t tk
(24)
¯ 0 X + G1 (X, Z))ds (Λ
For t ∈ [tk ,tk+1 ), we have
Then, one gets that
||X(t)|| ≤||X(tk )|| +
||Z f (τ)|| ≤ c2 c4 e−d4 τ kZ f (τ0 )k, which also implies that the solution of system (15) is globally exponentially stable. Remark 1: Under the condition of Theorem 1, there exist positive definition symmetric matrices Φ = ΦT > 0, Ψ = ΨT > 0 i = 1, 2 such that ¯
0
Ω2 (tk− ) =
¯
Since Rn eΛ0 h Rn is Schur stable, then exist positive constants c1 and d1 such that
||(Rm e
¯
eΛ0 h Rn
Ω1 (tk− ) =
Xs (t) = e
(20)
where
¯
¯ 0h Λ
(19)
Letting w(t) = (X T (t), Z T (t))T , we have
Xs (t) = eΛ0 (t−tk ) Xs (tk ) = eΛ0 (t−tk ) Rn Xs (tk− ),
¯ 0 (t−tk ) Λ
(18)
¯
(Rn eΛ0 h Rn )T ΦRn eΛ0 h Rn − Φ = −In ,
(17) ¯ ¯ Λ Λ 22 22 (Rm e ε h Rm )T ΨRm e ε h Rm − Ψ = −Im . Now, we mainly discuss the norm bound of the nonlinear uncertainties of system (12). Denoting
Z t tk
¯ 0 X|| + ||G1 (X, Z)||)ds. (||Λ
¯ 22 defined in ˆ 2 are Hurwitz stable, Λ Since A22 and Aˆ + BK (9) is Hurwitz stable. There exist positive constants c0 and d0 , independent of ε, it holds that ¯
||eΛ22
t−tk ε
|| ≤ c0 e−d0
t−tk ε
.
Then from the second equation of (12), it follows that Z ¯ (t−t ) Λ 1 t Λ¯ 22 (t−s) 22 k e ε G2 (X, Z)ds Z(t) =e ε Z(tk ) + ε tk
Gi (X, Z) = Pn γi1 (ε)Pn X + Pn γi2 (ε)Pm Z + gi (X, Z)
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t−tk
≤c0 e−d0 ε · ||Z(tk )|| Z t−s 1 t + c0 e−d0 ε · ||G2 (X, Z)||ds. ε tk
WeAIn1.8 By means of the property of norms for vectors and (18), we have ||w(t)|| ≤||X(t)|| + ||Z(t)|| ≤||X(tk )|| + c0 e−d0 +
Z t tk
+
· ||Z(tk )||
¯ 0 || + µ1 (ε))||X|| + ν1 (ε)||Z||]ds [(||Λ
Z t
1 ε
t−tk ε
tk
c0 e−d0
t−s ε
(µ2 (ε)||X|| + ν2 (ε)||Z||)ds
≤(1 + c0 )||w(tk )|| +
Z t tk
IV. AN EXAMPLE ¯ 0 || + µ1 (ε) (||Λ
In this section, we consider the plant described by Fig.1 is a modified armature-controlled DC motor [14], which consists of a mechanical torque equation and an equation for the electrical transient in the armature circuit, namely,
t−s c0 +ν1 (ε) + e−d0 ε (µ2 (ε) + ν2 (ε)))||w||ds. ε By the Gronwall inequality, one gets
Rt
||w(t)|| ≤(1 + c0 )||w(tk )|| · e tk
J w˙ = ki + (2 + ϑ1 )kw + f (w, i) Li˙ = −kw − Ri + ϑ2 u
¯ 0 ||+µ1 (ε )+ν1 (ε ))ds (||Λ
R t c0 −d t−s e 0 ε ·(µ2 (ε )+ν2 (ε ))ds t
·e
k ε
=(1 + c0 )||w(tk )|| · e
t−tk c0 (µ2 (ε)+ν2 (ε)) (1−e−d0 ε d0
)
≤ρ(ε)||w(tk− )||. From (22), one obtains that ||Ω1 (tk− )|| ≤ =
R h ||Λ¯ ||s 0 · (µ1 (ε) + ν1 (ε))||w||ds 0 e ¯
ρ (ε )(µ1 (ε )+ν1 (ε ))(e||Λ0 ||h −1) ||w(tk− )|| ¯ 0 || ||Λ
and ||Ω2 (tk− )|| ≤ =
(t
x(t) ˙ = (2 + ϑ1 )x(t) + z(t) + f (x, z), ε z˙(t) = −x(t) − z(t) + ϑ2 u(t),
−s)
k+1 1 R tk+1 c0 e−d0 ε (µ2 (ε) + ν2 (ε))||w||ds ε tk h ρ (ε )c0 (µ2 (ε )+ν2 (ε )) (1 − e−d0 ε )||w(tk− )||. d0
||Ω1 (tk− )||
Therefore, it holds that ≤ and ||Ω2 (tk− )|| ≤ δ2 (ε)||w(tk− )||. Based on Theorem 1 and Theorem 2, the closed-loop system (12) can be robustly stabilized by the controller (6) which is designed in Theorem 1. Now, we give the result in the following. The proof is omitted for saving paper space here. Theorem 3: Consider the closed-loop system (12) with tk+1 − tk = h,t ∈ [tk ,tk+1 ). Under the conditions of Theorem 1, and for an arbitrary given constant 0 < θ < 1, if δi , i = 1, 2 defined in (24) with ε = 0 satisfy (25)
1 xz where f (x, z) = 1+z 4 . It is obvious that | f (x, z) |≤ 2 | x |. One may get the following equivalent system for all t ∈ [tk ,tk+1 ) via the transformation, which is defined in (3). ξ˙ (t) =[1 + ϑ1 + ε(1 + ϑ1 )] · ξ (t) + (1 + ε) f¯(ξ , η)
+ ϑ2 u(t) − [ϑ1 ε + (1 + ϑ1 )ε 2 ] · η(t), ˙ =ε(1 + ϑ1 ) · ξ (t) + ε f¯(ξ , η) ε η(t)
¯
σ1 = ||eΛ0 h Rn ||, σ2 = ||e
Rm ||,
(ξ − εη)[(1 + ε)η − ξ ]2 . Omitting the nonwith f¯(ξ , η) = 1 + [(1 + ε)η − ξ ]4 linear part of (29), we take an approximate slow and fast systems as the plant model. ˙ ξˆ (t) = ξˆ (t) + u(t) ˙ˆ = η(t) ˆ + u(t) ε η(t)
(26)
Rn , Rm are defined in (16) and Φ, Ψ are defined in (17). Then there exists an ε ∗ > 0 such that the states of system (12) are globally exponentially stable for all ε ∈ (0, ε ∗ ]. Remark 2: If the feedback controller gains K1 , K2 of the model and the bound of nonlinear uncertainties of the plant are given, the approximate bound ε ∗ can be determined by solving the following inequality δ1 (ε)||Φ||(2σ1 + δ1 (ε)) + δ2 (ε)||Ψ||(2σ2 + δ2 (ε)) ≤ 1
(29)
+ ϑ2 u(t) + [−1 + ε − ε 2 (1 + ϑ1 )] · η(t).
where ¯ h Λ 22 ε
(28)
2
δ1 (ε)||w(tk− )||
δ1 ||Φ||(2σ1 + δ1 ) + δ2 ||Ψ||(2σ2 + δ2 ) ≤ θ ,
(27)
where i, u, R and L are the armature current, voltage, resistance and inductance respectively, J is the moment of inertia, w is the angular speed, and ki, kw are respectively the torque and the back electromotive force (e.m.f.) developed with constant excitation flux φ . Practically, L is small and can play the role of our parameter ε. ϑs , s = 1, 2 is two modulatory parameters in the original systems. For simplicity, the nonlinear uncertainties of the systems are taken as w(ki)2 f (w, i)) = 1+(ki) 4 . And we choose J = k = R = 1, and assume L = ε, w = x, i = z, then (27) can be rewritten as the standard form of singularly perturbed control systems
¯ 0 ||+µ1 (ε )+ν1 (ε ))h (||Λ
·e
It is obvious that there is a contradiction between the norm bound of nonlinear uncertainties and the approximate bound ε ∗. Remark 3: It is worth to point out that our results are based on the condition that the fast systems A22 of the plant is Hurwitz stable. That implies that the slow systems A0 of the plant don’t need to be stable. This case will be shown in the following example.
(30)
and the controller designed is ˆ u(t) = k1 · ξˆ (t) + k2 · η(t)
(31)
Define the errors of the slow and states between the plant and the model as ˆ e1 (t) = ξ (t) − ξˆ (t), e2 (t) = η(t) − η(t). Let the initial value of system (28) be [x(0), z(0)]T = [2, 1]T and the sampling period h = 1. The controller gains
688
WeAIn1.8 are designed as k1 = −1, k2 = −2 and the parameter is chosen as ε = 0.1. Fig.2 shows that the error states and the states of the responding closed-loop system with ϑ1 = −2 and ϑ2 = 1. 2 state x error of slow states state z error of fast states
States of whole control systems
1.5
1
0.5
0
f0.5
0
0.5
1
1.5
2
2.5
3
3.5
Time
Fig. 2. The states and error states of the networked control system.
While ϑ1 = −1 and ϑ2 = 1, one may get A0 = 0. In this case, the error states and the states of the responding closedloop system are also asymptotically stable. The simulation result is shown in Fig.3. 2 state x error of slow states state z error of fast states
States of whole control systems
1.5
1
0.5
[4] A. S. Matveev and A. V. Savkin, “Optimal Control of Networked Systems via Asynchronous Communication Channels with Irregular Delays,” Proc. The 40th IEEE on Conference Decision and Control, Orland Florida, USA, pp. 2323-2332, 2001. [5] L. A. Montestruque and P. J. Antsaklis, “Model-Based Networked Control Systems: Necessary and Sufficient Conditions for Stability,” in the 10th Mediterrannean Conf. Control and Automation, Lisbon, Portugal, 2002. [6] L. A. Montestruque, and P. J. Antsaklis, “Stability of Model-Based Networked Control Systems with Time-Varying Transmission Times,” IEEE Trans. Automat. Contr., vol. 49, pp. 1562-1572, 2004. [7] S. Mastellon, C. T. Abdallah and P. Dorato, “Model-Based Networked Control for Nonlinear Systems with Stochastic Packet Dropped”, in the 2005 American Control Conference, Portland, USA. , pp. 23652370, 2005. [8] L. A. Montestruque, and P. J. Antsaklis, “On the Model-Based Control of Networked Systems,” Automatica, vol. 39, pp. 1837-1843, 2003. [9] H. W. Yu and Y. F. Zheng, “On Model-Based Networked Control for Singularly Perturbed Systems,” in the IEEE International Conference on Control and Automation, Guangzhou, China, pp. 1548-1552, 2007. [10] G. X. Wang, Z. M. Wang and D. S. Naidu, “On Model-Based Networked Control of Singularly Perturbed Systems,” in the 27th Chinese Control Conference, Kunming, Yunnan, China, pp. 53-57, 2008. [11] Z. H. Shao and M. E. Sawan, “Stabilisation of Uncertain Singularly Perturbed Systems,” IEE Proc. Control Theory Appl., vol. 153, pp. 99-103, 2006. [12] A. R. Teel, L. Moreau and D. Nesic, “An Unified Framework for Inputto-State Stability in Systems with Two Time Scales,” IEEE Trans. Automat. Contr., vol. 48, pp. 1526-1544, 2003. [13] Z. H. Shao, “Robust Stability of Two-Time-Scale Systems with Nonlinear Uncertainties,” IEEE Trans. Automat. Contr., vol. 49, pp. 258-261, 2004. [14] P. V. Kokotovi´c, H. K. Khalil and J. O’reilly, Singular Perturbation Methods in the Control: Analysis and Design, Academic Press, London, UK, 1986. [15] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, UK, 1985.
0
f0.5
0
5
10
15
20
25
Time
Fig. 3. The states and error states of the networked control system.
V. CONCLUSION In this paper the robust control of SPS with nonlinear uncertainties over a network is addressed. We use an approximate slow and fast systems of the plant as a model plant in the actuator side of the plant to produce an estimation of the plant state behavior between transmission times. Under the assumption that the fast systems of the plant is stable, a state feedback control law has been designed such that the networked control system is robustly globally exponentially stable. Finally, an example is given to illustrate the results proposed in this paper. The work under this paper can be extended to the cases where network-induced delay and/or output feedback control are considered. R EFERENCES [1] D. Nesic and A. R. Teel, “Input-to-State Stability of Networked Control Systems,” Automatica, vol. 40, pp. 2123-2128, 2004. [2] D. Yue, Q. L. Han and C. Peng, “State Feedback Controller Design of Networked Control Systems,” IEEE Trans. circuits and systems-II: express briefs, vol. 51, pp. 640-644, 2004. [3] P. V. Zhivoglyadov, and R. H. Middleton, “Networked Control Design for Linear Systems,” Automatica, vol. 39, pp. 743-750, 2003.
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On the Model-Based Networked Control for Singularly Perturbed Systems with Nonlinear Uncertainties Hongwang Yu, Xiaomei Zhang, Guoping Lu and Yufan Zheng Abstract— The model-based networked control for a class of singularly perturbed control systems with nonlinear uncertainties is addressed in this paper. The approximate slow and fast systems of the plant, which are obtained by omitting the nonlinear uncertainties, are used as a model to estimate the state behavior of the plant between transmission times. The stability of model-based networked control systems is investigated under the assumption that the controller/actuator is updated with the sensor information at constant time intervals. It is shown that there exists the allowable upper bound of the singular perturbation parameter such that the model-based networked control system is global exponentially stable.
I. INTRODUCTION Networked control systems (NCSs) are currently receiving considerable attention in the literature ([1] ∼ [8]). The insertion of the real time network in the feedback control loop has many advantages and meanwhile can also make the analysis and design of the systems more complicated. Recently, several specific control techniques have been developed. One of them is model-based control proposed by Luis A. Montestruque and Panos J. Antsaklis in [5] ∼ [8]. A model-based NCS provides an explicit model of the plant in the actuator side to estimate the plant’s state between transmission times and generate the appropriate control signals when sensor data are not available (see Fig.1). At each transmission time the model is updated with the measured state of the plant. A model-based NCS has as its main objective the reduction of the data packets transmitted over the network. There are several works are valid for linear or nonlinear plant when the controller/actuator is updated with the sensor information at constant or nonconstant time intervals. The study of singularly perturbed systems (SPS) has received much attention during the past decades, and the analysis and synthesis of uncertain SPS have been investigated by many researchers; see, [11] ∼ [13]. The modelbased networked control for linear SPS was first discussed
in [9]. The slow and fast systems of the plant are used to produce an estimation of the plant’s state between transmission instants. When the slow and fast systems are both stable, the model-based networked control system is shown to be asymptotically stable by means of deriving their approximate solutions. In [10], general linear SPS are used as a model to estimate the plant’s state. To the authors’ best knowledge, there is no literature about networked control for nonlinear SPS. Thus, we consider the plant governed by SPS with uncertainties. By omitting the uncertain terms, we obtain their approximate slow and fast systems and take them as a model in the actuator side of the plant in this paper. It is assumed that the controller/actuator is updated with the sensor information at constant time intervals. A sufficient condition on searching for the allowable upper bound of the singular perturbation parameter is proposed, under which the model-based networked control system is global exponentially stable. This paper is organized as follows. In Section II some notations are reviewed and the state feedback networked control architecture shown in Fig. 1 is stated. In Section III sufficient conditions of robust stability are derived. A numerical example is presented in Section IV. Conclusions are given in the last section. Plant
Sensor
Model
Controller
Fig. 1. Networked control for SPS. This work was supported by National Natural Science Foundation of China under Grants No. 60674046 and 60874021, Natural Science Foundation Grant BK2007061 from Jiangsu Province, Qing Lan Project from the Jiangsu Provincial Department for Education, the Fund for Doctorial Program at Nantong University No. 07B14 and the Nantong University Fund for Natural Science No. 08Z009. H. Yu is with School of Mathematics and Statistics, Nanjing Audit University, Nanjing 211815, China [email protected] X. Zhang is with School of Sciences, Nantong University, Nantong 226007, Jiangsu, China [email protected] G. Lu is with School of Electrical Engineering, Nantong University, Nantong 226019, Jiangsu, China [email protected] Y. Zheng is with Department of Mathematics, Shanghai University, Shanghai 200436, China [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
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II. PRELIMINARIES AND PROBLEM STATEMENT The following notations will be employed in this paper. • ||x|| denotes the standard Euclidean norm of the vector x ∈ Rn . • λmax (A) denotes the largest eigenvalue of matrix A. • λmin (A) denotes the smallest eigenvalue of matrix A. • det(A) denotes the determinant of matrix A. • p ||A|| denotes the norm of the matrix A with ||A|| = λmax (AT A).
WeAIn1.8 •
A vector function f (t, ε) is said to be O(ε) over an interval [t1 ,t2 ] if there exist positive constants k and ε ∗ such that || f (t, ε)|| ≤ kε,
∀ε ∈ [0, ε ∗ ], ∀t ∈ [t1 ,t2 ].
Consider networked singularly perturbed control systems shown in Fig. 1, where the plant behavior is described by x(t) ˙ =A11 x(t) + A12 z(t) + B1 u(t) + f1 (x(t), z(t)), (1) ε z˙(t) =A21 x(t) + A22 z(t) + B2 u(t) + f2 (x(t), z(t)),
with t ∈ [tk ,tk+1 ), where x ∈ Rn are z ∈ Rm are state vectors of the slow and fast systems, respectively, u ∈ R p is the control input, Ai j , Bi (i, j = 1, 2) are constant real matrices with appropriate dimensions, the scalar ε > 0 is a singular perturbation parameter. The functions fi (x(t), z(t)), i = 1, 2 may be nonlinear uncertainties and are bounded by i = 1, 2, k fi (x(t), z(t)) k≤ ai k x(t) k +bi k z(t) k with ai and bi are positive constants. Define [xT (t), zT (t)]T = T · ξ T (t), η T (t) with In εH , T= −L Im − εLH
(2)
where ξˆ ∈ Rn are ηˆ ∈ Rm are state vectors of the slow and fast systems, respectively, u ∈ R p is the control input, Aˆ 0 , Aˆ 22 , Bˆ 0 , Bˆ 2 are constant real matrices with appropriate dimensions. Consider the following state feedback control law: ˆ u(t) = K1 · ξˆ (t) + K2 · η(t),
t ∈ [tk ,tk+1 ).
(6)
where K1 and K2 are control gains to be designed. Since T is non-singular, system (1) is stable if and only if system (4) is stable. In this paper, our purpose is to find a appropriate controller in the form of (6) such that the plant (4) with the model (5) is robustly stabilized under the condition of network transmission. In order to achieve it, we assume that the fast system of the plant with control-free is asymptotically stable. That is to say, A22 is supposed to be Hurwitz stable in this work. III. ROBUST STABILITY FOR UNCERTAIN SPS WITH NETWORKED CONTROL
(3)
−1 where H = A12 A−1 22 and L = A22 A21 . Here the transformation −1 matrix T is non-singular if A22 exists. One may also obtain T that ξ T (t), η T (t) = T −1 · [xT (t), zT (t)] with −εH In − εHL −1 T = L Im
and the equivalent form of system (1) as follows. ˙ ξ =[A0 + α11 (ε)]ξ + α12 (ε)η + (B0 + β1 (ε))u + (In − εHL) f¯1 (ξ , η) − H f¯2 (ξ , η) ε η˙ =α21 (ε)ξ + [A22 + α22 (ε)]η + (B2 + β2 (ε))u + εL f¯1 (ξ , η) + f¯2 (ξ , η)
in real control system. One may take their approximate slow and fast control systems as a model. ( ˙ ξˆ (t) = Aˆ 0 · ξˆ (t) + Bˆ 0 u t ∈ [tk ,tk+1 ), (5) ˙ˆ = Aˆ 22 · η(t) ˆ + Bˆ 2 u ε η(t)
Define the state error between the plant (4) and the model (5) as e1 (t) = ξ (t) − ξˆ (t),
ˆ e2 (t) = η(t) − η(t),
with t ∈ [tk ,tk+1 ). Assume that the sensor sends the state information through the network every time h, the update times are at tk second where tk+1 − tk = h for all k = 0, 1, 2, · · · . Since the model’s state is updated at tk , it holds that e1 (tk ) = e2 (tk ) = 0,
(4)
with t ∈ [tk ,tk+1 ), where
k = 0, 1, 2, · · ·
B0 = B1 − A12 A−1 22 B2 , α11 (ε) = −εHLA0 , α21 (ε) = εLA0 , α12 (ε) = ε(A0 H − HLA12 ) − ε 2 HLA0 H, α22 (ε) = εLA12 + ε 2 LA0 H, β1 (ε) = −εHLB1 , β2 (ε) = εLB1 , ¯f1 (ξ , η) = f1 (ξ + εHη, −Lξ + (Im − εLH)η), f¯2 (ξ , η) = f2 (ξ + εHη, −Lξ + (Im − εLH)η). Omitting the nonlinear uncertainties of the plant, we would get the slow and fast control systems of the plant if the accurate dynamic of the plant is given. But it is impossible
(8)
This resetting of the state error at every update time is a key for model-based NCSs. Let # # " " ξ (t) η(t) , ζ2 (t) = . ζ1 (t) = e1 (t) e2 (t) From (7), one may get " # ξ (t) ζ1 (t) = Pn · ˆ , ξ (t)
A0 = A11 − A12 A−1 22 A21 ,
(7)
"
# η(t) ζ2 (t) = Pm · , ˆ η(t) Im 0m In 0n . , Pm = with Pn = Im − Im In − In By using the Kronecker product, we can obtain the following dynamics of ζ1 (t) and ζ2 (t) for t ∈ [tk ,tk+1 ). ζ˙1 (t) =Pn (Λ11 + α¯ 11 (ε))Pn ζ1 (t) + Pn (Λ12 + α¯ 12 (ε))Pm ζ2 (t) + 12 ⊗ F1 (ζ1 , ζ2 ) (9) ˙ ε ζ2 (t) =Pm (Λ21 + α¯ 21 (ε))Pn ζ1 (t) + Pm (Λ22 + α¯ (ε))P ζ (t) + 1 ⊗ F (ζ , ζ )
685
22
m 2
2
2
1
2
WeAIn1.8 with t ∈ [tk ,tk+1 ), where A0 A¯ 12 Λ0 = , 0n×n A∗0 ¯ + α¯ 11 (ε) ¯ 0 + α¯ 11 (ε) − α¯ 12 (ε)L) γ11 (ε) = − ε H¯ L(Λ ¯ ¯ α¯ 22 (ε)L¯ − α¯ 21 (ε)) − α¯ 12 (ε)L, + H( 2 ¯ 12 ) − ε H¯ LΛ ¯ 0 H¯ − H¯ α¯ 22 (ε) γ12 (ε) =ε(Λ0 H¯ − H¯ LΛ ¯ α¯ 22 (ε)L¯ − α¯ 21 (ε))H¯ − ε H¯ L¯ α¯ 12 (ε) + ε H( ¯ α¯ 11 (ε) − α¯ 12 (ε)L) ¯ H¯ + ε(1 − ε H¯ L)(
where Λ11 =
Λ22 =
A0 0
B0 K1 Aˆ 0 + Bˆ 0 K1
A22 0
B2 K2 Aˆ 22 + Bˆ 2 K2
,
Λ12 =
, Λ21 =
0 B2 K2 0 Bˆ 0 K2
,
0 B2 K1 0 Bˆ 0 K1
,
α¯ 11 (ε) =
α11 (ε) β1 (ε)K1 0 0
,
α¯ 12 (ε) =
α12 (ε) β1 (ε)K2 0 0
,
α¯ 21 (ε) =
α21 (ε) β2 (ε)K1 0 0
,
α¯ 22 (ε) =
α22 (ε) β2 (ε)K2 0 0
,
+ α¯ 12 (ε), ¯ 0 + α¯ 11 (ε) − α¯ 12 (ε)L) ¯ γ21 (ε) =ε L(Λ ¯ + α¯ 21 (ε) − α¯ 22 (ε)L, 2 ¯ 12 + ε LΛ ¯ 0 H¯ + ε L¯ α¯ 12 (ε) γ22 (ε) =ε LΛ ¯ H, ¯ ¯ α¯ 11 (ε) − α¯ 12 (ε)L) + ε 2 L( ¯ H¯ + α¯ 22 (ε), + ε(α¯ 21 (ε) − α¯ 22 (ε)L) ¯ F¯1 (X, Z) − H¯ F¯2 (X, Z), g1 (X, Z) = (I2n − ε H¯ L) g2 (X, Z) = ε L¯ F¯1 (X, Z) + F¯2 (X, Z),
and F1 = (In − εHL) f¯1 (E1 ζ1 , E2 ζ2 ) − H f¯2 (E1 ζ1 , E2 ζ2 ) F2 = εL f¯1 (E1 ζ1 , E2 ζ2 ) + f¯2 (E1 ζ1 , E2 ζ2 )
with
with 12 = [1 1]T , E1 = [In 0], E2 = [Im 0]. The nonlinear functions Fi , i = 1, 2 represent the uncertainties of system (9). Under the conditions of the bounded functions fi , i = 1, 2, one can get the bounds of their norms as follows. k Fi (ζ1 (t), ζ2 (t)) k≤ a¯i k ζ1 (t) k +b¯ i k ζ2 (t) k,
a¯2 =εa1 k L k +a2 + εb1 k L k2 +b2 k L k, b¯ 1 = k H k (εa1 λ1 + εa2 k H k +b2 λ2 ) + b1 λ1 λ2 , b¯ 2 =ε k H k (a2 + εa1 k L k) + λ2 (b2 + εb1 k L k)
with
ε H¯ I2m − ε L¯ H¯
,
k gi (X, Z) k≤ a˜i k X k +b˜ i k Z k,
(13)
b˜ 2 =ε k H¯ k (a¯2 + ε a¯1 k L¯ k) + λ¯ 2 (b¯ 2 + ε b¯ 1 k L¯ k)
with λ1 =k In − εHL k and λ2 =k Im − εLH k. If the controller gain K2 is designed such that (Aˆ 22 + ˆ B2 K2 )−1 exists, it is obvious that Λ−1 22 exists and one may define [ζ1T (t), ζ2T (t)]T = T¯ · X T (t), Z T (t) I2n −L¯
The nonlinear functions gi , i = 1, 2 may be the uncertainties of the equivalent system (12) and are bounded by
where a˜1 =λ¯ 1 (a¯1 + b¯ 1 k L¯ k)+ k H¯ k (a¯2 + b¯ 2 k L¯ k), a˜2 =ε a¯1 k L¯ k +a¯2 + ε b¯ 1 k L¯ k2 +b¯ 2 k L¯ k, b˜ 1 = k H¯ k (ε a¯1 λ¯ 1 + ε a¯2 k H¯ k +b¯ 2 λ2 ) + b¯ 1 λ¯ 1 λ¯ 2 ,
a¯1 =λ1 (a1 + b1 k L k)+ k H k (a2 + b2 k L k),
¯ −LX ¯ + (I2m − ε L¯ H)Z), ¯ F¯1 = 12 ⊗ F1 (X + ε HZ, ¯ −LX ¯ + (I2m − ε L¯ H)Z). ¯ F¯2 = 12 ⊗ F2 (X + ε HZ,
(10)
where
T¯ =
A∗0 = Aˆ 0 + Bˆ 0 K1 − Bˆ 0 K2 (Aˆ 22 + Bˆ 2 K2 )−1 B2 K1 , A¯ 12 = B0 K1 − B2 K2 (Aˆ 22 + Bˆ 2 K2 )−1 Bˆ 2 K1
(11)
−1 ¯ where H¯ = Λ12 Λ−1 22 and L = Λ22 Λ21 . Applying transformation (11) into system (9), we can get the following equivalent form of system (9). ˙ X =Pn [Λ0 + γ11 (ε)]Pn X + Pn γ12 (ε)Pm Z + g1 (X, Z) (12) ˙ ε Z =Pm γ21 (ε)Pn X + Pm [Λ22 + γ22 (ε)]Pm Z + g2 (X, Z)
with λ¯ 1 =k I2n − ε H¯ L¯ k and λ¯ 2 =k I2m − ε L¯ H¯ k. Suppose that Λ−1 22 exists, one may take the dynamic system (12) as a standard singularly perturbed system with nonlinear uncertainties at t ∈ [tk , tk+1 ). We will design a control law based on the slow and fast systems of system (12) such that system (12) is robustly stabilized. One can get their slow and fast systems of the linear part by omitting the nonlinear uncertainties. The slow system is derived as ¯ 0 Xs , t ∈ [tk , tk+1 ) X˙s = Λ
(14)
¯ 0 = Pn Λ0 Pn , and the fast system is described with Λ dZ f ¯ 22 Z f , τ ∈ [τk , τk+1 ) (15) =Λ dτ t ¯ 22 = Pm Λ22 Pm . with τ = and Λ ε For linear slow and fast systems, controller gains K1 and K2 may be designed such that they are both exponentially stable. Let In 0n Im 0m Rn = , Rm = , (16) 0n 0n 0m 0m
686
WeAIn1.8 we have the following result. Theorem 1: Consider systems (14) and (15) with t ∈ [tk , tk+1 ). Suppose that A22 is Hurwitz stable. If there exist ˆ 2 is Hurwitz controller gains K1 and K2 such that Aˆ + BK ¯ stable and Rn eΛ0 h Rn is Schur stable, then the solutions of systems (14) and (15) are globally exponentially stable. Proof Considering the fact that the model’s state is updated at tk , we can obtain the solution of system (14) ¯
i = 1, 2, one obtains k Gi (X, Z) k≤ µi (ε) k X k +νi (ε) k Z k, where i = 1, 2, µi (ε) = a˜i + k γi1 (ε) k, νi (ε) = b˜ i + k γi2 (ε) k . − w(tk+1 ) = H · w(tk− ) + Ω(tk− ),
and ¯
Xs (tk− ) = eΛ0 h Xs (tk−1 ).
with
H =
Then one may get (Rn e
Rn )k Xs (t0 ).
d1
¯
||(Rn eΛ0 h Rn )k || ≤ c1 e−d1 k ≤ c1 ed1 e− h t . Therefore, one gets that d1
¯
||Xs (t)|| ≤ e||Λ0 ||h+d1 c1 e− h t kXs (t0 )k, which also implies that the solution of system (14) is globally exponentially stable. Next, one may get the solution of system (15) ¯
Z f (τ) = eΛ22 (τ −τk ) (Rm e
¯ h Λ 22 ε
e
¯ h Λ 22 ε
Rm
,
Ω(tk− ) =
Z tk+1 ¯ 0 (tk+1 −s) Λ
e
tk
1 ε
Z tk+1
e
# Ω1 (tk− )
(21)
Ω2 (tk− )
G1 (X, Z)ds,
¯ Λ 22 ε (tk+1 −s)
tk
"
(22) G2 (X, Z)ds.
||Ω1 (tk− )|| ≤δ1 (ε)||w(tk− )||,
(23)
||Ω2 (tk− )|| ≤δ2 (ε)||w(tk− )||, where c
¯ 0 ||+µ1 (ε )+ν1 (ε ))h+ 0 (µ2 (ε )+ν2 (ε )) (||Λ d
ρ(ε) =(1 + c0 )e
¯
||eΛ22 (τ −τk ) || ≤ c2 e−d2 (τ −τk ) ≤ c2 . ¯ h Λ 22
We also obtain that Rm e ε Rm is Schur stable. So there exist positive constants c4 , c5 , d3 and d4 such that ¯ h Λ 22 ε
0
Considering system (20) with the nonlinear term Ω(tk− ), we will get the following result. Theorem 2: Consider system (12) and (20) with tk+1 − ˆ 2 are Hurwitz stable. There tk = h. Suppose A22 and Aˆ + BK exist positive constants c0 and d0 , independent of ε, such that ||w(t)|| ≤ρ(ε)||w(tk− )||,
Rm )k Z f (τ0 ).
¯ 22 is Hurwitz stable since A22 and A+ ˆ BK ˆ 2 It is obvious that Λ are Hurwitz stable. So there exist positive constants c2 and d2 , independent of ε, it holds that
0
,
¯ (ε))(e||Λ0 ||h − 1)
ρ(ε)(µ1 (ε) + ν1 , ¯ 0 || ||Λ ρ(ε)c0 (µ2 (ε) + ν2 (ε)) δ2 (ε) = d0 with µi (ε), νi (ε), i = 1, 2 are defined in (19). Proof From the first equation of (12), one gets δ1 (ε) =
X(t) =X(tk ) +
Rm )k || ≤ c3 e−d3 k ≤ c4 e−d4 τ .
Z t tk
(24)
¯ 0 X + G1 (X, Z))ds (Λ
For t ∈ [tk ,tk+1 ), we have
Then, one gets that
||X(t)|| ≤||X(tk )|| +
||Z f (τ)|| ≤ c2 c4 e−d4 τ kZ f (τ0 )k, which also implies that the solution of system (15) is globally exponentially stable. Remark 1: Under the condition of Theorem 1, there exist positive definition symmetric matrices Φ = ΦT > 0, Ψ = ΨT > 0 i = 1, 2 such that ¯
0
Ω2 (tk− ) =
¯
Since Rn eΛ0 h Rn is Schur stable, then exist positive constants c1 and d1 such that
||(Rm e
¯
eΛ0 h Rn
Ω1 (tk− ) =
Xs (t) = e
(20)
where
¯
¯ 0h Λ
(19)
Letting w(t) = (X T (t), Z T (t))T , we have
Xs (t) = eΛ0 (t−tk ) Xs (tk ) = eΛ0 (t−tk ) Rn Xs (tk− ),
¯ 0 (t−tk ) Λ
(18)
¯
(Rn eΛ0 h Rn )T ΦRn eΛ0 h Rn − Φ = −In ,
(17) ¯ ¯ Λ Λ 22 22 (Rm e ε h Rm )T ΨRm e ε h Rm − Ψ = −Im . Now, we mainly discuss the norm bound of the nonlinear uncertainties of system (12). Denoting
Z t tk
¯ 0 X|| + ||G1 (X, Z)||)ds. (||Λ
¯ 22 defined in ˆ 2 are Hurwitz stable, Λ Since A22 and Aˆ + BK (9) is Hurwitz stable. There exist positive constants c0 and d0 , independent of ε, it holds that ¯
||eΛ22
t−tk ε
|| ≤ c0 e−d0
t−tk ε
.
Then from the second equation of (12), it follows that Z ¯ (t−t ) Λ 1 t Λ¯ 22 (t−s) 22 k e ε G2 (X, Z)ds Z(t) =e ε Z(tk ) + ε tk
Gi (X, Z) = Pn γi1 (ε)Pn X + Pn γi2 (ε)Pm Z + gi (X, Z)
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t−tk
≤c0 e−d0 ε · ||Z(tk )|| Z t−s 1 t + c0 e−d0 ε · ||G2 (X, Z)||ds. ε tk
WeAIn1.8 By means of the property of norms for vectors and (18), we have ||w(t)|| ≤||X(t)|| + ||Z(t)|| ≤||X(tk )|| + c0 e−d0 +
Z t tk
+
· ||Z(tk )||
¯ 0 || + µ1 (ε))||X|| + ν1 (ε)||Z||]ds [(||Λ
Z t
1 ε
t−tk ε
tk
c0 e−d0
t−s ε
(µ2 (ε)||X|| + ν2 (ε)||Z||)ds
≤(1 + c0 )||w(tk )|| +
Z t tk
IV. AN EXAMPLE ¯ 0 || + µ1 (ε) (||Λ
In this section, we consider the plant described by Fig.1 is a modified armature-controlled DC motor [14], which consists of a mechanical torque equation and an equation for the electrical transient in the armature circuit, namely,
t−s c0 +ν1 (ε) + e−d0 ε (µ2 (ε) + ν2 (ε)))||w||ds. ε By the Gronwall inequality, one gets
Rt
||w(t)|| ≤(1 + c0 )||w(tk )|| · e tk
J w˙ = ki + (2 + ϑ1 )kw + f (w, i) Li˙ = −kw − Ri + ϑ2 u
¯ 0 ||+µ1 (ε )+ν1 (ε ))ds (||Λ
R t c0 −d t−s e 0 ε ·(µ2 (ε )+ν2 (ε ))ds t
·e
k ε
=(1 + c0 )||w(tk )|| · e
t−tk c0 (µ2 (ε)+ν2 (ε)) (1−e−d0 ε d0
)
≤ρ(ε)||w(tk− )||. From (22), one obtains that ||Ω1 (tk− )|| ≤ =
R h ||Λ¯ ||s 0 · (µ1 (ε) + ν1 (ε))||w||ds 0 e ¯
ρ (ε )(µ1 (ε )+ν1 (ε ))(e||Λ0 ||h −1) ||w(tk− )|| ¯ 0 || ||Λ
and ||Ω2 (tk− )|| ≤ =
(t
x(t) ˙ = (2 + ϑ1 )x(t) + z(t) + f (x, z), ε z˙(t) = −x(t) − z(t) + ϑ2 u(t),
−s)
k+1 1 R tk+1 c0 e−d0 ε (µ2 (ε) + ν2 (ε))||w||ds ε tk h ρ (ε )c0 (µ2 (ε )+ν2 (ε )) (1 − e−d0 ε )||w(tk− )||. d0
||Ω1 (tk− )||
Therefore, it holds that ≤ and ||Ω2 (tk− )|| ≤ δ2 (ε)||w(tk− )||. Based on Theorem 1 and Theorem 2, the closed-loop system (12) can be robustly stabilized by the controller (6) which is designed in Theorem 1. Now, we give the result in the following. The proof is omitted for saving paper space here. Theorem 3: Consider the closed-loop system (12) with tk+1 − tk = h,t ∈ [tk ,tk+1 ). Under the conditions of Theorem 1, and for an arbitrary given constant 0 < θ < 1, if δi , i = 1, 2 defined in (24) with ε = 0 satisfy (25)
1 xz where f (x, z) = 1+z 4 . It is obvious that | f (x, z) |≤ 2 | x |. One may get the following equivalent system for all t ∈ [tk ,tk+1 ) via the transformation, which is defined in (3). ξ˙ (t) =[1 + ϑ1 + ε(1 + ϑ1 )] · ξ (t) + (1 + ε) f¯(ξ , η)
+ ϑ2 u(t) − [ϑ1 ε + (1 + ϑ1 )ε 2 ] · η(t), ˙ =ε(1 + ϑ1 ) · ξ (t) + ε f¯(ξ , η) ε η(t)
¯
σ1 = ||eΛ0 h Rn ||, σ2 = ||e
Rm ||,
(ξ − εη)[(1 + ε)η − ξ ]2 . Omitting the nonwith f¯(ξ , η) = 1 + [(1 + ε)η − ξ ]4 linear part of (29), we take an approximate slow and fast systems as the plant model. ˙ ξˆ (t) = ξˆ (t) + u(t) ˙ˆ = η(t) ˆ + u(t) ε η(t)
(26)
Rn , Rm are defined in (16) and Φ, Ψ are defined in (17). Then there exists an ε ∗ > 0 such that the states of system (12) are globally exponentially stable for all ε ∈ (0, ε ∗ ]. Remark 2: If the feedback controller gains K1 , K2 of the model and the bound of nonlinear uncertainties of the plant are given, the approximate bound ε ∗ can be determined by solving the following inequality δ1 (ε)||Φ||(2σ1 + δ1 (ε)) + δ2 (ε)||Ψ||(2σ2 + δ2 (ε)) ≤ 1
(29)
+ ϑ2 u(t) + [−1 + ε − ε 2 (1 + ϑ1 )] · η(t).
where ¯ h Λ 22 ε
(28)
2
δ1 (ε)||w(tk− )||
δ1 ||Φ||(2σ1 + δ1 ) + δ2 ||Ψ||(2σ2 + δ2 ) ≤ θ ,
(27)
where i, u, R and L are the armature current, voltage, resistance and inductance respectively, J is the moment of inertia, w is the angular speed, and ki, kw are respectively the torque and the back electromotive force (e.m.f.) developed with constant excitation flux φ . Practically, L is small and can play the role of our parameter ε. ϑs , s = 1, 2 is two modulatory parameters in the original systems. For simplicity, the nonlinear uncertainties of the systems are taken as w(ki)2 f (w, i)) = 1+(ki) 4 . And we choose J = k = R = 1, and assume L = ε, w = x, i = z, then (27) can be rewritten as the standard form of singularly perturbed control systems
¯ 0 ||+µ1 (ε )+ν1 (ε ))h (||Λ
·e
It is obvious that there is a contradiction between the norm bound of nonlinear uncertainties and the approximate bound ε ∗. Remark 3: It is worth to point out that our results are based on the condition that the fast systems A22 of the plant is Hurwitz stable. That implies that the slow systems A0 of the plant don’t need to be stable. This case will be shown in the following example.
(30)
and the controller designed is ˆ u(t) = k1 · ξˆ (t) + k2 · η(t)
(31)
Define the errors of the slow and states between the plant and the model as ˆ e1 (t) = ξ (t) − ξˆ (t), e2 (t) = η(t) − η(t). Let the initial value of system (28) be [x(0), z(0)]T = [2, 1]T and the sampling period h = 1. The controller gains
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WeAIn1.8 are designed as k1 = −1, k2 = −2 and the parameter is chosen as ε = 0.1. Fig.2 shows that the error states and the states of the responding closed-loop system with ϑ1 = −2 and ϑ2 = 1. 2 state x error of slow states state z error of fast states
States of whole control systems
1.5
1
0.5
0
f0.5
0
0.5
1
1.5
2
2.5
3
3.5
Time
Fig. 2. The states and error states of the networked control system.
While ϑ1 = −1 and ϑ2 = 1, one may get A0 = 0. In this case, the error states and the states of the responding closedloop system are also asymptotically stable. The simulation result is shown in Fig.3. 2 state x error of slow states state z error of fast states
States of whole control systems
1.5
1
0.5
[4] A. S. Matveev and A. V. Savkin, “Optimal Control of Networked Systems via Asynchronous Communication Channels with Irregular Delays,” Proc. The 40th IEEE on Conference Decision and Control, Orland Florida, USA, pp. 2323-2332, 2001. [5] L. A. Montestruque and P. J. Antsaklis, “Model-Based Networked Control Systems: Necessary and Sufficient Conditions for Stability,” in the 10th Mediterrannean Conf. Control and Automation, Lisbon, Portugal, 2002. [6] L. A. Montestruque, and P. J. Antsaklis, “Stability of Model-Based Networked Control Systems with Time-Varying Transmission Times,” IEEE Trans. Automat. Contr., vol. 49, pp. 1562-1572, 2004. [7] S. Mastellon, C. T. Abdallah and P. Dorato, “Model-Based Networked Control for Nonlinear Systems with Stochastic Packet Dropped”, in the 2005 American Control Conference, Portland, USA. , pp. 23652370, 2005. [8] L. A. Montestruque, and P. J. Antsaklis, “On the Model-Based Control of Networked Systems,” Automatica, vol. 39, pp. 1837-1843, 2003. [9] H. W. Yu and Y. F. Zheng, “On Model-Based Networked Control for Singularly Perturbed Systems,” in the IEEE International Conference on Control and Automation, Guangzhou, China, pp. 1548-1552, 2007. [10] G. X. Wang, Z. M. Wang and D. S. Naidu, “On Model-Based Networked Control of Singularly Perturbed Systems,” in the 27th Chinese Control Conference, Kunming, Yunnan, China, pp. 53-57, 2008. [11] Z. H. Shao and M. E. Sawan, “Stabilisation of Uncertain Singularly Perturbed Systems,” IEE Proc. Control Theory Appl., vol. 153, pp. 99-103, 2006. [12] A. R. Teel, L. Moreau and D. Nesic, “An Unified Framework for Inputto-State Stability in Systems with Two Time Scales,” IEEE Trans. Automat. Contr., vol. 48, pp. 1526-1544, 2003. [13] Z. H. Shao, “Robust Stability of Two-Time-Scale Systems with Nonlinear Uncertainties,” IEEE Trans. Automat. Contr., vol. 49, pp. 258-261, 2004. [14] P. V. Kokotovi´c, H. K. Khalil and J. O’reilly, Singular Perturbation Methods in the Control: Analysis and Design, Academic Press, London, UK, 1986. [15] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, UK, 1985.
0
f0.5
0
5
10
15
20
25
Time
Fig. 3. The states and error states of the networked control system.
V. CONCLUSION In this paper the robust control of SPS with nonlinear uncertainties over a network is addressed. We use an approximate slow and fast systems of the plant as a model plant in the actuator side of the plant to produce an estimation of the plant state behavior between transmission times. Under the assumption that the fast systems of the plant is stable, a state feedback control law has been designed such that the networked control system is robustly globally exponentially stable. Finally, an example is given to illustrate the results proposed in this paper. The work under this paper can be extended to the cases where network-induced delay and/or output feedback control are considered. R EFERENCES [1] D. Nesic and A. R. Teel, “Input-to-State Stability of Networked Control Systems,” Automatica, vol. 40, pp. 2123-2128, 2004. [2] D. Yue, Q. L. Han and C. Peng, “State Feedback Controller Design of Networked Control Systems,” IEEE Trans. circuits and systems-II: express briefs, vol. 51, pp. 640-644, 2004. [3] P. V. Zhivoglyadov, and R. H. Middleton, “Networked Control Design for Linear Systems,” Automatica, vol. 39, pp. 743-750, 2003.
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