Guaranteed cost control of linear systems over networks with state and input quantisations D. Yue, C. Peng and G.Y. Tang Abstract: The guaranteed cost control design for linear systems connected over a common digital communication network is addressed. A new model is proposed that takes into consideration the effect of both the quantisation levels and the network conditions. A control design criterion is derived on the basis of the Lyapunov functional method and the idea of the cone complementary linearisation algorithm. A numerical example is given to show the application of the method proposed.
1
Introduction
Recently, much attention has been paid to the study of the control problems of systems connected over a digital communication network, also called network-based control systems (NCSs). Different to the classical control theory, the effect of the communication network on the system must be considered when discussing the control problems for NCSs. Considering the limited communication capacity in the network, the quantisation problem of the feedback controller has been investigated by many researchers [1 – 6]. The time-invariant [2, 3] or time-varying [1, 5, 6] quantisation methods were proposed although only the case where the communication over the channel was in one direction was considered. For the NCS with the communication over the channel being in two directions, the quantised control design problem has been discussed by Ishii and Basar [7] using two quantisers in the network, both from sensor to controller and from controller to actuator. However, it was assumed that the network was error-free and that no network-induced delay and data dropout occurred in the data transmission. This assumption is obviously not true of some real NCSs [8, 9]. The effects of the network-induced delay and data dropout on the system performance, stability and control design problems for NCSs have been investigated by many researchers in recent years [8 –12]. In these works, however, the quantisation of the state signal from sensors and the control input was neglected. To date, to the best of the authors’ knowledge, the problem of how to design a quantised feedback controller for NCSs when considering the effect of the network conditions, such as the network-induced delay and data dropout, is still to be solved.
In this paper, the guaranteed cost control design (GCCD) problem for NCSs with the communication over the channel being in two directions, is examined. The quantised feedback controller is designed with regard to effect of network-induced delay and data dropout on the system. As the computation available in the coder and the decoder on the plant side is often very limited, the quantisers employed in this paper are of the time-invariant type. Using the sector bound expression of the quantisation density and the Lyapunov functional method, a criterion for the controller design is derived, which is then solved in terms of a set of matrix inequalities. To minimise the bound of the cost function, an optimisation problem based on the variation bound of the initial function and the idea of the cone complementary linearisation algorithm is proposed. To show the applications of the method proposed, a numerical example is given. Notation: Rn denotes the n-dimensional Euclidean space, Rnm the set of n m real matrices, I the identity matrix of appropriate dimensions, and k.k is the Euclidean vector norm or the induced matrix 2-norm as appropriate. The notation X . 0 (respectively X 0), for X [ Rnm, means that the matrix X is a real symmetric positive definite (respectively positive semi-definite). lmax(P) (lmin(P)) denotes the maximum (minimum) of the eigenvalue of matrix P. For an arbitrary matrix B and two symmetric A matrices A and C, denotes a symmetric matrix, B C where ‘’ denotes the entries implied by symmetry. 2
System description and preliminaries
Consider the following linear system with parameter uncertainty # The Institution of Engineering and Technology 2006 IEE Proceedings online no. 20050294 doi:10.1049/ip-cta:20050294
x_ ðtÞ ¼ ½A þ DAðtÞ xðtÞ þ BuðtÞ
ð1Þ
D. Yue and C. Peng are with the Institute of Information and Control Engineering Technology, Nanjing Normal University, 78 Bancang Street, Nanjing, Jiangsu 210042, People’s Republic of China
where x(t) [ Rn and u(t) [ Rm are the state vector and control input vector, respectively, A and B are two constant matrices, x0 [ Rn is the initial condition, and DA(t) is the parameter uncertainty appearing in the state, which satisfies
G.Y. Tang is with College of Information Science and Engineering, Ocean University of China, Qingdao 266071, China
DAðtÞ ¼ DFðtÞEa
Paper first received 16th August and in revised form 6th December 2005
ð2Þ
E-mail:
[email protected]
658
IEE Proc.-Control Theory Appl., Vol. 153, No. 6, November 2006
where D and Ea are two constant matrices and F(t) satisfies kF(t)k 1. Associated with system (1), the following cost function ð1 T x ðtÞRxðtÞ þ uT ðtÞQuðtÞ dt ð3Þ J¼ t0
where t0 0, R . 0 and Q . 0. The aim of this paper is to design a control law u(t) such that system (1) under this controller is exponentially stable and satisfies the requirements on the cost performance, that is, J exists such that J J . This problem can be called GCCD. The GCCD problem for system (1) with a linear controller u(t) ¼ Kx(t) has been investigated by many researchers in the sense that the closed loop is a continuous-time or sampled-data system [3, 4]. However, all the results were derived by Mahmoud [13, 14] on the basis of an implicit assumption that the channel from sensor to controller and then to actuator is error-free and no delay exists. This assumption is obviously not true for the NCSs [8, 9]. Different from traditional control systems, the controller considered in this paper is imposed on system (1) through a common digital communication network medium. Moreover, considering the limited capacity in the communication channel and also for reducing the amount of data rate being transmitted in the network, which thus leads to an increase in the quality of service of the network, the state signal from the sensor and the control input signal are quantised by two quantisers, respectively. The NCS with quantisations is shown in Fig. 1. System (1) under a network-based quantised controller can be modelled as x_ ðtÞ ¼ ½A þ DAðtÞ xðtÞ þ BuðtÞ uðtÞ ¼ f ðvðtÞÞ vðtÞ ¼ Kgðxðik hÞÞ
ð4Þ
time delay, which denotes the time from the instant ikh when sensor nodes sample sensor data from a plant to the instant S 1 when actuators transfer data to the plant. Obviously, k¼1 [ikh þ tk , ikþ1h þ tkþ1) ¼ [t0 , 1), t0 0. In this paper, we assume that u(t) ¼ 0 before the first control signal reaches the plant and a constant h . 0 exists such that (ikþ1 2 ik)h þ tkþ1 h, k ¼ 1, 2, . . . . Remark 1: If f(v) ¼ v and g(x) ¼ x, (4) – (6) reduce to x_ ðtÞ ¼ ½A þ DAðtÞ xðtÞ þ BuðtÞ uðtÞ ¼ Kxðik hÞ;
fi(.) and gj(.) (i ¼ 1, 2, . . . , m; j ¼ 1, 2, . . . , n) are assumed to be symmetric, that is, fi(2vi) ¼ 2fi(vi) and gj(2xj) ¼ 2gj(xj). h is the sampling period, ik (k ¼ 1, 2, 3, . . .) are some integers and fi1 , i2 , i3 , . . .g , f0, 1, 2, 3, . . .g. tk is the
t [ ½ik h þ tk ; ikþ1 h þ tkþ1 Þ ð10Þ
which was investigated in Yue et al. [12]. Remark 2: In (4) – (6), fi1 , i2 , i3 , . . .g is a subset of f0, 1, 2, 3, . . .g. Moreover, it is not required that ikþ1 . ik . When ik . ikþ1 , it indicates the situation of error sequence of the data packets. When fi1 , i2 , i3 , . . .g ¼ f0, 1, 2, 3, . . .g, it means no packet dropout occurs in the transmission. If ikþ1 ¼ ikþ1, it implies that h þ tkþ1 . tk , which includes tk ¼ t0 and tk , h as the special cases. Therefore (4) – (6) can be viewed as a general form of the model of the systems under a network-based quantised controller where the effect of the quantisation, network-induced delay and data packet dropout, are simultaneously considered. The set of quantisation levels is described by U ¼ f+ui ; i ¼ +1; +2; . . .g < f0g
ð11Þ
Denote by #g[1] the number of quantisation levels in the interval [1, 1/1]. The density of the quantiser is defined as follows [2]
ð5Þ t [ ½ik h þ tk ; ikþ1 h þ tkþ1 Þ ð6Þ
where K [ Rmn is the feedback gain to be determined and f(.) and g(.) are two quantisers defined as T f ðvÞ ¼ f 1 ðv1 Þ; f 2 ðv2 Þ; . . . ; f m ðvm Þ ð7Þ T gðxÞ ¼ g1 ðx1 Þ; g2 ðx2 Þ; . . . ; gn ðxn Þ ð8Þ
ð9Þ
h0 ¼ lim sup
#g½1
1!0 ln 1
ð12Þ
In this paper, the quantisers are logarithmic. That is, for fi or gj (i ¼ 1, 2, . . . , m; j ¼ 1, 2, . . . , n), the set of quantisation levels is chosen as n o Uzk ¼ +uðiÞ : uðiÞ ¼ rizk u0 ; i ¼ +1; +2; . . . < f+u0 g < f0g
ð13Þ
where zk is fk or gk , 0 , rzk , 1 is given for zk . It is known [4] that the smaller the rzk , the smaller the h0 . Therefore in the following, we also call rzk the quantisation density of the quantiser zk . The associated quantiser zk is defined as 8 1 1 > > if ui , b ui ; b . 0 < ui ; 1 þ d zk 1 d zk ð14Þ z k ð bÞ ¼ 0; if b ¼ 0 > > : zk ðbÞ; if b , 0 where
d zk ¼
Fig. 1 Structure of NCSs with quantisations IEE Proc.-Control Theory Appl., Vol. 153, No. 6, November 2006
1 r zk 1 þ r zk
ð15Þ
In terms of the method given in Fu and Xie [3], for any fi(.) or gj(.), a sector-bound expression can be given as follows fi ðvi Þ ¼ 1 þ Dfi ðvi Þ vi ð16Þ gj ðxj Þ ¼ 1 þ Dgj ðxj Þ xj ð17Þ 659
where jDfi(vi)j dfi and jDgj(xj)j dgj . For the sake of simplicity, in the following, we use Dfi and Dgj to denote Dfi(vi) and Dgj(xj), respectively. Define Df ¼ diag Df1 ; Df2 ; . . . ; Dfm Dg ¼ diag Dg1 ; Dg2 ; . . . ; Dgn
appropriate dimensions such that 2 L11 þ R þ W 6 L L þ ½ K þ DðKÞ T Q½K þ DðKÞ 21 22 6 6 4 L31 L32
ð18Þ
hN T1
Then, f (.) and g(.) can be written as f ðvÞ ¼ I þ Df v gðxÞ ¼ I þ Dg x
ð27Þ
hT k ¼ 1; 2; . . .
ðikþ1 ik Þh þ tkþ1 h;
ð21Þ
ð28Þ
where
Combining (20) and (21), (4) – (6) can be expressed as
L11 ¼ N 1 þ N T1 M 1 ½A þ DAðtÞ ½A þ DAðtÞT M T1 L21 ¼ N 2 N T1 M 2 ½A þ DAðtÞ ½BK þ BDðKÞT M T1 L31 ¼ N 3 M 3 ½A þ DAðtÞ þ M T1 þ P L22 ¼ N 2 N T2 M 2 ½BK þ BDðKÞ ½BK þ BDðKÞT M T2 L32 ¼ N 3 þ M T2 M 3 ½BK þ BDðKÞ
x_ ðtÞ ¼ ½A þ DAðtÞ xðtÞ þ B I þ Df K I þ Dg xðik hÞ xðtÞ ¼ Fðt; t0 hÞ xðt0 hÞ W fðtÞ t [ ½t0 h; t0
L33 þ W hN T3
ð20Þ
where I denotes the identity matrix of appropriate dimensions. For simplicity, it is assumed that dfi ¼ df and dgj ¼ dg , where df and dg are known numbers. Therefore from (15), the above assumption means that rfi ¼ rf ¼ (1 2 df)/(1 þ df ) and rgj ¼ rg ¼ (1 2 dg)/ (1 þ dg).
t [ ½ik h þ tk ; ikþ1 h þ tkþ1 Þ
3 7 7 7,0 5
ð19Þ
hN T2
ð22Þ
L33 ¼ M 3 þ M T3 þ hT
ð23Þ
then, systems (25) and (26) are exponentially stable and the upper bound J of cost function J is given as ð t0 f_ T ðsÞT f_ ðsÞ ds ð29Þ J ¼ xT ðt0 ÞPxðt0 Þ þ
˙ (t, t0 2 h) ¼ [A þ where F(t, t0 2 h) is a solution to F DA(t)]F(t, t0 2 h), t [ [t0 2 h, t0]. It can be seen from (22) that the network conditions and the quantisations of the state and control input will affect the performance of the system. Therefore the GCCD problem for system (1) under a network-based quantised controller in (5) and (6) is described as follows. For the given rf, rg and h, design the feedback gain K such that, if (ikþ1 2 ik)h þ tkþ1 h, k ¼ 1, 2, . . . , systems (22) and (23) are exponentially stable and a scalar J exists such that J J . In this case, the GCCD problem for system (1) under a network-based quantised controller in (5) and (6), is solvable.
t0 h
Proof: Construct the following Lyapunov function ðt ðt V ðtÞ ¼ xT ðtÞPxðtÞ þ x_ T ðvÞT x_ ðvÞ dv ds
ð30Þ
th s
where P . 0 and T .Ð 0. As x(t) 2 x(ikh) 2 tikh x˙(s) ds ¼ 0 and (25), one can see that for arbitrary matrices Ni and Mi (i ¼ 1, 2, 3) of appropriate dimensions ðt x_ ðsÞds ¼ 0 ð31Þ eT ðtÞ N xðtÞ xðik hÞ ik h
3
Performance analysis
and
In this section, we present a result for the performance analysis of systems (22) and (23). Note that
eT ðtÞ Mf½A þ DAðtÞ xðtÞ ½BK þ BDðKÞxðik hÞ þ x_ ðtÞg ¼ 0 ð32Þ where eT ðtÞ ¼ ½xT ðtÞ xT ðik hÞ x_ T ðtÞ
ðI þ Df ÞKðI þ Dg Þ ¼ K þ Df K þ KDg þ Df KDg W K þ DðKÞ
ð24Þ and
Therefore (22) and (23) can be rewritten as x_ ðtÞ ¼ ½A þ DAðtÞ xðtÞ þ ½BK þ BDðKÞ xðik hÞ t [ ½ik h þ tk ; ikþ1 h þ tkþ1 Þ
ð25Þ
xðtÞ ¼ Fðt; t0 hÞ xðt0 hÞ W fðtÞ t [ ½t0 h; t0 ð26Þ On the basis of (25) and (26), we can give a performance analysis result. Lemma 1: Given h and a matrix K, if there exist matrices P . 0, T . 0, W . 0 and Nj and Mj ( j ¼ 1, 2, 3) of 660
N T ¼ ½N T1
N T2
M T ¼ ½M T1
M T2
N T3 M T3
Taking the time derivative of V(t) for t [ [ikh þ tk , ikþ1h þ tkþ1) and using (31) and (32), we obtain ðt T T _ V ðtÞ ¼ 2x ðtÞ P x_ ðtÞ þ 2e ðtÞ N xðtÞ xðik hÞ x_ ðsÞ ds þ 2e ðtÞM ½A þ DAðtÞ xðtÞ ½BK þ BDðKÞ xðik hÞ þ x_ ðtÞ ðt þ hx_ T ðtÞT x_ ðtÞ x_ T ðsÞT x_ ðsÞ ds
ik h
T
ð33Þ
th
where V˙(t) ¼ lim supd ! 0 þ (1/d)[V(t þ d) 2 V(t)] [15, 16]. IEE Proc.-Control Theory Appl., Vol. 153, No. 6, November 2006
Noting (28), it is easy to see that when t [ [ikh þ tk , ikþ1h þ tkþ1) ðt 2eT ðtÞN x_ ðsÞ ds heT ðtÞNT 1 N T eðtÞ ik h
ðt þ
x_ T ðsÞT x_ ðsÞ ds
( j ¼ 2, 3) and g, if there exist matrices P~ . 0, T~ . 0, ~ . 0 and a symmetric matrix X and matrices N~ i (i ¼ 1, M 2, 3) and Y of appropriate dimensions and positive scalars 1k (k ¼ 1, 2, . . . , 7) such that ~ XX M " # ~ M ,0 Y gI
ð34Þ
th
Then, combining (33) and (34), we obtain V_ ðtÞ eT ðtÞ V1 eðtÞ;
L11 V1 ¼ 4 L21 L31
ð41Þ
t [ ½ik h þ tk ; ikþ1 h þ tkþ1 Þ ð35Þ 2
where 2
ð40Þ
3
L22 L32
5 þ hNT 1 N T L33
From (27), it can be shown that V1 , diagðW
0 W Þ
ð36Þ
G~ 11 þ 11 DDT þ 1a BBT 6~ ~G22 þ 11 l2 DDT þ 1a l2 BBT 6 G21 þ 11 l2 DDT þ 1a l2 BBT 2 2 6 6~ T T T ~ 6 G31 þ 11 l3 DD þ 1a l3 BB G32 þ 11 l2 l3 DD þ 1a l2 l3 BBT 6 6 hN~ T1 hN~ T2 6 6 Q1 0 4
V_ ðtÞ ljjxðtÞjj2 ljj_xðtÞjj2 ; t [ ½ik h þ tk ; ikþ1 h þ tkþ1 Þ
Q2
0
Combining (35) and (36), we have
ð37Þ
where l ¼ lmin(2W). Following the method given in Yue et al. [12], we can conclude the exponential stability of systems (25) and (26). Note that for t [ [ikh þ tk , ikþ1h þ tkþ1), u(t) can be expressed as
G~ 33 þ 11 l32 DDT þ 1a l32 BBT hN~ T3 hT~ 0 0 J1 0
0
0
3 7 7 7 7 7,0 7 7 7 5 J2 ð42Þ
uðtÞ ¼ ½K þ DðKÞ xðik hÞ From (35), we can obtain the following estimate of V˙(t) for t [ [ikh þ tk , ikþ1h þ tkþ1) V_ ðtÞ eT ðtÞV1 eðtÞ þ xT ðtÞRxðtÞ
where
þ x ðik hÞ½K þ DðKÞ Q½K þ DðKÞ xðik hÞ
G~ 11 ¼ N~ 1 þ N~ T1 AX T XAT ;
T
x ðtÞ RxðtÞ u ðtÞQuðtÞ T
T
G~ 21 ¼ N~ 2 N~ T1 l2 AX T Y T BT
T
¼ e ðtÞV2 eðtÞ x ðtÞRxðtÞ u ðtÞQuðtÞ ð38Þ
G~ 32 ¼ N~ 3 þ l2 X l3 BY
V_ ðtÞ xT ðtÞRxðtÞ uT ðtÞQuðtÞ; t [ ½ik h þ tk ; ikþ1 h þ tkþ1 Þ S1
Note that ikþ1h þ tkþ1) ¼ [t0 , k¼1 [ikh þ tk , Integrating both sides of (39) from t0 to 1 yields ð1 ½xT ðtÞRxðtÞ þ uT ðtÞQuðtÞ dt V ðt0 Þ W J
~ G~ 31 ¼ N~ 3 l3 AX T þ X þ P; G~ 22 ¼ N~ 2 N~ T2 l2 BY l2 Y T BT
where V2 ¼ V1 þ diag(R, [K þ D(K)]TQ[K þ D(K)], 0). From (27), V2 , 0. Therefore (38) means that
ð39Þ 1).
A
t0
G~ 33 ¼ l3 X þ l3 X T þ hT~ Q1 ¼ ½ X XETa T Q2 ¼ ½ Y T Y T J1 ¼ diagð R1
Design of feedback gain
In this section, on the basis of Lemma 1, we derive a criterion that is expressed in terms of a set of matrix inequalities and can describe the relationship between the feedback gain K and the quantisation density of quantisers. Theorem 1: Consider system (1), with a network-based quantised controller in (5) and (6), where f (.) and g(.) are given in (14) with quantisation densities of rf and rg , respectively. For a given h, choose positive scalars lj IEE Proc.-Control Theory Appl., Vol. 153, No. 6, November 2006
gd2g X 11 I Þ
J2 ¼ diagðQ1 1b I 15 I
4
(43)
T
T T
ðikþ1 ik Þh þ tkþ1 h; k ¼ 1; 2; . . .
gd2g 16 I
gd2g X 12 I
YT
gd2g 13 I
gd2g X
gd2g X T
gd2g 14 I
gd2g 17 IÞ
1a ¼ 12 d2f þ 13 þ 14 d2f ;
1b ¼ 15 d2f þ 16 þ 17 d2f
then the GCCD problem for system (1) with a network-based quantised controller in (5) and (6) is solvable. The feedback gain is given as K ¼ YX21 and the upper bound of J is ð t0 ~ 1 xðt0 Þ þ ~ 1 f_ ðsÞ ds f_ T ðsÞX 1 TX J ¼ xT ðt0 Þ X 1 PX t 0 h
ð44Þ 661
Proof: Define 2
L11 þ R 6 L21 6 P¼6 6 L31T 4 hN 1 0
Then, from (45) and (47), we have L22 L32 hN T2 K þ DðKÞ
L33 hN T3 0
hT 0
3 7 7 7 7 5 Q1
Then
T P P0 þ 11 LTD LD þ 11 1 Ea Ea T þ ð12 d2f þ 13 þ 14 d2f ÞLTB LB þ 11 2 K K 2 2 2 2 1 1 1 T þ ð11 3 gdg þ 14 gdg þ 16 gdg þ 17 gdg Þ I 2 I 2 T T þ 11 5 I 2 K KI 2 D
þ ð15 d2f þ 16 þ 17 d2f Þ I T1 I 1 ¼ P00
P ¼ P0 þ LTD FðtÞE a þ E Ta FT ðtÞLD þ LTB Df K þ KT Df LB þ LTB KT g þ T Tg K T LB þ LTB Df KT g þ T Tg K T Df LB þ T Tf KI 2 þ I T2 K T T f þ I T1 KT g þ T Tg K T I 1
ð48Þ
Setting M1 ¼ X21, M2 ¼ l2X21 and M3 ¼ l3X21 and by common matrix computation, we obtain ^ 00 ¼ diagð X X X X I ÞP00 diagð X X X X I Þ P ^ 0 þ 11 F T F 1 þ 11 F T F 2 þ ð12 d2 þ 13 þ 14 d2 Þ F T F 3 ¼P f f 1 1 2 3
þ T Tf KT g þ T Tg K T T f
1 T T þ ð11 2 þ 15 Þ I 2 Y Y I 2
where 2
G11 6 G21 6 P0 ¼ 6 6 G31T 4 hN 1 0
G22 G32 hN T2 K
G33 hN T3 0
hT 0
2 1 2 1 2 1 2 T þ ð11 3 gdg þ 14 gdg þ 16 gdg þ 17 gdg Þ I 2 XX I 2
3 7 7 7 7 5 Q1
þ ð15 d2f þ 16 þ 17 d2f Þ I T1 I 1 where 2
G~ 11 þ XRX 6 6 G~ 21 6 0 ^ ¼6 P G~ 31 6 6 4 hN~ T1
G11 ¼ N 1 þ N T1 M 1 A AT M T1 þ R G21 ¼ N 2 N T1 M 2 A K T BT M T1
0
~G22 G~ 32 hN~ T 2
Y
~G33
hT~
0
Q1
hN~ T3 0
G31 ¼ N 3 M 3 A þ M T1 þ P
N~ i ¼ XN i X ði ¼ 1; 2; 3Þ;
G22 ¼ N 2 N T2 M 2 BK K T BT M T2
F 1 ¼ ½ DT l2 DT l3 DT 0 0 F 2 ¼ ½ Ea X T 0 0 0 0 F 3 ¼ ½ BT l2 BT l3 BT 0 0
G32 ¼ N 3 þ M T2 M 3 BK G33 ¼ M 3 þ M T3 þ hT LD ¼ ½ DT M T1 DT M T2 DT M T3 0 0 LB ¼ ½ BT M T1 BT M T2 BT M T3 0 0 E a ¼ ½ Ea 0 0 0 0 ; K ¼ ½ 0 K 0 0 I1 ¼ ½ 0 T g ¼ ½0
0
0 0 0 I ; I 2 ¼ ½ 0 I 0 0 0 Dg 0 0 0 ; T f ¼ ½ 0 0 0 0 Df
It is easy to show that T T 2 1 T P P0 þ 11 LTD LD þ 11 1 E a E a þ 12 L B D f L B þ 12 K K T T T 2 þ 13 LTB LB þ 11 3 T g K KT g þ 14 LB Df LB T T T 1 T T þ 11 4 T g K KT g þ 15 T f T f þ 15 I 2 K KI 2
P~ ¼ XPX ;
3 7 7 7 7 7 7 5
T~ ¼ XTX
By Schur complement, we can show that (42) is equiˆ 00 , 0, which ˆ 00 , 0. Therefore (42) implies P valent to P 00 ˆ further means that P , 0. Define l ¼ lmin(2P00 ) and W ¼ lI. Thus, from (48), it is easy to see that P þ diagð W ; 0; W ; 0; 0 Þ , 0
ð49Þ
As (49) is equivalent to (27), by Lemma 1, we can complete the proof. A Remark 3: From (47), it can be seen that the feedback gain K is bounded by g. As long as (40) –(42) are feasible for solvability, the requirement on the bound constraint of K can be assured by adjusting the value of g. This additional feature is also useful to limit the cost of controller implementation.
T T þ 16 I T1 I 1 þ 11 6 T g K KT g T T þ 17 T Tf T f þ 11 7 T g K KT g
ð45Þ
where e i . 0 (i ¼ 1, 2, . . . , 7). Using the Schur complement, (41) leads to ~ Y T Y gM
ð46Þ
As K ¼ YX21, from (40) and (46), we have K T K gI 662
ð47Þ
Remark 4: (43) is used to describe the network conditions. From (42) and (43), it can be seen that the quantisation density of quantisers and the network conditions will ~ which will directly affect the solution of X, Y, P~ and T, thus affect the feedback gain K and the upper bound of cost function. To minimise the bound J shown in (44), the commonly used method in Mahmoud [13, 14] is to transform it as an optimisation problem. However, to solve the optimisation problem in Mahmoud [13, 14], the initial condition function is needed to be known. In other words, the solution to the optimisation problem is dependent on the given initial IEE Proc.-Control Theory Appl., Vol. 153, No. 6, November 2006
function. Therefore such an optimisation solution has only a theoretical but not practical meaning. ˙ (t) ¼ [A þ DA(t)]f(t). It is known from (26) that f Suppose that there exist ai . 0 (i ¼ 1, 2) such that ~ 1 a1 I; X 1 PX
~ 1 a2 I X 1 TX
ð50Þ
Then, from (44), we have J a1 xT ðt0 Þxðt0 Þ þ qa a2
ð t0
fT ðtÞfðtÞ dt
ð51Þ
t 0 h
where qa ¼ (kAk þ kDk kEak)2. In practice, it is possible to know the variation range of the initial condition. That is, the upper bound of estimate of the initial condition can be obtained. Suppose that qb 0 exists such that fT(t)f(t) qb , t [ [t0 2 h, t0]. Therefore from (51), we have J q b a1 þ hq a q b a2
ð52Þ
21
Then, defining SX ¼ X , ST ¼ T~ 21, SP ¼ P~ 21 and ~ 21 and combining (40) – (42), (50) and (52) and SM ¼ M using the idea of the cone complementary linearisation algorithm [17], the GCCD problem for system (1) with a network-based quantised controller in (5) and (6) can be transformed by solving the following optimisation problem ~ Trace ðSP P~ þ ST T~ þ SX X þ SM MÞ þqb a1 þ hqa qb a2 a1 I SX Subject to : ð41Þ and ð42Þ; , 0; SX SP I SX a2 I S X , 0; ,0 SX ST SX SM SM I SP I ST I 0; 0; 0 ~ I M I P~ I T~ I SX ~ .0 0; P~ . 0; T~ . 0; M I X Minimise :
5
ð53Þ
Example 1: In this example, we consider an application of the method proposed in this paper to the following linear system with state uncertainty x_ ðtÞ ¼ ½A þ DAðtÞ xðtÞ þ BuðtÞ
ð54Þ
with 2 A¼ 1
0 ; 1
0 B¼ 0:5
Table 1: Computational results for different h h
Feedback gain K
and kDA(t)k 0.1. Suppose the initial function f(t) satisfies fT(t)f(t) 1, t [ [t0 2 h, t0]. The quantisers f(.) and g(.) in (5) and (6) are chosen as logarithmic with rf ¼ rg ¼ 0.818. It is also known that the network condition satisfies (ikþ1 2 ik)h þ tkþ1 h. The parameter matrices of cost function (3) are R ¼ 0.1I and Q ¼ 0.1. For different network conditions, that is, h ¼ 0.1 or 0.2, solving the optimisation problem (53) by choosing l2 ¼ 0.2, l3 ¼ 2 and g ¼ 20, the derived feedback gains and the upper bounds of cost function are p shown in Table 1. Fig. 2 shows the variations of (x21(t) þ x22(t)) for the case h ¼ 0.1 or 0.2 under feedback gain of K ¼ ½ 1:2466 4:3625 or K ¼ ½ 1:4039 4:2350 , respectively. 6
Numerical example
Fig. 2 State responses for different network conditions
Conclusion
In this paper, we have investigated the guaranteed cost control problem for systems under a network-based quantised controller. In the system performance analysis, the effect of not only the quantisation levels but also the network conditions, such as network-induced delay and data dropout, on the system has been considered. As two quantisers are employed in the controller design, it is more available for the NCSs than the method only considering one quantiser in one direction from sensor to controller. From the criterion derived for control design, it is easy to see the relationship between the feedback gain K and the quantisation levels and the network conditions. In this paper, the control design has been developed on the basis of quantisers with an infinite number of quantisation levels. With a finite number of quantisation levels, the consideration of the GCCD problem for system (1) will be derived on the basis of the practical stability concept, which is left for future research. In addition, the GCCD problem for the nonlinear systems or the dynamic output GCCD problem is also of interest not only in theoretical but also in practical applications.
Upper bound of cost function
0.1
[21.2466 24.3625]
38.8
0.2
[21.4039 24.2350]
39.7
IEE Proc.-Control Theory Appl., Vol. 153, No. 6, November 2006
7
Acknowledgments
The authors would like to thank the associate editor and the anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper. The 663
research work of D. Yue was partially supported by the National Natural Science Foundation of China (60474079), the Teaching and Research Award Program for Outstanding Young Teachers at Nanjing Normal University and the Key Scientific Research Foundation by the Ministry of Education of China (03045). 8
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IEE Proc.-Control Theory Appl., Vol. 153, No. 6, November 2006
