International Journal of Control Vol. 79, No. 7, July 2006, 752–763
Stabilization of remote control systems with unknown time varying delays by LMI techniques Y.-J. PAN*y, H. J. MARQUEZz and T. CHENz yDepartment of Mechanical Engineering, Dalhousie University, Halifax, Nova Scotia, Canada B3J 2X4 zDepartment of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2V4 (Received 27 September 2005; in final form 11 January 2006) In this paper, the stabilization of one class of remote control systems with unknown time varying delays is analysed and discussed using LMI techniques. A discrete time state space model under a static control law for remote control systems is first introduced based on some assumptions on the uncertain term. The time delay is unknown, time varying, and can be decomposed into two parts: one fixed part which is unknown and is an integer multiple of the sampling time; the other part which is randomly varying but bounded by one sampling time. Static controller designs based on delay dependent stability conditions are presented. This system is then extended to a more general case when the randomly varying part of the time delay is not limited to one sampling time. The derivative of the time delay is not limited to be bounded. Hence, the contributions are as follow: (i) for a given controller, we can use these stability criteria to test stability of the resulted system; (ii) we can design a remote controller to stabilize an unstable system. Finally, simulation examples are presented to show the effectiveness of the proposed method and to demonstrate remote stabilization of open loop unstable systems.
1. Introduction Over the past few years, increasing attention has been drawn to the control of time delay systems (Dugard and Verriest 1998, Nilsson 1998, Mahmoud 2000). The stability problem is one of the most important issues in time delay systems since delays caused by transmission may result in instability, especially when there are uncertainties (Zhang et al. 2001, Pan et al. 2004b). Many delay independent and delay dependent stability criteria, which are mainly concerning continuous systems, have been presented (Xie and de Souza 1997, Cao et al. 1998). However, most practical applications of remote controllers through network transmission are implemented digitally. Up until now, little attention has been paid to discrete time systems with delays and controlled in a remote way. One possible reason is that, for a known delay, the delay difference equation can be rewritten to be a high order system without delay by augmentation (Lozano et al. 2004). However, in a case *Corresponding author. Email:
[email protected]
with large known delay, this augmentation approach can lead to high dimensional systems; and moreover, it is not applicable for systems with unknown delays. In several references (Song et al. 1999, Lee and Kwon 2002, Xu et al. 2004), no input delay was considered in the problem formulation. This simplifies the design of complex controllers for robust stabilization. In Lee and Lee (1999), two delay dependent conditions were developed for discrete time systems but no control design and uncertainties were considered in the system parameters. The stability conditions in Lee and Lee (1999) are the analogues corresponding to the results for continuous time delay systems as in Xie and de Souza (1997), Cao et al. (1998) and Park (1999). In Nilsson (1998), stochastic control of time delay systems in discrete time is considered while the limitation is that delays in both channels should be shorter than one sampling time. Xiao et al. (2000) model the problem via a discretetime jump linear system. However the assumption of this approach is that the transition probability matrix for the time delay should be known as a priori. It is usually
International Journal of Control ISSN 0020–7179 print/ISSN 1366–5820 online 2006 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00207170600654554
753
Stabilization of remote control systems difficult to get this matrix in reality. In Stipanovic and Siljak (2001), discrete time delay systems with non-linear perturbations are considered and it is proposed for the case with constant time delay. Most recently, Yue et al. (2005) propose a robust H1 controller for continuous time delay systems while the lower bound of the network induced delay can be used in the design. However, Assumption 3 on the time delay distribution in this paper is a strong condition. In Yu et al. (2005), only one channel time delay is considered. In these two recent approaches, sampled-data approach is incorporated in analysing the communication channel(s) which reflects the physical point of real network transmission. Based on the above analysis and from a novel aspect, here we look further into the stability problem of remote control systems with unknown time varying delays in the control input in the discrete time domain. In this case, the existing theory using augmentation cannot be applied because of the unknown time varying delay. One approach is to decompose the time delay into two parts: one fixed part which is unknown and is an integer multiple of the sampling time; the other part which is randomly varying but bounded by one sampling time. Then the stability condition of the discrete time state space model under a static control law for the remote control system is first introduced based on a certain assumption on the uncertain term. Note that one motivation to study the problem in discrete time is that most systems with a communication channel involved are implemented digitally in practice. This system is then extended to a more general case when the randomly varying part of the time delay has a bound which is more than one sampling time. Delay dependent stability conditions are presented for the static controller design by using LMI techniques. In the delay dependent case, conditions based on linear matrix inequalities (LMIs) are derived by introducing some slack matrix variables which are less conservative than the conventional approaches (Mahmoud 2000). A similar approach for continuous time systems is presented in Wu et al. (2004). Accordingly, we can design a static remote controller to stabilize a given open loop unstable system if there exist solutions from the derived LMIs. There is no assumption on the variation rate of the time delay, i.e., the derivative of the time delay is not necessary to be bounded. Another advantage is that for a given controller, we can use these stability criteria to test stability of the resulted closed loop systems. The paper is organized as follows. In x 2, the problem formulation is first presented with the analysis on the discrete time system when the delay variation part is less than one sampling time. In x 3, static controllers based on delay dependent conditions are analysed in detail. In x 4, the system description of the general case and the
controller design are analysed when the delay variation part is longer than one sampling time. Section 6 draws the conclusions. Notations: Rn denotes an n-dimensional real vector space; kk is the Euclidean norm or induced matrix norm.
2. Problem formulation The following system with uncertain delay is considered x_ ðtÞ ¼ Ap xðtÞ þ Bp uðt 2 ðtÞÞ,
ð1Þ
where x 2 Rn is the measurable state vector, u 2 Rm denotes the remote control signal, Ap 2 Rnn and Bp 2 Rnm are known constant matrices. The system in (1) can be illustrated as in figure 1, where 1 ðtÞ is the time delay from the system sensor to the remote controller and uc 2 Rm is the control input to the physical system. In this paper, the sensor is assumed to be time driven; the controller and the actuator are event driven. Assumption 1: The time delays in the two channels can be represented as i ðtÞ ¼ hi Ts þ "i ðtÞ, i ¼ 1, 2, where Ts is the sampling time and "i ðtÞ is unknown but is bounded as 0 "i ðtÞ < Li Ts with Li hi , where Li is a known integer. hi is an unknown integer constant and is assumed to be: hi 2 ½0, hi , where hi is known. Remark 1: Note that the controller and the actuator are event driven; thus there is no holding-up on the controller (pure gain) and actuator side. The total delay from the sensor to the actuator can be represented as ðtÞ ¼ 2 ðtÞ þ 1 ðt 2 ðtÞÞ. According to Assumption 1, similarly ðtÞ can be rewritten as ðtÞ ¼ hTs þ "ðtÞ, with h 2 ½h1 þ h2 , h1 þ h2 þ 1 being an unknown integer and "ðtÞ being unknown but bounded with 0 "ðtÞ < LTs where L and h are known integers. and L < h h,
uc Actuator
τ2(t)
Sensor
Network Transmission Channel
u
Figure 1.
x System
τ1(t)
Remote Controller
Block diagram of the remote control systems.
754
Y.-J. Pan et al.
Here we first consider the case when L ¼ 1 which means that the randomly varying part of the time delay is bounded by one sampling time. The case when h > L > 1 is extended in x 4. The timing flow of the control system is shown in figure 2. Suppose the measured state xkh is sent at time (k h)Ts from the sensor, it arrives at the controller in the sampling interval [(k h2)Ts, (k h2 þ 1)Ts) according to the delay structure in Assumption 1. At the controller side, the control signal is then denoted as ukh2 ¼ Kxkh if a static gain is designed and the sampled state arrives at the controller at time t 2 ½ðk h2 ÞTs , ðk h2 þ 1ÞTs : This control signal arrives at the actuator at time t ¼ kTs þ "ðtÞ: Figure 3 shows an example of a time delay profile satisfying ðtÞ ¼ 3Ts þ "ðtÞ with 0 < "ðtÞ < Ts , measured in the teleoperation experiment (Pan et al. 2004a); this can be used to validate the delay assumptions introduced earlier. Integrating the open loop system in the interval [kTs, (k þ 1)Ts], and using figure 2, we have Z
ðkþ1ÞTs
xkþ1 ¼ Axk þ
where A ¼ eAp Ts , B ¼ Z
Z
ðkþ1ÞTs
kTs ðkþ1ÞTs "ðkÞ
B0 ¼
e kTs ðkþ1ÞTs
Z
eAp ½ðkþ1ÞTs s dsBp ¼
Ap ½ðkþ1ÞTs s
Z
ðkþ1ÞTs "ðkÞ
Ts
0 Ts
dsBp ¼
eAp ½ðkþ1ÞTs s dsBp ¼
B1 ¼
Z
eAps dsBp ,
eAps dsBp ,
"ðkÞ "ðkÞ
Z
eAps dsBp , 0
and B0 þ B1 ¼ B. Because there are uncertainties in the time delay, we cannot use the prediction method to stabilize the system which requires accurate information of the delay in order to predict the future states. Hence a static controller is designed as u(t) ¼ Kx(t 1 (t)), where T 1 (t) is the delay from the sensor to the controller in the continuous time domain. The main task here is how to design a proper K such that the closed loop control system is stable. Furthermore, as previously discussed in figure 2, in discrete time domain we have ukh2 ¼ Kxkh . Hence it is straightforward that (2) becomes
eAp ½ðkþ1ÞTs s Bp uðs 2 ðsÞÞds
kTs
Z
ðkþ1ÞTs
¼ Axk þ
xkþ1 ¼ Axk þ ðB þ B0 ÞKxkh þ ðB þ B1 ÞKxkh1 , ð3Þ
eAp ½ðkþ1ÞTs s dsBp ukh2
ð2Þ
ðkþ1ÞTs "ðkÞ
Z
ðkþ1ÞTs "ðkÞ
þ
where A and B can be derived directly from Ap and Bp in (1). Based on the above analysis, though Bi, i ¼ 0, 1, are uncertain, embedding them into standard uncertain terms would allow us to treat the problem as a robust
eAp ½ðkþ1ÞTs s dsBp ukh2 1
kTs
¼ Axk þ ðB þ B0 Þukh2 þ ðB þ B1 Þukh2 1 ,
k−h
k−h2+1
k−h2
k
k+1
Measured x(t)
uk−h
2
Controller
ε(t)
Actuator
uk−h2
τ(t)
uc(k) Input to plant
uc(k−1) k−h
Figure 2.
k−h2
k−h2+1
k
Timing flow of the control system.
k+1
755
Stabilization of remote control systems 0.2 0.19 0.18
Time delay (sec)
0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1
0
5
10
15
20
25
30
35
40
Time (sec)
Figure 3.
The time delay when h ¼ 3, L ¼ 1 and Ts ¼ 0.05 sec.
stabilization problem. Hence in this paper, we assume that Bi satisfies the following assumption. Assumption: Bi can be expressed as Bi, Ei ¼ (k)Fi, i ¼ 0, 1, where E, Fi are known with appropriate dimensions and i(k)T i(k) < I. Note that Assumption 2 on uncertainties is a commonly used condition for many existing approaches for uncertain time delay systems (Mahmoud 2000, Xu et al. 2004, Yue et al. 2005). There always exist E and Fi such that i(k)T i (k) < I for Bi, i ¼ 0, 1. However, in this paper, for a specific system pair (Ap, Bp) and sampling time Ts, we can use numerical methods to get the matrices E and Fi which are required to be known for the controller design. Lemma 1 (Mahmoud 2000): Let 1 and 2 be real constant matrices of compatible dimensions, and H(t) be a real matrix function satisfying HðtÞT HðtÞ I: Then the following inequality holds T1 HðtÞ2 þ T2 HðtÞT 1 "T1 1 þ "1 T2 2 ,
ð4Þ
1 ¼ 2 xTk N1 þ xTkh N2 þ xTkh1 N3 þ xTkþ1 N4 " # k X ðxj xj1 Þ ¼ 0, xk xkh
ð6Þ
j¼khþ1
2 ¼ 2 xTk S1 þ xTkh S2 þ xTkh1 S3 þ xTkþ1 S4 " # k X ðxj xj1 Þ ¼ 0, xk xkh1
ð7Þ
j¼kh
where " is a positive constant.
3 ¼ 2 xTk M1 þ xTkh M2 þ xTkh1 M3 þ xTkþ1 M4
3. Controller design
½xkþ1 Axk A0 xkh A1 xkh1 ¼ 0:
Rewrite the system in (3) as xkþ1 ¼ Axk þ A0 Xkh þ A1 xkh1 ,
where A0 ¼ BK þ B0 K ¼ A0 þ 0 with A0 ¼ BK and 0 ¼ B0 K, and A1 ¼ BK þ B1 K ¼ A1 þ 1 with A1 ¼ BK and 1 ¼ B1 K: In this part, we focus on the controller design based on the delay department condition by introducing some slack matrix variables which result in a less conservative condition. The similar approach for the continuous time system is introduced in Wu et al. (2004). At first, we extend the method in the continuous time domain to the discrete time domain. For any matrices Ni, Si and Mi (i ¼ 1, 2, 3, 4) of appropriate dimensions, the following equations hold
ð5Þ
ð8Þ
Based on the Lyapunov functional analysis while combining the equations (6)–(8) in the derivation, first we have the following proposition.
756
Y.-J. Pan et al.
Proposition 1: For a given control gain K and a given upperbound, h of h the system in (3) is asymptotically stable if there exist symmetric positive definite matrices P, Q1 , Q2 , Z1 , Z2 2 Rnn , and matrices Ni , Si , Mi , i ¼ 1, 2, 3, 4 with appropriate dimensions such that the following inequality holds, 2 6 6 6 6 6 6 6 6 4
D11
D21 D31
D22 D32
D33
D41 D42 D43 D44 T T T T 1 hN hN hN hN hZ 1 2 3 4 h þ 1 ST1 h þ 1 ST2 h þ 1 ST3 h þ 1 ST4 0 h þ 1 Z1
Then we have V1,k ¼ V1,kþ1 V1,k ¼ xTkþ1 Pxkþ1 xk Pxk , V2,k ¼
1 kþ1 X X i¼h j¼kþiþ2
1 k X X
3
T xj xj1 Z1 xj xj1
i¼h j¼kþiþ1
7 7 7 7 7 > > > > > > 6D 7 > > D > > 21 22 7 = > > > > > D D D D > > 41 42 43 44 > > > 1 > ; : T þ hNZ1 N þ h þ 1 SZ ST 1 2 ð13Þ
2
7 7 7 7 7 7 7 7 0 such that the following inequality holds,
hZ1 0 0 0 0
ðh þ 1ÞZ2 0 0 0
3 7 7 7 7 7 7 7 7 < 0, 7 7 7 I 7 7 0 I 5 0 0 I
ð16Þ
758
Y.-J. Pan et al.
where
represented as 2
0
D11 ¼ D11 ¼ P þ hZ1 þ Q1 þ ðh þ 1ÞZ2
6 6 6 6 ð9Þ ¼ 6 6 6 6 4
þ Q2 þ N1 þ NT1 þ S1 þ ST1 M1 A AT MT1 , 0
D21 ¼ N2 NT1 þ S2 M2 A KT BT MT1 ,
where
0
D32 ¼ N3 ST3 M3 BK KT BT MT3 ,
0
0
0
D33 ¼ Q2 S3 ST3 M3 BK KT BT MT3 ,
3
7 D21 D22 7 7 0 0 0 7 D32 D33 D31 7 0 0 0 0 7 D41 D42 D43 D44 7 7 5 hNT1 hNT2 hNT3 hNT4 hZ1 T T T T h þ 1 S1 h þ 1 S2 h þ 1 S3 h þ 1 S4 0 ðh þ 1ÞZ2
ð17Þ
3 0 6 AT 7 6 7 4 ¼ 6 0 7 MT1 4 AT1 5
D22 ¼ Q1 N2 NT2 M2 BK KT BT MT2 ,
2
MT2
MT3
MT4
0 3 M1 6 M 7 6 27 6 7 0 A0 A1 0 4 M3 5 M4 2 3 0 6 GT 7 6 7 1 MEðMEÞT þ 6 0T 7 0 G0 4 G1 5 2
0
D41 ¼ D41 ¼ hZ1 ðh þ 1ÞZ2 þ N4 þ S4 M4 A þ MT1 0
D42 ¼ N4 þ MT2 M4 BK, 0
D43 ¼ S4 þ MT3 M4 BK, D44 ¼ D44 ¼ P þ hZ1 þ ðh þ 1ÞZ2 þ M4 þ
0
þ 4 ,
0
D31 ¼ N3 ST1 þ S3 M3 A KT BT MT1 ,
0
0
D11
MT4 :
G1
0 :
ð18Þ
0 Proof: The uncertainty part in (9) can be dealt with by using the property in Lemma 1. The left side of the inequality in (9) can be further 2
0
D11
Using (18) and Schur complement, it is obvious to obtain that the following condition is a sufficient condition of inequality (9),
3
0 6 D0 D22 7 7 6 21 7 6 0 0 0 7 6 D31 D D 32 33 7 6 0 0 0 6 D0 D42 D43 D44 7 7 6 41 7 6 T T T 7 6 hNT1 hN hN hN hZ 1 2 3 4 7 6 6 ðh þ 1ÞST ðh þ 1ÞST ðh þ 1ÞST ðh þ 1ÞST 0 ðh þ 1ÞZ 7 7 6 1 2 3 4 7 6 T T T T T T T T 4 E M1 E M2 E M3 E M4 0 0 I 5 0 G0 G1 0 0 0 0 I 2 0 D11 0 0 6 D22 6 D21 6 0 0 6 D0 D32 D33 31 6 0 0 0 0 6 D42 D43 D44 6 D41 6 T T T T ¼6 hN hN hN hN hZ 1 1 2 3 4 6 6 T T T T 6 ðh þ 1ÞS1 ðh þ 1ÞS2 ðh þ 1ÞS3 ðh þ 1ÞS4 0 ðh þ 1ÞZ2 6 T T T T T T 6 ET MT E M2 E M3 E M4 0 0 I 1 6 6 4 0 F0 K 0 0 0 0 0 I
0
0
F1 K
0
0
0
0
0
3 7 7 7 7 7 7 7 7 7 7>0: 7 7 7 7 7 7 5
ð19Þ
I œ
759
Stabilization of remote control systems Based on the analysis in Propositions 1 and 2, we are now able to design the feedback gain K which can ensure the asymptotical stability of the networked control system in (3). Theorem 1: For given scalars i , i ¼ 1, 2, 3, 4, and a given upperbound h of h, if there exist symmetric positive b2 , Z b1 , Q b1 , Z b2 2 Rnn, matrices bQ definite matrices P, b b Y, Ni , Si , i ¼ 1, 2, 3, 4, non-singular matrix X with appropriate dimensions and constant > 0 such that the following inequality holds, 2
11
6 21 6 6 6 31 6 6 41 6 6 bT 6 hN 1 6 6 bT 6 ðh þ 1ÞS 1 6 6 1 ET 6 6 4 0
that Mi ¼ i M0 where M0 is non-singular and i is known and given. Define X ¼ M1 0 , W ¼ 1 and Y ¼ KXT . diagðX, X, X, X, X, X, I, I Þ, ¼ Then pre-multiplying the inequality in (16) by W and post-multiplying by WT, the inequality in (20) can be obtained. Note that the inequality in (20) is only a sufficient condition for the solvability of (16) based on these derivations. Remark 2: Note that the diagonal term 44 in (20) ^ Z^ 1 , Z^ 2 . contains three positive definite matrices P,
22 32
33
42 bT hN 2
43 bT hN 3
44 bT hN 4
b1 hZ
bT ðh þ 1ÞS 2 T 2 E
bT ðh þ 1ÞS 3 T 3 E
bT ðh þ 1ÞS 4 T 4 E
0 0
b2 ðh þ 1ÞZ 0
I
F0 Y
0
0
0
0
0
I
0
F1 Y
0
0
0
0
0
I
0
3 7 7 7 7 7 7 7 7 7 L > 1, i.e., the varying part of the time delay has a bound which is more than one sampling time. When L > 1, the discretized system becomes more complicated. It can be derived by the same procedure as in x 2 and is represented as follows: xkþ1 ¼ Axk þ ðB þ B0 Þkxkh þ ðB þ B1 Þkxkh1 þ þ ðB þ BL ÞkxkhL ,
ð22Þ
where A and B are the same as in (3). The Bi term is similar as in Assumption 2 but with i ¼ 0, 1, . . . , L: Then the corresponding results on the control gain design can be extended from Theorem 1. The remarks and proofs
760
Y.-J. Pan et al. 3,3 ¼ Q^ 2 N^ 2,3 NT2,3 3 BY 3 YT BT ,
are similar and hence they are omitted here. Rewrite the system in (22) as xkþ1 ¼ Axk þ A0 xkh þ A1 xkh1 þ þ AL xkhL ,
.. .
ð23Þ
Lþ3,1 ¼
h þ i Z^ iþ1 þ N^ TLþ3,Lþ3
i¼0
where Ai ¼ BK þ Bi K ¼ Ai þ i with Ai ¼ BK and i ¼ Bi K, i ¼ 0, 1, . . . , L: From Theorem 1, the controller design can be concluded as follows.
þ N^ Lþ3,Lþ3 Lþ3 AXT þ 1 X, Lþ3,2 ¼ N^ Lþ3,Lþ3 2 X Lþ3 BY, .. .
Theorem 2: For given scalars i , i ¼ 1, . . . , L þ 3, and a given upperbound h of h, if there exit symmetric positive ^ Q^ j , Z^ j 2 Rnn , j ¼ 1, . . . , L þ 1, definite matrices P, matrices Y, N^ j,i , non-singular matrix X with appropriate dimensions and constants > 0 such that the following inequality holds, 2
L X
1,1 2,1 3,1 .. .
6 6 6 6 6 6 6 6 Lþ3,1 6 T 6 hN^ 1,1 6 6 .. 6 . 6 6 h N^ T 6 L Lþ1,1 6 6 1 ET 6 0 6 6 .. 4 . 0
2,2 3,2 .. .
3,3 .. .
.. .
Lþ3,2 hN^ T1,2 .. . hL N^ T 2 ET F0 Y .. .
... ... .. . ... ... ... .. .
0
...
Lþ1,2
L X
Lþ3,Lþ3 ¼ P^ þ
L X
h þ i Z^ iþ1 þ Lþ3 XT þ Lþ3 X,
i¼0
Lþ3,Lþ2 hN^ T1,Lþ2 .. . hL N^ T
Lþ3,Lþ3 hN^ T1,Lþ3 .. . hL N^ T
Lþ2 ET 0 .. .
Lþ3 ET 0 .. .
hZ^ 1 .. . 0 0 0 .. .
FL Y
0
...
Lþ1,Lþ2
Lþ1,Lþ3
where hL ¼ h þ L and 1,1 ¼ P^ þ
Lþ3,Lþ2 ¼ S^Lþ3 þ Lþ2 X Lþ3 BY,
h þ i Z^ iþ1
..
. . . . hL Z^ Lþ1 ... ... ... ... .. .. . . ... ...
I . . . I .. .. . . ... ...
3 7 7 7 7 7 7 7 7 7 7 7 7 < 0, ð24Þ 7 7 7 7 7 7 7 7 7 .. . 5 . . . I
then with the control law u ¼ KxðÞ, K ¼ YT , the system in (3) is asymptotically stable for all admissible networkinduced delays.
i¼0
þ
Lþ1 h X
i Q^ i þ N^ i,1 þ N^ Ti,1 1 AXT 1 XAT ,
5. Illustrative examples
i¼1
2,1 ¼
Lþ1 X
N^ i,2 N^ T1,1 2 AXT 1 YT BT ,
Consider the following open loop unstable continuous system with "
i¼1
3,1 ¼
Lþ1 X
N^ i,3 N^ T2,1 3 AXT 1 YT BT ,
Ap ¼
0:01999
3:048
0
13:86
#
" , Bp ¼
1:698 27:73
# :
ð25Þ
i¼1
3,2 ¼ N^ 1,3 NT2,3 3 BY 2 YT BT , 2,2 ¼ Q^ 1 N^ 1,2 NT1,2 2 BY 2 YT BT ,
The time delay profiles of the two channels: from sensor to controller and from controller to actuator, are set to be same as shown in figure 3. Hence the time delay parameters are h ¼ 3, L ¼ 1 with the sampling time
761
Stabilization of remote control systems Ts ¼ 0:05 sec. By zero-order holder method, the corresponding unstable discrete system in (3) is
1:001 0:11 0 A¼ , B¼ , 0 0:5 1 where the initial condition is xð0Þ ¼ ½0:5, 0:5T : Choosing E¼
0:1 0:02 0:01 0:1
and F0 ¼ F1 ¼ ½1, 1T , Tk k < I is proven to be satisfied by calculating the terms B0 and B1 according to their expressions, 8"ðkÞ 2 ½0, Ts , in Matlab. The initial condition is xð0Þ ¼ ½0:5, 0:5T : At first, by pole placement without considering the time delays, a gain K ¼ ½0:2, 0:45 gives the poles of the (A þ BK) within the unit circle as: 0.9755 0.1461i. However, this is not sufficient to stabilize the system. As shown in figure 4, the system response is still unstable. Secondly, through solving the corresponding LMI (20) in Theorem 1 on the delay dependent analysis, and with given values 1 ¼ 2 ¼ 3 ¼ 1, 4 ¼ 45 and h ¼ 3, we have X¼
1:7198 0:3270
0:069 : , Y¼ 0:0144 0:8741 0:2112
Hence K ¼ YXT ¼ ½0:0399, 0:0016: The resulted system response is stable as in figure 5. Thus the
The evolution of x1(k)
(a)
unstable system is stabilized by the remote controller designed according to Theorem 1 which is based on discrete time domain analysis. Furthermore, as discussed in the introduction, most existing works consider designs in continuous time domain or for sample-data systems. In discrete time domain, less attention has been paid to the problem than in this paper, e.g., Xiao et al. (2000) model the problem via a discrete-time jump linear system and Lee and Kwon (2002) is for a system with state delay only. Nilsson (1998) considers the networked control problem with delays shorter than one sampling time. Hence, here we compare the proposed scheme with the stability criterion in Lee and Lee (1999). In Lee and Lee (1999), the discrete time delay system is of form xðk þ 1Þ ¼ AxðkÞ þ Ad xðk hÞ
and the delay dependent criterion in Theorem 2.1 is developed for discrete systems with analogues corresponding to the results obtained for continuous time delay systems in Xie and de Souza (1997) and in Cao et al. (1998). Let 0 0 Ad ¼ BK ¼ 0:0399 0:0016 for (26) which is identical to the proposed case when B0 ¼ 0 and B1 ¼ B since B0 þ B1 ¼ B in this paper. The maximum time delay h ¼ 68 sampling steps for (26) in Lee and Lee (1999). However, in the proposed
20 10 0 −10 −20
0
5
10
15
20
25
30
35
40
25
30
35
40
Time (Sec) The evolution of x2(k)
(b) 20 15 10 5 0 −5 −10 −15
0
5
10
15
20 Time (Sec)
Figure 4.
ð26Þ
The evolution of the system states with K ¼ [0.2, 0.45].
762
Y.-J. Pan et al.
The evolution of x1(k)
(a) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
5
10
15
20
25
30
35
40
45
50
30
35
40
45
50
Time (Sec)
The evolution of x2(k)
(b) 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1
0
5
10
15
20
25 Time (Sec)
Figure 5.
The evolution of the system states with K ¼ [0.0399, 0.0016] designed according to Theorem 1.
The evolution of x1(k)
(a) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
5
10
15
20
25 30 Time (Sec)
35
40
45
50
0
5
10
15
20
25 30 Time (Sec)
35
40
45
50
The evolution of x2(k)
(b) 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1
Figure 6.
The evolution of the system states with K ¼ ½0:0399, 0:0016 when h ¼ 136Ts :
scheme, after checking the delay-dependent condition in Proposition 2, the LMI in 16 is still feasible when we have times delay steps as h 136. When checking the response of the closed-loop system as shown in figure 6,
it is still stable though the settling time is longer. Hence, the delay-dependent criterion in Proposition 2 is less conservative than the one in Theorem 2.1 in Lee and Lee (1999).
Stabilization of remote control systems 6. Conclusions The controller designs based on delay dependent stability conditions for discrete time remote control systems have been proposed in this paper by using LMI technique. Two cases classified according to the knowledge of the time delay bound have been discussed. Future work may be in the following areas: (i) on the robust stabilization of the remote control systems with uncertain external disturbances and (ii) on the case with data dropouts.
Acknowledgements This research was supported by NSERC and Syncrude Canada.
References Y.Y. Cao, Y.X. Sun and J. Lam, ‘‘Delay dependent robust H1 control for uncertain systems with time varying delays’’, in proceedings of the IEE Proceedings on Control Theory and Applications, 145(3), 1998, pp. 338–344. L. Dugard and E.I. Verriest, Stability and Control of Time-Delay Systems, Lecture Notes in Control and Information Science, Vol. 228, Berlin: Springer, 1998. J.K. Hale, Introduction to Functional Differential Equations, New York: Springer-Verlag, 1993. B. Lee and J.G. Lee, ‘‘Delay dependent stability criteria for discrete time delays systems’’, in Proceedings of the 1999 American Control Conference, San Diego, California, USA, 1999, pp. 319–320. Y.S. Lee and W.H. Kwon, ‘‘Delay dependent robust stabilization of uncertain discrete time state delayed systems’’, in Proceedings of the 15th IFAC World Congress, Barcelona, Spain, 2002. R. Lozano, P. Castillo, P. Garcia and A. Dzul, ‘‘Robust prediction based control for unstable delay systems: Application to the
763
yaw control of a mini-helicopter’’, Automatica, 40, pp. 603–612, 2004. M.S. Mahmoud, Robust Control and Filtering for Time Delay Systems, New York: Marcel Dekker, Inc., 2000. J. Nilsson, ‘‘Real time control systems with delays’’, PhD thesis, Lund Institute of Technology, 1998. Y.J. Pan, C. Canudas-de-Wit and O. Sename, ‘‘Bilateral teleoperation control on vehicle systems – networked control system project (NECS)’’, Technical report, Laboratoire d’Automatique de Grenoble, France, 2004a. Y.J. Pan, C. Canudas-de-Wit and O. Sename, ‘‘Predictive controller design for bilateral teleoperation systems with time varying delays’’, in Proceedings of the 43rd IEEE Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, 2004b, pp. 3521–3526. P. Park, ‘‘A delay-dependent stability criterion for systems with uncertain time-invariant delays’’, IEEE Transactions on Automatic Control, 44, pp. 876–877, 1999. S.H. Song, J.K. Kim, C.H. Yim and H.C. Kim, ‘‘H1 control of discrete time linear systems with time varying delays in state’’, Automatica, 35, pp. 1587–1591, 1999. D.M. Stipanovic and D.D. Siljak, ‘‘Robust stability and stabilization of discrete time nonlinear systems: the lmi approach’’, International Journal of Control, 74, pp. 873–879, 2001. M. Wu, Y. He, J.H. She and G.P. Liu, ‘‘Delay-dependent criteria for robust stability of time-varying delay systems’’, Automatica, 40, pp. 1435–1439, 2004a. L. Xiao, A. Hassibi and J.P. How, ‘‘Control with random communication delays via a discrete-time jump system approach’’, in Proceedings of the 2000 American Control Conference, Chicago, Illinois, USA, 2000, pp. 2199–2204. L. Xie and C.E. de Souza, ‘‘Delay dependent robust stability and stabilization of uncertain linear delay systems: a linear matrix inequality approach’’, IEEE Transactions on Automatic Control, 42, pp. 1144–1148, 1997. S. Su, J. Lam and T. Chen, ‘‘Robust H1 control for certain discrete stochastic time delays systems’’, Systems and Control Letters, 51, pp. 203–215, 2004. M. Yu, L. Wang and T. Chu, ‘‘Sampled-data stabilization of networked control systems with nonlinearity’’, IEE Proceedings on Control Theory and Applications, 152, pp. 609–614, 2005. D. Yue, Q.L. Han and J. Lam, ‘‘Network-based robust H1 control of systems with uncertainty’’, Automatica, 41, pp. 999–1007, 2005. W. Zhang, M.S. Branicky and S.M. Phillips, ‘‘Stability of networked control systems’’, IEEE Control Systems Magzines, 21, pp. 84–99, 2001.
