Browse \u3e Conferences\u3e Intelligent Control. 2003 IEEE ...    Networked control systems: a sampled-data approach .1254671 abstract   . Hokayem, P.F.;   Abdallah, C.T.;    Dept. of EECE, Univ. of New Mexico, Albuquerque, NM, USA    This paper appears in: Intelligent Control. 2003 IEEE International Symposium on  Issue Date: 8-8 Oct. 2003  On page(s): 415 - 419  Location: Houston, TX, USA  Print ISBN: 0-7803-7891-1  INSPEC Accession Number: 7929432  Digital Object Identifier: 10.1109/ISIC.2003.1254671   Date of Current Version: 08 January 2004  Abstract In this paper we present a novel modelling method for networked control systems, motivated from a sampled-data approach. We study sufficient conditions that guarantee exponential stability for the closed-loop system and illustrate our results via a numerical example

Abdallah, Chaouki T.

Hokayem, Peter E.

English

University of New Mexico

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10-5-2003
Networked control systems: a sampled-data
approach
Chaouki T. Abdallah
Peter E. Hokayem
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Recommended Citation
Abdallah, Chaouki T. and Peter E. Hokayem. "Networked control systems: a sampled-data approach." 2003 IEEE International
Symposium on Intelligent Control (2003): 415-419. doi:10.1109/ISIC.2003.1254671.
Proceedings of the 2003 IEEE 
International Symposium on Intelligent Control 
Houston, Texas *October 5-8,2003 
Networked Control Systems: A Sampled-Data 
Approach? 
Peter E Hokayem and Chaouki T. Abdallah 
EECE Department, MSCOI I100 
I Universiry of New Mexico 
Albuquerque, NM 87131-0001, USA 
{hokayem,chaouki)@eece.unm.edu 
Absnact-In thfs paper we present a novel modelling method 
for networked control systems, motivated from a sampled- 
data approach. We study sufficient conditions that guarantee 
exponential stability for the closed-loop system and illustrate our 
results via a numerical example. 
Index Terms-Networked Control Systems, sampled-data sys- 
tems, lifting, exponential stability. 
I. INTRODUCTION 
Over the past decade, major advancements in the area 
of communication and computer networks 191 have made it 
possible for control engineers to include them in feedback 
systems in order to achieve real-time requirements. This gave 
rise to a new paradigm in control systems where instantaneous 
flow of the control signals is no longer sufficient, and the 
feedback loop is closed through a real-time network. Such 
control systems that utilize networks to achieve closed loop 
performance are called Networked Control Sysfem (NCS). 
Several examples of NCSs are available in automobile in- 
dustry, teleoperation of robots, and automated manufacturing 
systems. Including the networks into the design of such 
systems has made it possible to increase mobility, reduce 
the cost of dedicated cabling, and render easier and cheaper 
maintenance. 
This paper starts by reviewing some basic trends in the 
study of stability of networked control systems in Section 
11. Then we present our new approach for modelling such 
systems in Section 111. In Section IV, we address the issue 
of stability, of such models, using Lyapunov techniques for 
discrete-time systems. Finally, we illustrate our results via a 
numerical example in Section V. 
11. REVIEW OF PREVIOUS WORK 
in the past decade, several methods of modelling networked 
control systems have been proposed. and the stability of such 
models was the main concern of their analysis. In this section 
we provide an overview of basic approaches and results. 
A .  Structural Approach 
The authors of [7] present an extended structural analysis 
of networked control systems, using an eigenvalue approach. 
In their model, the network resides between the sensors that 
are attached to the plant, and the actuators. The network is 
$This work h a  been supported by NSF Grant INT-98183 12 and NSF Grant 
0233205. 
modelled as a fixed-rate sampling of the continuous plant. 
They also present 8 model plant that provides state estimate, 
and the error between the actual plant and the model plant 
is used to augment the state-vector. Then, the analysis is 
applied to the augmented system in order to obtain necessary 
conditions for guaranteeing stability of the closed-loop system. 
They analyze the performance of the system when full state 
and partial state are available for feedback. 
B. Perturbation Approach 
In [IO], a try-once-discard (TOD) protocol is introduced, 
where the next node to transmit data on a multi-node net- 
work is decided dynamically based on the highest weighted 
error from the last transmission. The goal is to find a 
maximum transmission interval that guarantees satisfactory 
stability performance. The network resides between the plant 
and controller and introduces the error between successive 
transmissions. The resulting state-space system is comprised 
of the plant state-vector, and the error stale-vector. The error 
is considered as a perturbation of the original plant, and 
methods presented in [5] are utilized to derive conditions for 
the stability of the closed-loop system. 
C. De/ay Appmack 
Nilsson [SI includes the following cases for modeling the 
effects of introducing the network into the control-loop, ren- 
dering an NCS: . Constant delay 
Random independent delays 
Random delays govemed by an underlying Markov chain 
Then for each model he solves an LQG optimal control 
problem, to generate a controller that guarantees stability. 
D. Hybrid Systems Appmach 
Zhang el. a1 [ I l l ,  1121 utilize results previously derived 
for the stability of hybrid systems, to find bounds on the 
delay introduced by the network. In particular, [ l l ]  models 
the network as a constant delay introduced into the full state 
feedback as follows: 
k(t) = Az(t) - BKj.(t) ,  
j.(t+) = z(t - T ) ,  
t E [kh + T ,  (k + 1)h + T ]  
t E [kh + 7, k = 0,1,2,.  . ]  
(1) 
where h is the sampling period. Then the trajectory of the 
delayed state vector z(t  - T )  is solved for, in terms of z(t) 
0-7803-7891-1103/$17.00 0 2003 IEEE 41 5 
and 2(t). The bound on the delay r results from imposing 
Schur stability conditions on the following matrix. 
where for a given matrix M ,  E(h)M = $ eA(h-n)Mdq 
An extensive study has recently appeared in [4] where 
the NCS has limited data rate available in order to maintain 
stability. The problem is tackled from different perspectives: 
Variable-rate sampling, various quantization schemes, dis- 
tributed control, and switching control with sufficient dwell- 
time. The main objective is to reduce the amount of data to 
be transmitted via the network. 
111. N E W  MODELLING OF NCS 
As seen in the previous sections, there are several trends 
in modeling networked control systems. In this section we 
are going to introduce yet another modelling method and 
manipulate it to obtain a generalized LTI sampled-data system. 
The proposed model allows us to avoid the tedious analysis 
of the effect of the delay introduced by the network. This is 
achieved through incorporating the delay into the model of 
the system, and it is sufficient to study the stability of the 
overall system, without explicitly addressing the actual value 
and nature of the delay. Before we introduce the new model, 
we present few assumptions: 
I. The controller and actuators are directly attached to the 
plant, i.e. no transport delay exists between the controller 
and plant actuators. 
11. The sensors are part of the plant model. 
111. The network effect is recognized only between the sen- 
Proposition I :  We model the network as a variable-rate 
ideal sampler (&), between the plant (G) and the controller 
(C), and a corresponding zero-order hold (HTk), as shown in 
Figure 1. 
sors and controller. 
Q 
U 
Fig. I. System Model 
Consider the following plant model, 
2(t) = Az(t) + Blw(t) + B p ( t )  
~ ( t )  = Ci~( t )  + Dliw(t) f D124t) 
Y(t) = C Z N  13) 
where z( t )  E R" is the state vector, u(t)  E Rm is the control 
input vector, w(t) E R' is the vector of exogenous inputs, 
z ( t )  E RP is the vector of controlled outputs, and y ( t )  E Etq 
is the vector of measurable outputs. Finally, 
We assume that D z ~  = 0 2 2  = 0, i.e. the transfer functions 
from the control input, u(t) ,  and from the exogenous input, 
w(t), to the measured output, ~ ( t ) ,  are strictly proper. The 
latter condition provides continuity in the measured output 
vector [I], i.e. avoiding impulses in the output. 
The above framework results in a time-varying system, 
that has both continuous and discrete signals, hence a hybrid 
system. The study of such systems is in'general complex, and 
a unified theoty for such systems is not yet available [6].  For 
such reasons, we need to manipulate the model in order to 
obtain a generalized LTI sampled-data system. In order to do 
so, we employ the lifting technique [I], [Z], and incorporate 
the ideal sampler and hold devices into the plant model in the 
following manner: 
where 7k = t k  - tk-1 is the variable samplingrate, L, and 
L;: are the lifting and inverse lifting operators, respectively. 
The transformed system is shown in Figure 2. 
/ I  U 
;. y rk i 
...... ................................................... ...... 
Fig. 2. The Recanfigured NCS 
Next we present the above transformations mathematically. 
i. Gll -+GI,  
The transfer function GII relates w ( t )  to z ( t ) ,  in con- 
tinuous time. d l l  on the other hand relates &.to i both 
being the lifted signals, corresponding Lo w(t) and z ( t ) .  
Consequently the linear operators of GII are given as 
follows: 
A = eAT* 
TL 
B I G  = eA(rk-S)Blw(p)dq (6) 
(&z)(t)  = CleA'z 
(&2i))(t) = D11w(t) + Cl 
41 6 
ii. G 1 2  -+ 8 1 2  
In a similar fashion, we transform BIZ and DIZ into BIZ 
and 0 1 2 ,  respectively. And 8 1 2  relates the discrete input 
U k  and the lifted Output z k .  
B 2  = LTkeA"dqBz (7) 
(&iL)(t) = DlzO + CI 
... 
111. GZI --t GZI and GZZ - &Z 
Both transformations follow from (6) and (7), 
After applying the above transformations to (4) we obtain 
an LTI sampled-data system (c), which is shown in Figure 2. 
Then we refer back to the usual 7im (see [2]) design to obtain 
the controller (C). Assuming that the controller (C) has been 
designed, we present stability analysis results of the overall 
system in the next section. 
IV. STABILITY ANALYSIS 
In this section we study the stability of the model presented 
in the previous section. We shall start by deriving the closed- 
loop system that involves 8 2 2  and the controller C. Note that 
we only need to stabilize Gzz due to the following theorem. 
Theorem I: [I] The controller C internally stabilizes the 
hybrid system in figure 2,jf and only if it intemally stabilizes 
the discrete-time system GZZ in (5) .  
H 
The plant model of GZZ is described as follows, 
I*+] = A X k + & U k  
y k  = 6 Z X k  = C 2 X k  (8) 
and the controller C is described by the following state-space 
realization 
Vk+l = A c % + & Y k  
U k  = C c u k  + D c Y k  (9) 
Combining (8) and (9) we get the following augmented state 
space representation 
( X k t l  ) 
S t f l  = vk+l 
- ( A+BzD,Cz 
BzC, ) ( ; ) B S h  A,  
H k S x  (10) 
Notice that the above system does not take into account the 
effects of disturbances. Consequently, we shall introduce the 
effects of disturbances, through w(t) in (3). into (10) as 
Defrnirion 1: The origin of the system Z k + l  = A k x k  is 
exponentially stable if there exists an U > 0, and for every 
E > 0 there exists a a(€) > 0, such that 
llXkll 5 Ee-u(''-'o) 11~011 (12) 
whenever 11x011 < a(€) and to 2 0. If 6 ( e )  - CO then the 
system is exponentially stable in the large. 
w 
The following theorem utilizes results in [3], and specializes 
Theorem 2: The origin of the closed loop discrete-time 
them to solve the problem at hand. 
system ( IO)  is exponentially stable in the large provided, 
i. supvkEN Tk < CO 
ii. l l H k l l  < &,Vk E N 
Proof. Given IlHkll  < a < 1,Vk E N ,  then there exist a 
syIlUYletriC matrix P k  > 0, such that H r P k H k  - P k  -1. 
Then 11pk11 5 IlIll + IIHkTpKffkII  5 1 + a Z I I P k l l  * 1 5 
l l p k l l  5 &, since o < a  < 1. 
Let v ( S k )  = s ( k ) T p k - ] S ( k ) ,  then 
AV V ( s k + l )  - v ( S k )  
T T = s k + 1 P k s k + l  - S k p k - l s k  
= S E ( H F P k H k  - P k ) s k  + S T ( p k  - p C - 1 ) S k  
Since I l P k - P k - l l l m o z  = &,-I = A. For thesystemto 
be stable, AV must be less than zero. Therefore, < 0 
The above result guarantees that the system (IO) is stable. 
Still required to prove that it is exponentially stable. Since 
(1- 1 
, a < & .  
v ( S k )  A S ( k ) T P k - l S ( k )  then 
Combining (14) and (15) we get, 
- . .  
follows 
Let a = m t n { l , - l n ( m ) }  and E = 6, the result 
In the above analysis we have ignored the effect of the 
disturbances on the system. So we are going to extend the 
result of Theorem 2 to compensate for bounded and vanishing, 
state-bounded disturbances and in what follows. 
s k + 1  = H k S k  + ( ) H k S k  f r k  (11) follows. w 
Before we plunge into the stability analysis, we shall present a 
general formal definition of exponential stability for discrete- 
time systems. 
41 7 
Theorem 3: (Bounded Disturbance) Given that the origin of 
the discrete-time system (10) is exponentially stable, and that 
l l r k l l  5 y < +a, (bounded-input), then the system (11) has 
a bounded-state output. 
Proof. The proof is simple through analyzing the time 
progression of the state-vector. 
Taking the limit of k on both sides: 
z(k + 1) = ng, H , ~ ( o )  + (n,"=,+, H,)  .rj. 
limk-, (z(k + 1)) = limk,, ( n f = , ~ , . z ( ~ ) )  + 
limk-, (E:=, (nej+, H , )  .rj) 
* 11z(00)II 5 ilrll. (A) = < 00. Since the first limit 
tends to zero as k - 00 and l l f f k l l  < a < 1 + we take 
the maximum of H k  = a and form a geometric progression 
Theorem 4: (Vanishing Disturbance) The origin of the 
closed loop discrete-time system (1 1) is exponentially stable 
in the large provided, 
i. sup,kEN Tk c 00 
whose answer is (6). , 1 
ii. llHk11 < -$j,Vk E N 
iii. < ~ ~ S k ~ ~ j V k  E N 
Proof. We follow a similar analysis as in Theorem 2.  Let 
I l rk l l  < Y l b k l l .  where 7 > 0. 
AV V ( S k + l ) - V ( S k )  
= ( H k S k  + r k ) T P k ( H k S k  + r k )  - S ; P k - l S k  
= S % ( f f r P k f f k  - % ) S k  + S r ( P k  - P k - 1 ) S k  
+2srH,TPkrk + r;pkrk 
2aZ - 1 
- < (3) llSk112 
(17) 
(3 + m) Y2 1bkl12 
And AV in (17) is always negative provided that y < 1.  The 
rest follows as in Theorem 2. 1 
V. NUMERICAL EXAMPLE 
In this section we will consider a numerical example to 
illustrate the theoretical stability results derived in Section IV, 
specifically in Theorem 2. 
Consider the following scalar continuous-time LTI plant 
model 
i ( t )  = 0.5z(t) t 10u(t) 
Y ( t )  = 4 t )  (18) 
whose discrete version is that described in (8). Consider also 
the following discrete-time LTI controller C 
Uk+1 = 0 . h k  - 0 . 5 y k  
U k  = - 0 . 5 ~ k  - l / x  (19) 
Consequently, the closed-loop system matrix f f k  in (lo), 
corresponding to (18) and (19), is given by 
I 1 
9 . 2  
1.1 :i 
By Theorem 2, we need to keep the norm of Hb less than 5. Since we fixed the values for the controller parameters, 
we can vary Tk to meet the required condition on H A .  The 
range of Tk for which the induced Euclidean norm' of H k  is 
less than 5 is shown in Figure 3, where 
0.061 < Tk < 0.126. (21) 
In order to fully understand the implications of varying the 
sampling time Tk on the stability of the system, we will first 
study the behavior of the closed loop system in (20) g' iven a 
constant T k .  
The response of the closed-loop system at the boundary of 
the range given in (21). i.e. ~k = 0.126, is shown in Figure 4 
where the system retains its stability. 
We further increase the value to T k  beyond 0.126 until we hit 
the first instability point. As seen in Figure 5, the response of 
the closed loop system diverges for Tk = 0.164. This conveys 
the conservativeness of the stability analysis, since the results 
are sufficient but not necessary. 
Finally, we test the system response for a variable sampling 
time given by 
Tk = 0.126 + € x U (22) 
where U is a uniformly distributed random number between 
0 and 1, and 6 E R. This representation of Tk allows us to 
see how far can we sample randomly beyond the theoretical 
bound and still maintain stability. As seen in Figure 6, the 
system diverges for c = 0.076. 
It is interesting to compare the two results presented in 
Figures 5 and 6. For the random case, the value of TA. depends 
on the outcome of the random number U given in (22), 
whose mean is 0.038 for the simulation in Figure 6. Hence, 
average(7k) = 0.126+ average(€ x U )  = 0.164 which is the 
'The induced Euclidean norm of any matrix M is given by 
X,,.(MTM) ' I z ,  where denotes the maximum eigenvalue. 
41 8 
2 1 
Fig. 4. System response for T* = 0.126 
0, , I I 
I 
0 1 2  3 . 5 1  7 8 P 10 
unn 1-1 
-21 
Fig. 5.  System response for ~k = 0.164 
same as the fixed r k  in Figure 5 .  Consequently, the random 
~k behaves like the fixed one on average. 
VI. CONCLUSION 
In this paper we have presented a new method for modelling 
Networked Control Systems. The main idea is viewing NCS as 
a variable-rate, sampled-data system. Then, we utilized some 
results petraining to the stability of such sampled-data systems 
and extended them to the problem at hand. The bounds derived 
for guaranteing stability are conservative, and further work 
should aim at developing new bounds that eliminate that kind 
of conservativeness. 
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Fig. 6. System response for T& = 0.126 + L x U 
[2] Chcn, T. and Francis, B., Optimal Sampled-Data Conrml System. 
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41 9 


Networked control systems: a sampled-data approach

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