This paper addresses the stabilization problem for unmanned air vehicles over digital and wireless communication channels with time delay. In particular, the case with band-limited channels is considered. An observer-based state feedback control policy is employed to stabilize the linear control system of unmanned air vehicles. A sufficient condition on the minimum data rate for mean square stabilization is derived, and a new quantization, coding, and control policy is presented. Simulation results show the validity of the proposed scheme

Liu, Qing-Quan

English

TELKOMNIKA (Telecommunication Computing Electronics and Control)

TELKOMNIKA, Vol.11, No.1, March 2013, pp. 79~86 
ISSN: 1693-6930 
accredited by DGHE (DIKTI), Decree No: 51/Dikti/Kep/2010     79 
  
Received November 6, 2012; Revised January 6, 2013; Accepted January 27, 2013 
Stabilization of Unmanned Air Vehicles over Wireless 
Communication Channels  
 
 
Qing-Quan Liu 
College of Equipment Engineering, Shenyang Ligong University 
No. 6, Nan Ping Zhong Road, Hun Nan Xin District, Shenyang, 110159, China 
e-mail: lqqneu@163.com 
  
 
Abstrak 
Makalah ini membahas masalah stabilisasi untuk kendaraan udara tak berawak melalui saluran 
komunikasi digital dan nirkabel dengan waktu tunda. Secara khusus, kasus ini dilakukan pada saluran 
band-terbatas. Kebijakan kendali umpak-balik keadan berbasis berbasis pengamat digunakan untuk 
menstabilkan sistem kendali linier kendaraan udara tak berawak tersebut. Sebuah kondisi yang cukup 
pada laju data minimum untuk stabilisasi kuadrat rerata diturunkan, dan kuantisasi, penyandian, dan 
kebijakan kendali baru dihadirkan. Hasil simulasi menunjukkan validitas dari skema yang diusulkan. 
 
Kata kunci: kendaraan udara tak berawak, waktu tunda, data rate, berarti stabilisasi persegi 
 
 
Abstract 
This paper addresses the stabilization problem for unmanned air vehicles over digital and 
wireless communication channels with time delay. In particular, the case with band-limited channels is 
considered. An observer-based state feedback control policy is employed to stabilize the linear control 
system of unmanned air vehicles. A sufficient condition on the minimum data rate for mean square 
stabilization is derived, and a new quantization, coding, and control policy is presented. Simulation results 
show the validity of the proposed scheme. 
 
Keywords: unmanned air vehicles, time delay, data rate, mean square stabilization 
 
 
1. Introduction  
The research on networked control systems has received resurgent interests in recent 
years [1-3]. Some salient examples include unmanned air vehicles (UAVs), unmanned ground 
vehicles (UGVs), and unmanned underwater vehicles (UUVs) [4-6]. The presence of such digital 
and wireless communication channels which connect the sensor and the controller or the 
actuator and the controller brings up many challenges, and makes some traditional control 
approaches inapplicable or inefficient.  
One of the earliest papers on the topic is [7], where a weaker stability concept called 
containability was introduced, and the issue of coding and communication protocol became an 
integral part of the analysis. The research on stabilization of linear time-invariant systems with 
limited data rate was addressed. The fundamental problem of finding the lower bound on the 
data rate for stabilization was solved in [8-10]. There has been a lot of new interest in quantized 
feedback control where the measurements are quantized, coded and transmitted over a digital 
communication channel. A fundamental problem is how to design a quantization, coding and 
control scheme in order to achieve some given control performances.  
The research on quantized feedback control can be categorized depending on whether 
the quantizer is static or dynamic. Static quantizers were employed in [11-13]. In particular, it 
was shown in [8] that the coarsest quantizer is logarithmic. The results were generalized in [13] 
to a number of output feedback problems using a sector bound approach. Furthermore, 
dynamic quantizers were employed in [14-18]. It was shown in [14] that stabilization of a single-
input single-output (SISO) linear time-invariant (LTI) system could be achieved by employing 
only a finite number of quantization levels. It was shown in [15] that a feedback policy could be 
designed to bring the closed-loop state arbitrarily close to zero for an arbitrarily long time by 
employing a quantizer with various sensitivity. 
We addressed the observer-based, dynamic state feedback stabilization problem for 
                 ISSN: 1693-6930 
TELKOMNIKA  Vol. 11, No. 1, March 2013 :  79  – 86 
80
networked control systems with limited data rates over noisy communication channels [19]. The 
case with both measurement quantization and control signal quantization was considered in [20] 
and the case of LQG systems subject to both input and output quantization was addressed [21].  
In this paper, we investigate the stabilization problem for unmanned air vehicles, where 
the sensors and the controllers are connected via digital and wireless communication channels. 
The case with random channel propagation delay is considered. Here, a buffer is employed in 
order to achieve better control performance. Our work here differs in that we present an 
observer-based state feedback control policy to ensure stabilization of unmanned air vehicles 
over digital and wireless communication channels with random time delay, and derive the 
sufficient condition on the data rate for stabilization. 
The remainder of this paper is organized as follows: Section 2 introduces problem 
formulation. Section 3 presents the sufficient condition on the data rate for stabilization. Section 
4 gives a numerical example. Conclusions are stated in Section 5. 
 
 
2. Problem Formulation 
Here, we examine unmanned air vehicles and consider a realization of the feedback 
linear time-invariant system of the form 
 
)()()(
),()()()1(
kDGkCXkY
kFWkBUkAXkX


  (1) 
 
where  X(k)
 
nR  is the plant state, U(k)
 
pR  is the control input, Y (k)
 
qR  is the 
observation output, G(k) lR is the bounded additive disturbance, and W(k) mR  is the 
process disturbance, respectively. A, B, C, D, and F are known constant matrices with 
appropriate dimensions. Here, we address the case that the sensors and the controller are 
geographically separated and connected via digital and wireless communication channels with 
time delay. 
 
 
 
 
Figure 1. Networked control systems 
 
 
Without loss of generality, assume that the pair (A, B) is controllable, and the pair (A, C) 
is observable. The initial condition X(0) and process disturbance W(k) are mutually independent 
random variables with zero mean, satisfying  0
2
)0( XE  and  WkXE 
2
)( , 
respectively. Here, assume that W(0), ,W(k) are independent and identically distributive (iid). 
Here, we try to implement an observer-based state feedback control law of the form 
 
).(ˆ)(
,))]'((,)),(([)(ˆ
)),()(()()()1(
1
kXKkU
kxqkxqkX
kXCkYLkBUkXAkX
n







    (2) 
 
As stated in [22], we define a quantizer ZRq  :)(  with sensitivity  i(k) and saturation value 
TELKOMNIKA  ISSN: 1693-6930  
 
Stabilization of Unmanned Air Vehicles over Wireless Communication…(Qing-Quan Liu) 
81
Mi by the formula 
 
q(zi(k)) =
1
, ( ) ( ) ( )
2
1
, ( ) ( ) ( )
2
( ) 1 1 1
, ( ) ( ) ( ) ( ) ( )
( ) 2 2 2
i i i
i i i
i
i i i i i
i
M if z k M k
M if z k M k
z k
if M k z k M k
k



  

    

         
     (3) 
 
where we define  
 
   RzzkZkz  ,:max: . 
 
the indexes M  and M  will be employed if the quantizer saturates [22].                                                                   
The closed-loop system (1) can be written as  
 
       )()()()1( kQkNkMk  

 (4) 
 
where 
 
.
0
0
:,
)(
)(
:)(,
)(
)(
:)(
,
0
0
:,:,
)(
)(
:)(







































LD
F
Q
kG
kW
k
kX
kX
k
BK
BK
N
LCALC
BKA
M
kX
kX
k





 
 
In this paper, our main task is to present a quantization, coding and control policy to 
stabilize the system (1) in the mean square sense  
 

2
)(suplim kXEk          (5) 
 
by employing the limited data rate of the channels, and to derive the sufficient condition on the 
data rate for stabilization of the system (1). 
 
 
3. Control over Wireless and Digital Channels with Time Delay 
This section deals with the stabilization problem for linear control systems of unmanned 
air vehicles where the sensors and the controllers are geographically separated and connected 
via wireless and digital communication channels with time delay.  
First, we give a lemma which comes from [23]. 
 
3.1. Lemma 
Let Rz  denote a random variable and z  denote an estimate of z. Define R(D) as 
the information rate distortion function between z

 and z. The expected distortion constraint is 
defined as  Rd . Given  2D E z z   , there must exist a quantization and coding scheme if 
the data rate R satisfies the following condition: 
 
2
2
1 ( )
( ) log
2
z
R R D
D

      (bits/sample)      (6) 
where  
 
                 ISSN: 1693-6930 
TELKOMNIKA  Vol. 11, No. 1, March 2013 :  79  – 86 
82
 22 ( )z E z Ez   . 
 
Proof: See [23]. 
The following theorem is our main result. 
 
3.2. Theorem 
Consider the closed-loop system (4) where the sensors and the controller are 
connected via the digital and wireless communication channel with time delay. Let )(i denote 
the ith eigenvalue of a matrix. Then, there exists the quantization, coding, and control policy to 
stabilize the closed-loop system (4) in the mean square sense (5) if the data rate R of the 
channel satisfies the following condition: 
 
)/(
)'(
log
2
1
2 samplebits
MM
R
i
i  

     (7)
 
 
where 
   )'(:),,1(: MMni i . 
 
Proof: Notice that the closed-loop system (4) can also be rewritten as 
 
)()1()(
)())()(())()(()1(
kQkkMV
kQkNkMkkMk






   (8)
 
 
where we define 
 
)()(:)( kkkV 

 . 
 
Here, )(k  denotes the centroid of the uncertain region of the plant states at time k . 
Then it follows that 
 
))1(')((2)')(')((2))1(')((2
)'()()'()1( )()1()(
2

 
kkQEtrQkkMEVtrkkMEVtr
QQtrtrMMtrkE kkkV

 


 (9)
 
 
where we define 
 
).(')(:
),1(')1(:
),(')(:
)(
)1(
)(
kkE
kkE
kVkEV
k
k
kV










 
 
Notice that )(kV , )1(' k and )(k  are mutually independent. Then, it follows that 
 
.0)1(')(
,0)(')(
,0)1(')(



kkE
kkEV
kkEV





 
 
Thus, we have 
TELKOMNIKA  ISSN: 1693-6930  
 
Stabilization of Unmanned Air Vehicles over Wireless Communication…(Qing-Quan Liu) 
83
 
)'()()'()1( )()1()(
2
kkkV
QQtrtrMMtrkE    .          (10) 
 
There exists a quantization, coding, and control policy to make the following inequality hold: 
 
)()'()(
)1()(
2
 kkV trMMtrkE   .                       
 
It means that for any )1,0( , it holds that 
 
)()'()(
)1()(
2
 kkV trMMtrkE   .                      (11) 
 
Substitute (11) into (10) and obtain 
 
)'()()1( )(
22
kQQtrkEkE   . 
 
Thus, it follows that 
 
.
1
1
)'()()(
2
1
0 )(
122
W
k
O
k
k
i k
ikk
Q
QQtroEkE




   


 
 
 
Thus, we have 
 


 Wk QkE
22
1
1
)(suplim

 . 
 
It states that there exists the quantization scheme (3) and control policy (2) to stabilize 
the system (4) in the mean square sense (5).   
Notice that 
 
.)()(
)()()(
1
1
1







d
i
d
d
i
d
idkZAkX
idkZdkXAkX


 
 
Here, we define 







)(
0
)(
kZ
kZ . Then, )()()( kkkZ 

 . 
 
 
 
It follows that 
 
)()()( kVkZkZ 

                                      
 
where )(kZ

 is the quantization value of )(kZ .  By the quantization scheme (3), we know that  
  
                 ISSN: 1693-6930 
TELKOMNIKA  Vol. 11, No. 1, March 2013 :  79  – 86 
84
))((
)'(
))(( 2
1 1
2 kv
MM
ikz i
n
i
n
i
i
i 

   . 
 
Then, it follows from Lemma 3.1 that 
 
)/(
)'(
log
2
1
))((
log
2
1
2
2
2
samplebits
MM
D
kz
R
i
i
i
i
i









 
 
where 
   )'(:),,1(: MMni i . 
 
3.3. Remark 
It is shown in Theorem 3.2 that, the exists that a quantization, coding, and control policy 
such that the system (1) is stabilizable in the mean square sense (5) if the data rate of the digital 
and wireless communication channel that connects the sensors and the controller is larger than 
the lower bound given by Theorem 3.2. Clearly, the data rate has important effects on 
stabilization of the control system of unmanned air vehicles.  
 
 
4. Numerical Example 
In this section, we examine a class of networked control problems of unmanned air 
vehicles, and consider linear control systems over digital and wireless communication channels 
with time delay. In particular, we present a numerical example to illustrate the proposed 
quantization, coding and control scheme. Here, we consider an open-loop unstable system as 
follows: 
 
).(24.2)(
52.145.126.0
35.053.123.2
)(
),(23.0)(
51.2
50.0
25.2
)(
72.127.013.0
25.073.232.0
61.024.041.3
)1(
kGkXkY
kWkUkXkX






























 
 
Let '2000,5000] [5000,  X(0)  and 0001 W  . Here, we try to implement an observer-
based state feedback control law of the form 
 
  ).(ˆ81.251.034.1)(
,))]'((,)),(([)(ˆ
)),(
52.145.126.0
35.053.123.2
)((
32.151.2
12.043.1
31.022.1
)(
51.2
50.0
25.2
)(
72.127.013.0
25.073.232.0
61.024.041.3
)1(
1
kXkU
kxqkxqkX
kXkYkUkXkX
n














































 
We first set 180bits/s  R  . A corresponding simulation is given in Figure 2. 
 
 
TELKOMNIKA  ISSN: 1693-6930  
 
Stabilization of Unmanned Air Vehicles over Wireless Communication…(Qing-Quan Liu) 
85
 
 
Figure 2. The responses of system states when bits/s 180  R   
 
 
It is shown in Fig.2 that there exists no quantization, coding and control scheme to 
stabilize the system if the data rate of the channel is smaller than the lower bound given by 
Theorem 3.2. If we assume that bits/s 240  R , a corresponding simulation is given in Fig.3. It is 
shown in Fig.3 that the quantization, coding, and control scheme can stabilize the unstable 
system if the data rate R of the channel is greater than the lower bound given by Theorem 3.2. 
 
 
 
 
Figure 3. The responses of system states when bits/s 240  R   
 
 
If we assume that bits/s 480  R , a corresponding simulation is given in Fig.4. It is shown 
in Fig.4 that the system is stabilizable in the mean square. The data rate has important effects 
on the control performance. That is, the greater the data rate R of the channel, the better the 
control performance. 
 
 
 
Figure 4. The responses of system states when bits/s 480  R   
                 ISSN: 1693-6930 
TELKOMNIKA  Vol. 11, No. 1, March 2013 :  79  – 86 
86
5. Conclusion 
In this paper, we addressed the stabilization problem of linear control system of 
unmanned air vehicles. Our results stated that the data rate of the digital and wireless 
communication channel has important effects on the control performance of unmanned air 
vehicles. Thus, the data rate of the channel that connects the sensors and the controller would 
be larger than the lower bound given in our results. The simulation results have illustrated the 
effectiveness of the proposed scheme. 
 
 
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Stabilization of Unmanned Air Vehicles over Wireless Communication Channels

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Stabilization of Unmanned Air Vehicles over Wireless Communication Channels

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