Journal of Robotics, Networking and Artificial Life
Vol. 6(4); March (2020), pp. 235–239
DOI: https://doi.org/10.2991/jrnal.k.200222.006; ISSN 2405-9021; eISSN 2352-6386
https://www.atlantis-press.com/journals/jrnal

Research Article

Data-based Analysis Methods for the State Controllability
and State Observability of Discrete-time LTI Systems
with Time-delays
Zhuo Wang1,*, Ruigang Wang2
The Research Institute of Frontier Science, Beihang University, 37 Xueyuan Road, Haidian District, Beijing 100191, People’s Republic of China
The School of Instrumentation and Optoelectronic Engineering, Beihang University, 37 Xueyuan Road, Haidian District,
Beijing 100191, People’s Republic of China
1
2

ARTICLE INFOABSTRACT

Article History

We present a couple of data-based methods to analyze the state controllability and state observability of discrete-time Linear
time-Invariant (LTI) systems with time-delays, which have unknown parameter matrices. They first augment the system into ahigh dimensional LTI model, then apply the measured state and output data to directly build the controllability and observability
matrices respectively of this high dimensional model, whose ranks are used as the criteria of the corresponding properties of thesystem before augmentation. These data-based methods have low computational load and calculation complexity.

Received 21 October 2019
Accepted 10 December 2019
Keywords
Data-based analysis methods
discrete-time LTI systems with
time-delays
high dimensional LTI model
measured state and output datastate controllability and state
observability© 2020 The Authors. Published by Atlantis Press SARL.
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION
In the control theory and engineering area, time delay universallyand frequently happens in daily life and practical industries, whichmakes the time-delay systems a class of important research objects[1,2]. There are some typical examples of time-delay systems, suchas the artificial neural networks [2], the multi-agent systems [3],the electronic and optical systems [4], etc. As an essential propertyof the real world, even by using the state-of-the-art technology, theimpact of time delays cannot be completely eliminated. Hence, theextensive research on characteristics of systems with time delays,
for instance, the stability and robustness [5], the controllability
and observability [6], etc. is both theoretically and practicallysignificant.
But when it is put into practice, people always encounter difficulties in both theoretical and practical areas. One typical difficulty
is about the system dimension: when it is in the continuous-time
domain, it can be regarded as an infinite dimensional system,
which has a transcendental equation with infinite solutions as its
characteristic equation [7]; when the system is discrete-time, its
dimension increases dramatically with the increase of the time-­delay terms [1,3,8]. In addition, nowadays modern industries havebecome so large-scale and complex that correctly building theirdynamic models has become more and more difficult and even
Corresponding author. Email: zhuowang@buaa.edu.cn*

impractical [9,10]. All these problems have prevented analyzing thecharacteristics of time-delay systems.
Nevertheless, it is fortunate that the modern sensor technologies,
the wireless communication techniques, the computer and internet science and technologies, etc. have been developed rapidly, andhave been widely used in more and more fields. People acquire allkinds of data from industrial production and daily life in more
effective and systematic ways [10]. The information contained in
the measured data usually reflects the intrinsic nature of the systems [11]. In the big data era, this situation is very obvious, so
the idea of developing data-based methods is naturally generated,which only use the measured data to analyze the characteristics oftime-delay systems.
In most of the published research reports, the data-based analysis
methods for system characteristics could only deal with particulardelay model structures, but may not be able to analyze the systems
with a general form of time delays. For the sake of overcoming
these problems, in this paper we will develop a couple of data-based
methods to analyze the state controllability and state observabilityof a general class of discrete-time delayed Linear time-Invariant(LTI) systems. They can directly determine the system characteristics just by utilizing a simple augmented state-space model together
with the measured historical and current testing data, without
identifying the unknown parameter matrices. Their advantages are
discussed from the aspects of identification workload and computational complexity.

236

Z. Wang and R. Wang / Journal of Robotics, Networking and Artificial Life 6(4) 235–239

2. DESCRIPTION OF THE PROBLEM
This study concerns the analysis problem of the state controllability and state observability of the general discrete-time LTI systems
with time delays in the following form:NxNu x (k + 1) = ∑ Ai x(k − i) + ∑ B ju(k − j),


i =0
j=0

Ny
 y(k) = C x(k − l ),
∑
l

l =0

(1)

This paper supposes that these Ai, Bj and Cl are all unknown andnone of them has random elements. Next below, we will propose acouple of analysis methods to determine the above system characteristics without the necessity of identifying the parameter matrices, but just by utilizing the measured data. The advantage of thesedata-based characteristics determination methods will also be
illustrated.
To begin with, system (1) is first expanded into a high dimensional
LTI model by state-space augmentation:(k ≥ 0), (2) u(k)  x(k) 
where U (k) =  u(k − 1) , X (k) =  x(k − 1)  are the input and










u(k − N u )
 x(k − N x )
the state of system (2), respectively; and the augmented parameter
matrices are defined as
A1
A2
 AN x 
 A0
 I
0
0
0 


 n

0
0 ,
In

A= 0
 



 


0
0
0
I

n
 

	   
 B0 B1  BNu 
0 0  0 B=,    
 0 0  0 
C = C0 C1  C N y 0  0  .
We start with the state controllability in this subsection, where acommonly used criterion is first given below.
Lemma 1. The necessary and sufficient condition for LTI systemz(k + 1) = Hz(k) + Gv(k) (k ≥ 0) with the input v ∈ m and the statez ∈n , to be completely state controllable, isrank([H n −1G, H n − 2G,..., HG, G]) = n.

where u ∈  m , x ∈ n , y ∈  q are the input, the state and the output
of system (1), respectively; the time delays 1 ≤ Nu, Ny ≤ Nx < ∞ are
integer constants and the time index k ≥ –Nx; among each groupof the parameter matrices A1 ,…, AN x ∈  n × n , B1 ,¼, BNu Î n ´ m andC1 ,¼, CN y Î  q ´ n , there is at least one nonzero matrix.

 (k ) + BU
 (k) X (k + 1) = AX
	   
 (k)
y(k) = CX
2.1. State Controllability Analysis with
Data-based Criterion

Because the state controllability of system (2) is equivalent to thatof system (1), based on Lemma 1, we can determine this characteristic by checking the rank of n( N x +1) −1 B ,…, AB , B ]. (4)		 WC = [ AIn order to obtain the augmented controllability matrix WC , thetraditional analyzing approaches need to identify the unknown A0,
A1, ..., ANx, B0, B1, ..., BNu in system (1) first, and construct matrices
2B
 , ..., A, B as in (3), then calculate AB , A n( N x +1) −1 B
 , and build W
ACwith these identification and calculation results. It is natural thatthe identification errors are often introduced in the above process,and the calculation workload is really heavy especially when thestate dimension n and the time delay Nx in state are large.
Due to this situation, to avoid the identification errors and to
reduce the workload effectively, it is intended to find a more simple
and direct method to construct WC . This goal can be achieved byutilizing the data-based analysis method given below.
This method starts with a set of n(Nx + 1) + 1 tests on system (2).
In the first n(Nx + 1) tests, we design the input sequences as U{p}(0),..., U{p}(n(Nx + 1) – 1) (p = 1, 2, ..., n(Nx + 1)), which can make thefollowing n(Nx + 1) vectors
U { p} (0)


{ p}
1
(
)
U
	 
 (1 ≤ p ≤ n(N x + 1)) V { p} = U { p} (n(N + 1) − 1)
x
linearly independent. Therefore, the matrix(3) ∈  q × n( N x +1) and ∈ n( N x +1) × n( N x +1) , B ∈ n( N x +1) × m( Nu +1) , C
where AI n ∈ n × n is the identity matrix.
Obviously, the state controllability as well as the state observabilityof both systems (1) and (2), are equivalent.

	 V = [V {1} ,…,V {n( N x +1)} ] ∈  mn( Nu +1)( N x +1) × n( N x +1) (5)

has the rank of n(Nx + 1). On the other hand, the last test is doneon system (2), where the input sequence is set as U{n(Nx + 1) + 1}(k) ≡ 0.
Next, let all the corresponding states of system (2) have the same
initial value:
X {1} (0) = X {2} (0) =  = X {n( N x +1) +1} (0) = X0.
Measure and record the values of X{1} (n(Nx + 1)), ..., X{n(Nx + 1)}(n(Nx + 1)) and X{n(Nx + 1) + 1}(n(Nx + 1)). With the above particular augmented states, we present the following data-based state
controllability criterion.

Z. Wang and R. Wang / Journal of Robotics, Networking and Artificial Life 6(4) 235–239

Theorem 1. Define
Z X{ } = X { } (n(N x + 1)) − Xp{ n ( N x + 1) + 1}p{1}

{ n ( N x + 1)}{2}

Z X = [ Z X , Z X , …, Z X].

(n(N x + 1)), (1 ≤ p ≤ n(N x + 1)),  (6)Then, both system (1) and system (2) are completely state controllable, if and only if rank (ZX) = n(Nx + 1).
Proof. With the particular n(Nx + 1) + 1 testing input sequences,
the same initial state X0, and the measured augmented state data,
we should obtain
 n( N x +1) X , X { p} (n(N + 1))
X {n( N x +1) +1} (n(N x + 1)) = A
0
x
=
  

n( N x +1) -1k { } (n(N + 1) - 1 - k)å A BUk =0n( N x +1)

+A
Z X{ p} =n( N x +1) -1

p

x

 (7)

X0 ,

k

 { } (n(N + 1) - 1 - k),å A BUpn( N x +1) −1

{1}

{1 }

x

U








(n( N + 1) − 1){ n ( N x + 1 )}(9)

Similar to the state controllability case, it is necessary to find a newcriterion of the state observability only based on the measured data,
if the identification errors and a large amount of calculations are to
be avoided.
We still start with some tests on system (2), where n(Nx + 1) linearly
independent initial states X{1}(0), X{2}(0), ..., X{n(Nx + 1)} (0) are particularly set first. Such that the following matrix{n( N +1)}
{1}
{2 }	    F X = [ X (0), X (0),¼, X x (0)] 

(10)

(11)

At the same time, sample the corresponding output data y{1}(k),y{2} (k), ..., y{n(Nx + 1)}(k) at the sampling instants k = 0, 1, ..., n(Nx + 1)
– 1, with which we let , B
]
B,…, ABU (0 )× U (n(N + 1) − 1)
= WCV .

T , A
T C
 T ,…,( A n( N x +1) -1 )T C
 T ]T . 	   WOB = [C    U {1} (k ) = U {2} (k ) =  = U {n( N x +1)} (k ) º 0 

for all 1 ≤ p ≤ n(Nx + 1). Based on Equations (4)–(7),Z X = [AAs aforementioned, the state observability of system (1) and system(2) are equivalent. According to Lemma 2, the rank of the following matrix can be used as the criterion of the state observability ofsystem (2):has rank of n(Nx + 1). Then, set corresponding inputs as

x

k =0

237

(0 )



U

{ n ( N x + 1 )}x

Since V has the rank of n(Nx + 1) because of the linear independence of V{1}, V{2}, ..., V{n(Nx + 1)}, thenrank(WC ) + rank(V ) − n(N x + 1) ≤ rank(WCV )≤ min{rank(WC ), rank(V )}  (8)= rank(WC )y { p] (0)


{ p}
(
)
y
1
 (1 ≤ p ≤ n(N x + 1)),
Yp = 	    y { p} (n(N + 1) − 1)
x
Y = [Y1 ,Y2 ,…,Yn( N x +1) ] ∈  qn( N x +1) × n( N x +1)

(12)

  

Next, a data-based criterion on the state observability will be presented below.

Consequently, rank(Z X ) = rank(WCV ) = rank(WC ).

Theorem 2. Suppose that the initial states of system (2) can be set
as in (10). Then, system (1) and system (2) are both completely state
observable, if and only if

By Lemma 1, the sufficient and necessary condition for system (2)to be completely state controllable is rank (ZX) = n(Nx + 1), and thesame holds true for system (1). q.e.d.

where Y is defined in (12).

2.2. State Observability Analysis with
Data-based Criterion

Proof. Since the proving process is similar to that of Theorem 1,
this proof is omitted here for brevity. q.e.d.

After the introduction of data-based state controllability criterion,
this subsection will propose the method to determine the state
observability of both system (1) and system (2), only by utilizing themeasured data. Here, a relevant lemma is given first to make a start.
Lemma 2. The LTI systemz (k + 1) = Hz (k ) + Gv(k )(k ≥ 0), 		  w(k ) = Fz (k )mwith the input v ∈ , the state z ∈ and the output w ∈ , iscompletely state observable, if and only ifnrank([F T , H T F T ,…,(H n −1 )T F T ]T ) = n.

rank(Y ) = n(N x + 1),

q

Similar to the previously introduced data-based state controllability criterion, the data-based criterion of the state observability
introduced here has some advantages:(i) it can determine the state observability of discrete-timedelayed LTI systems either with (at least one of A1, ...,
ANx, C1, ..., CNy in system (1) is nonzero) or without timedelay;(ii) it does not need to identify these parameter matrices, such thatthe identification errors can be avoided;x + 1) -1 n ( N x + 1) -1  ,n( N...,CA(iii) it does not need to calculate CA
for the construction of the observability matrix WOB in Equation (9),which greatly reduces the workload.

Z. Wang and R. Wang / Journal of Robotics, Networking and Artificial Life 6(4) 235–239

238

3. COMPUTATIONAL COMPLEXITYANALYSIS
This section will take the data-based state controllability analysis
algorithm given in Theorem 1 as a representative, to show the meritof lower computational complexity. In order to make the discussing process simpler, this paper considers both a summation anda multiplication of two matrix elements as one operation. For thediscrete-time delayed LTI systems described by (1), when we mul , there should be n(N + 1)
 i (1 ≤ i ≤ n(N + 1) – 2) with A
tiply Axx
multiplication operations and n(Nx + 1) − 1 summation operations
for calculating each element, so that there are totally n2(Nx + 1)2
 i +1 , since the number of(2nNx + 2n – 1) operations to obtain A
 i +1 is n2 (N + 1)2.
elements in A

Our data-based methods provided a new perspective for the analysis of the time-delay systems, but there are still some shortcomings
to be improved. For the current stage, they can only analyze discrete-time LTI systems, but cannot deal with continuous-time, nonlinear or time-variant systems. Besides, if the system is stochastic,
we have no other way but to identify all the parameter matrices andthe developed data-based analysis methods are no longer feasible.
In the future researches, we would like to study these problems andhope to find appropriate methods to overcome these difficulties.

CONFLICTS OF INTEREST
The authors declare they have no conflicts interest.

x , there are n(N + 1)
 k (1 ≤ k ≤ n(N + 1) – 1) multiplies B
When Axx
multiplications and n(Nx + 1) − 1 summations for each calculated
element, while there are mn(Nu + 1) (Nx + 1) elements in the obtained
k B
 . Thus, there are mn(N + 1) (N + 1) (2nN + 2n – 1) operations.
A
u

x

x

And it is needed to do (nNx + n – 1) such kind of matrix multipli n( N x +1) −1 B ,…, AB , B ], it iscations in total. For computing W = [ ACneeded to performmn(N u + 1)(N x + 1)(2nN x + 2n - 1)(nN x + n - 1)+n( N x +1) - 2

å in (N + 1) (2nN + 2n - 1)
i =1

2

x

2

x



operations. In summary, the computational complexity of the traditional model-based state controllability analysis algorithms is(n5 N x 5 ).
As a comparison, by observing (5) and (6), our data-based state
controllability analysis algorithm performs only n(Nx + 1) + 1
groups of experiments as well as n2 (Nx + 1)2 element subtractions.
So that the developed data-based analyzing method only has thecomputational complexity of (n2 N x 2 ).
By extension, we may conclude that our data-based analysis methods have lower computational complexity than the traditional analysis algorithms.

4. CONCLUSION AND FUTURE WORKIn this paper, some new data-based methods are proposed to analyze and determine the state controllability and the state observability of a general class of discrete-time delayed LTI systems.
These data-based analyzing methods conducted some specificinput tests, using the measured historical and current data to
directly construct the system matrix, the state controllability matrix
and the state observability matrix, which are utilized to determine
the corresponding characteristics. Generally speaking, they havemerits in two major aspects over the traditional model-based analysis approaches: the less identification work and the lower computational complexity.

REFERENCES[1] L. Zou, Z. Wang, H. Gao, X. Liu, State estimation for discrete-time
dynamical networks with time-varying delays and stochastic disturbances under the round-robin protocol, IEEE Trans. NeuralNetw. Learn. Syst. 28 (2017), 1139–1151.
[2] S. Yin, H. Yang, H. Gao, J. Qiu, O. Kaynak, An adaptive NN-based
approach for fault-tolerant control of nonlinear time-varying
delay systems with unmodeled dynamics, IEEE Trans. NeuralNetw. Learn. Syst. 28 (2017), 1902–1913.
[3] Y. Chen, D.W.C. Ho, J. Lü, Z. Lin, Convergence rate for discrete-time multiagent systems with time-varying delays and general coupling coefficients, IEEE Trans. Neural Netw. Learn. Syst.
27 (2016), 178–189.
[4] H. Guerboukha, A. Markov, H. Qu, M. Skorobogatiy, Time
resolved dynamic measurements at THz frequencies using arotary optical delay line, IEEE Trans. Terahertz Sci. Technol.
5 (2015), 564–572.
[5] Z. Meng, W. Xia, K.H. Johansson, S. Hirche, Stability of positive switched linear systems: weak excitation and robustness totime-varying delay, IEEE Trans. Autom. Control 62 (2017), 399–405.
[6] L. Zhang, K. Zhang, Controllability and observability of boolean
control networks with time-variant delays in states, IEEE Trans.
Neural Netw. Learn. Syst. 24 (2013), 1478–1484.
[7] L. Zou, Z. Wang, H. Gao, X. Liu, Event-triggered state estimation
for complex networks with mixed time delays via sampled data
information: the continuous-time case, IEEE Trans. Cybern.
45 (2015), 2804–2815.
[8] J. Shen, J. Lam, On the decay rate of discrete-time linear delaysystems with cone invariance, IEEE Trans. Automat. Control
62 (2017), 3442–3447.
[9] G. Scarciotti, A. Astolfi, Model reduction of neutral linear and nonlinear time-invariant time-delay systems with discrete and distributed delays, IEEE Trans. Automat. Control 61 (2016), 1438–1451.
[10] W. Yan, D. Tang, Y. Lin, A data-driven soft sensor modelingmethod based on deep learning and its application, IEEE Trans.
Ind. Electron. 64 (2017), 4237–4245.
[11] S. Yin, Y. Jiang, Y. Tian, O. Kaynak, A data-driven fuzzy information
granulation approach for freight volume forecasting, IEEE Trans.
Ind. Electron. 64 (2017), 1447–1456.

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239

AUTHORS INTRODUCTION
Dr. Zhuo Wang

Mr. Ruigang Wang
He received the B.E. degree in automation from Beihang University, China,
in 2006, and the PhD degree in electrical and computer engineering from the
University of lllinois at Chicago, USA, in
2013. He is currently a Professor in the
School of Instrumentation Science andOptoelectronics Engineering, Beihang
University.

He received the B.E. degree from the
Hefei University of Technology, China, in2018. He is currently working toward the
PhD degree at Beihang University, Beijing,
China.