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Observer Design for Linear Time-Delay Systems
Md. Aminul Haq1 , Ibrahim Beklan Kucukdemiral2
1Department of Control and Automation Engineering, Yildiz Technical University, Istanbul, Turkey
ocean_blue04@yahoo.com
2Department of Control and Automation Engineering, Yildiz Technical University, Istanbul, Turkey
beklan@yildiz.edu.tr
Abstract
The paper present the design of an observer for a general
class of systems with delays in states. A state space model of
observer with delays is proposed. The novelty of the study is
to include the state derivatives in the design. The stability of
the observer is proved by Lyapunov approach. Linear
Matrix Inequality (LMI) approach is used in the analysis of
the problem. Numerical examples are studied to see the
validity of the approach.
1. Introduction
Time-delay system (TDS) is a system having delays in its states,
inputs or outputs and occurs in many natural and engineering
events. Time-delay is commonly encountered in chemical
processes, biological systems, long transmission line of
pneumatic, hydraulic system, steel rolling mills, space missions
and usually a very common source of instability.Time-delay
systems actually belongs to the class of functional differential
equation (FDE),which has infinite dimensions making it more
complex. Both to analysis and designs taking into the
consideration of deviating arguments or differential difference
term is necessary for engineers to make models to behave like
more to real process.
state observer problem has been studied for many years in
order to improve satisfactory observer action under exogenous
disturbance. Several methods for observer design for time
delay system such as Lyapunov–Krasovskii approach, algebraic
Riccati Equation approach, Fattouh et el method has been
discussed in [1].The delay-dependent design methods which are
suitable for systems with time delay being of known size have
been proposed in [2,3] based on Riccati Equation approach.A
Lyapunov approach to design an observer for discrete time
system in terms of Riccati Equation has been proposed in [4].
Based on Lyapunov stability theory, the design of observer with
internal delay and unknown input is formulated in terms of
Linear Matrix Inequality (LMI) in [5] and authors developed a
delay independent matrix representation. A reset observer
framework has been proposed in [6] for linear time-delay
systems to improve settling time and overshoot performance. It
is well known that filtering problem is dual to the
control one for linear systems without uncertainty.
Controller (observer) design procedure has been proposed and
developed in [7, 8, 9, 10, 11], which could be adopted for
observer design too because of duality.
Fig.1. Block diagram of proposed observer
To design an observer for TDSS we use simple Luenberger
approach, but we introduced here two feedback line instead of
one. The first feedback line contains a proportional gain matrix
and second feedback line has a gain matrix (given) followed by
a differentiator block. So here we are considering not only the
difference between real states and estimator states or error
signals but also the rate of change of error signals. Taking into
consideration both error and rate of change of error data would
make the observer more reliable than simple Luenberger type
one.
2. Problem Formulation
Consider the following linear time-delay system,
x(t)=Ax(t)+Adx(t-h)+Bu(t)+Nw(t)
y=Cx(t)
x(t+) = () , 0 (1)
where ,
xRn : The State vector.
w(t)Rq : The exogenous disturbance input which belongs to
L2[0,∞).
y(t)Rp : The output vector.
A, Ad , B , N , C .
The above matrices are constant and known system matrices.
h 0 :a positive scalar denoting the time delay.
(.) : a continuously differentiable function on [-,0]
representing the initial condition.
848
3. Main Results
Let us formulate an observer dynamics as follows,
(t)=F(t)+G(t-h)+Hu(t)+Mw(t)+L1(y(t)-(t))+L2((t)-(t))
(t)=C(t) (2)
Where,
(t)Rn : The estimator state vector
L1,L2 :The constant observer gain matrix to be selected
appropriately.
(t)Rp :The estimated output vector.
F, G , H , M , C .
Thorem: Observer in form of (2) can be constructed if there
exists matrices P=PT>0, R1=>0, R2=>0 and X for a given
noise attenuation level , satisfying the following LMI
0
00 00
0
000
00
200
AXC 0 0 0 2 0
00000
< 0 (3)
where =(ATPZ-1 -CTXTZ-1+AdTPZ-1+ Z-TPA- Z-TXC+ Z-TPAd +CTC)
3.1. Proof:
Subtracting equation (2) from equation (1) we get,
x(t) - x(t) = Ax(t)+Adx(t-h)+Bu(t)+Nw(t) - Fx(t)-Gx(t-h)
-Hu(t)-Mw(t)-L1(y(t)-y(t))-L2(y(t)-y(t))
e(t) =Ax(t)+Adx(t-h)+Bu(t)+Nw(t) - Fx(t)-Gx(t-h)-Hu(t)
-Mw(t)-L1(y(t)-y(t))-L2(y(t)-y(t))+ Fx(t)+Gx(t-h)
-Fx(t)-Gx(t-h)
e(t) = (A -F)x(t)+(Ad–G)x(t-h)+(B-H)u(t)+(N-M)w(t)
+F(x(t) x(t))+G(x(t-h)- x(t-h))-L1(Cx(t)- Cx(t))
-L2(Cx(t)-Cx(t))
e(t) = (A -F)x(t)+(Ad–G)x(t-h)+(B-H)u(t)+(N-M)w(t)
+Fe(t)+Ge(t-h)-L1C(x(t)- x(t)) -L2C(x(t)-x(t))
e(t)+L2Ce(t)= (A -F)x(t)+(Ad–G)x(t-h)+(B-H)u(t)+(N-M)w(t)
+Fe(t)+Ge(t-h)-L1Ce(t)
(I+ L2C)e(t) = (A -F)x(t)+(Ad–G)x(t-h)+(B-H)u(t)
+(N-M)w(t)+(F-L1C)e(t)+Ge(t-h)
e(t) =( I+ L2C)-1[(A -F)x(t)+(Ad–G)x(t-h)+(B-H)u(t)
+(N-M)w(t)+(F-L1C)e(t)+Ge(t-h)]
e(t) = Z[(A -F)x(t)+(Ad–G)x(t-h)+(B-H)u(t)
+(N-M)w(t)+(F-L1C)e(t)+Ge(t-h)]
Where , Z=( I+ L2C)-1
Here, we will choose L2 arbitrarily and calculate the gain L1
accordingly.
Obviously,
e(t)→0 as t→∞ if the following conditions are satisfied:
(1) The system is stable and observable.
(2) ( I+ L2C) is invertible.
(3) A=F,
Ad=G,
B=H,
N=M, then the error dynamics reduces to,
(t) =( I+ L2C)-1[(F-L1C)e(t)+Ge(t-h)]
(t) = Z[(F-L1C)e(t)+Ge(t-h)] (4)
We will utilize following the Leibniz rule
Lemma 1: A(t-h) = A(t)-αd
α
We will also use the following lemma in our proof
Lemma 2: -2UTV UTRU+VTRV
Then we have the error dynamics as follows,
(t) = Z[(F-L1C)e(t)+Ge(t-h)]
Using Leibniz rule given in Lemma 1, we can write,
e(t-h) = e(t)-αd
α
= e(t)-ZFCeα Geα h d
α
(t) =Z(F-L1C)e(t)
+ZG{e(t)-ZFCeα Geα h d
α}
The error dynamics (4) is now transformed into the following
equation.
849
(t) = Z[(F-L1C)+G]e(t)
FCetαGetαh d
α
(5)
(t)0 as t∞ means error in (5) tends to ‘0’ as time evolves.
Delay-dependent approach: Consider the following Lyapunov–
Krasovskii functional
V(e,t)= e(t)T Z-TPZ-1 e(t)
+ eθTFCTR F Ceθ
θ.
+ eθTGTG eθ
θ.
(e,t) = (t)T Z-TPZ-1e(t)+ e(t)TZ-TPZ-1 (t)
+h e(t)T FCT F C e(t)
-eθTFCTR F Ceθ
θ
+ h e(t)TGTR2Ge(t)
-eθTGTRGeθdθ
= e(t)T[(F-L1C)+G]TZT Z-TPZ-1e(t)
+ e(t)T Z-TPZ-1Z[(F-L1C)+G] e(t)
-2e(t)TZ-TPZ-1ZGZFCθθhθ
+ h e(t)T(F-L1C)TR1(F-L1C)e(t)
-eθTFCFCeθd
θ
+ h e(t)TGTR2Ge(t)- eθTGTRGeθdθ
e(t)T[(F-L1C)+G]TPZ-1e(t)+ e(t)T Z-TP[(F-L1C)+G] e(t)
+etTTPGZR
d θ
+θFCFCθθ
+etTPGZ
d θ
+θhθhθ
+ h e(t)T(F-L1C)TR1(F-L1C)e(t) + h e(t)TGTR2Ge(t)
tθFCFCetθd
θ
etθhTGRGetθhdθ
e(t)T[FTPZ-1 -CTL1TPZ-1+GTPZ-1+ Z-TPF- Z-TPL1C+ Z-TPG]e(t)
+ h e(t)TZ-TPGZR1-1ZTGTPZ-1e(t)+ h e(t)TZ-TPGZR2-1ZTGTPZ-1e(t)
+ h e(t)T(F-L1C)TR1(F-L1C)e(t)+ h e(t)TGTR2Ge(t)
Now applying Schur complement we get,
e(t)T
FC
000
0
00
0 0 0
FC 0 0 0
e(t)
Here, = (FTPZ-1 -CTL1TPZ-1+GTPZ-1+ Z-TPF- Z-TPL1C+ Z-TPG)
If above matrix is less than 0,then (e,t) is negative so e(t)0 as
t∞.
FC
000
0
00
0 0 0
FC000
< 0 (6)
Pre and post multiplying (6) by diag {I,I,I,I,P} and replacing
–hR2-1 by h(R2-2I) as we know –R2-1<(R2-2I) ,we get
FC
00 0
0
00
0 0 0
FC000
< 0
We can replace –hPR1-1P by h (R1-2P) as, h(R1-P) (R1-P) >0
hR1-hP-hP+ hPR1-1P>0 then –hPR1-1P<h(R1-2P)
hFC
00 0
0
00
0 0 2 0
FC 0 0 0 2
< 0
850
Now let PL1=X and defining the matrix (right hand side of the equation ) as Σ,
Σ =
00 0
0
00
0 0 2 0
FXC 0 0 0 2
< 0
Here =(FTPZ-1 -CTXTZ-1+GTPZ-1+ Z-TPF- Z-TXC+ Z-TPG)
For observer,it has to satisfy the following equation,
e,t
+z(t)Tz(t)-2w(t)Tw(t)]dt<0 (7)
If e,t+z(t)Tz(t)-2w(t)Tw(t)<0 then (7) will be true also.
e(t)T Σ e(t)+ e(t)TCTCe(t))-2w(t)Tw(t)<0 [here, z=y(t)- (t)= Cx(t)- Cx(t)= Ce(t)] (8)
if (t)=[e(t) ; w(t)] and applying Schur complement to (8) ,
(t)T
0
00 00
0
000
0 0 200
FXC 0 0 0 2 0
0000 0
(t)< 0
where =(FTPZ-1 -CTXTZ-1+GTPZ-1+ Z-TPF- Z-TXC+ Z-TPG +CTC)
0
00 00
0
000
0 0 200
FXC 0 0 0 2 0
0000 0
< 0
According to necessary condition, replacing F=A and G=Ad we get the following final LMI
0
00 00
0
000
00
200
AXC 0 0 0 2 0
00000
< 0
where =(ATPZ-1 -CTXTZ-1+AdTPZ-1+ Z-TPA- Z-TXC+ Z-TPAd +CTC)
Solving the LMI for P and X we can get L1=P-1X.
851
4. Numerical Example
In this section, we will demonstrate the theory developed in this
paper by means of simple examples. Here to solve problem we
have used Matlab software,Yalmip Optimization Toolbox and
Sedumi solver. Consider the linear continuous time-delay
system (9) and (10) with parameters given by
Example (1): A=2 0.5
0.5 3 Ad=0.5 0
00.5
(9)
C= [ 1 0 ] L2=[0.5 0.4]T (chosen) (10)
Where 0 < h0.77 is an unknown positive scalar.
The purpose is to design observer using equation (3)
according to the block diagram. The transfer function from
exogenous disturbances to error state outputs meets the
prescribed norm upper bound constraint |||| 0.8
Here, we take the value =0.3
Solving the LMI, we get P= 1.0374 0.6950
0.6950 0.9064
R1=0.3822 0.2531
0.2531 0.2936 R
2=0.9766 0.3675
0.3675 0.8124
X=0.0918
0.2156 L1 = 0.5095
0.6285
Here in the example plant initial state is [5;-2] and estimator
initial state is [0;0].
Fig.2. Trajectories of state (t) and (t)
Fig.3. Trajectories of state (t) and (t)
Example (2):
A=2.5 1.2
1.25 4.3 Ad=2.3 1.5
1.4 3.2 (11)
C= [ 0 1 ] L2=[0.5 0.4]T (chosen) (12)
Where 0 < h0.32 is an unknown positive scalar.
The transfer function from exogenous disturbances to error state
outputs meets the prescribed norm upper bound
constraint |||| 0.8
Here, we chose =0.3.
Solving the LMI, we can get the values as follows,
P= 0.9438 0.0715
0.0715 0.9943 R1= 0.9319 0.2139
0.2139 0.9992
R2= 0.8275 0.0966
0.0966 1.1380 X=2.5462
0.2880
L1 =2.7347
0.4862
852
Here in the example plant initial state is [4;-3] and estimator
initial state is [0;0].
Fig.4. Trajectories of state (t) and (t)
Fig.5. Trajectories of state (t) and (t)
From the simulation result shown on graphs,we can see that the
trajectories of plant states and observer states converge within
few seconds,which is pretty good performance by the observer
designed using the method developed in this paper.
5. Conclusions
In this paper an observer design procedure for systems with
delays in states has been studied. An appropriate gain matrix for
observer is calculated while the gain matrix for differentiator
block has been predetermined. Necessary and sufficient
conditions have also been derived. Numerical examples
provided here described the effectiveness of this method.
Future research will be focused on upgrading this observer
dynamics into a delay free observer design.
6. References
[1] Oliver Sename, New trends in design of observers for time-
delay systems, Kybernetika, Vol 37 no. 4(2001) 427-458.
[2] Zidong Wang, Biao Huang, H. Unbehauaen ,Robust
observer design of linear time-delay systems with
parametric uncertainty,Systems & Control Letters 42 (2001)
[3] A.Fattouh, O.Sename, J.M. Dion, observer design for
time-delay systems,in proc 37th IEEE Confer. on Decision
& Control (Tampa,Florida,USA),(1998) 4545-4546.
[4] M. Boutayeb, Obsevers design for linear time-delay
systems, Systems & Control Letters 44 (2001) 103-109.
[5] Yan-Ming,Fu,Guang-Ren Duan,Shen-Min Song,Design of
Unknown Input Observer for Linear Time-delay
Systems,InternationalJournal of Control,Automationn and
Systems,vol.2 no.4 (2004) 530-535.
[6] Guanglei Zhao,Jingcheng Wang,Reset obsevers for linear
time-varying systems: Delay dependent Approach,Journal
of the Franklin Institute(2014).
[7] Fatma Yildiz Tascikaraoglu,Levent Ucun,Ibrahim B
Kucukdemiral,Receding horizon control of time delay
systems,Transaction of the institute of measurement and
Control (2014) 1-10.
[8] M.N. Alpaslan Parklakci,Robust delay-free observer-based
controller design for uncertain neutral time delay,
Systems,Systems Science (2006).
[9] M.N. Alpaslan Parklakci,Robust delay-free observer-based
controller design for uncertain neutral time delay,
Systems,Systems Science (2006).
[10] Mai Viet Thuan, Vu Ngoc Phat, Hieu Trinh,Observer-based
controller design of time-delay systems with an interval
time-varying delay, Int. J. Appl. Math. Comput. Sci., 2012,
Vol. 22, No. 4, 921–927.
[11] Cao, Y.-Y., Sun, Y.-X., & Lam, J. ,Delay dependent robust
control for uncertain systems with time varying delays,
IEE Proceedings: Control Theory and Applications, (1998).
143, 338–344.
[12] Vladimir B. Kolmanovskii , On the Liapunov-Krasovskii
functionals for stability analysis of linear delay systems,
International Journal of Control,(1999) 72:4, 374-384.
[13] Ping Li,James Lam,Positive state-bounding observer for
positive interval continuous-time systems with time delay,
Int. J. Robust. Nonlinear Control 2012; 22:1244–1257.
[14] Jean-Pierre Richard, Time-delay systems: an overview of
some recent advances and open problems, Automatica 39
(2003) 1667-1694.
[15] Sheng yuan Xu & James Lam, A survey of linear matrix
inequality techniques in stability analysis of delay systems,
International Journal of Systems Science, (2008),
39:12,1095-1113.
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