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Control and Stability Analyzing of the Time-Delay Systems with Time-Varying Delays

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Systems with delays are a specific group of dynamic systems. The fact that some delays can be short and some can be long makes system analysis and control design much more complex. In this paper we present a proposal to solve the problem of stability and control design of continuous time systems with time-varying delays. Based on the Lyapunov-Krasovskii functional the stability condition is derived using the linear matrix inequality (LMI) techniques, and convex optimization principle to find LMI variables which simultaneously satisfies the limitations given by the theory of Lyapunov-Krasovskii functionals. Obtained solution is the feasible convex LMI problem for the static controller design. Finally the design method is demonstrated using a system model example.
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70 Acta Electrotechnica et Informatica, Vol. 11, No. 3, 2011, 70–74, DOI: 10.2478/v10198-011-0032-9
CONTROL AND STABILITY ANALYZING OF THE TIME-DELAY SYSTEMS WITH
TIME-VARYING DELAYS
Daniel GONTKOVI ˇ
C, R´
obert F ´
ONOD
Department of Cybernetics and Artificial Intelligence, Faculty of Electrical Engineering and Informatics, Technical University of
Koˇ
sice, Letn´
a 9, 042 00 Koˇ
sice, Slovak Republic, tel.: +421 55 625 2749, e-mail: daniel.gontkovic@tuke.sk,
robert.fonod@student.tuke.sk
ABSTRACT
Systems with delays are a specific group of dynamic systems. The fact that some delays can be short and some can be long
makes system analysis and control design much more complex. In this paper we present a proposal to solve the problem of stability
and control design of continuous time systems with time-varying delays. Based on the Lyapunov-Krasovskii functional the stability
condition is derived using the linear matrix inequality (LMI) techniques, and convex optimization principle to find LMI variables which
simultaneously satisfies the limitations given by the theory of Lyapunov-Krasovskii functionals. Obtained solution is the feasible convex
LMI problem for the static controller design. Finally the design method is demonstrated using a system model example.
Keywords: Lyapunov-Krasovskii functional, LMI, Lyapunov function, Time-delay systems, Time-varying delays, Schur complement
1. INTRODUCTION
Systems with delays frequently appear in engineering,
where time delays are the property of a physical system.
Whenever material, information or energy is physically
transmitted from one place to another, there is a delay asso-
ciated with the transmission. The value of the delay is de-
termined by the distance and the transmission speed, where
some delays are short, some are very long. Since delays
might lead to a system destabilization, additive conditions,
based on the assumption that in the system there is informa-
tion on delays states, are taking into account in design task
formulation. Such systems are called systems with delayed
state variables. However, the presence of long delays makes
system analysis and control design substantially much more
complex.
The study of functional differential equations started
long before 1900, but the mathematical formulation of the
problems were developed in the 20th century. Thus, the
notion of a functional differential equation was introduced
by Myshkis [13] in 1949. A further progress have been
made to study systems with delays in the past 50 years
( [14], [2] and the references therein); main ideas concern-
ing the work in this area can be found in e.g. Krasovskii [8]
(time-domain approach, extension of the Lyapunov second
method to functional differential equations). Burton [1] (re-
finements of the Lyapunov-Krasovskii theory and periodic
solutions) During the last decades, we have witnessed sig-
nificant development in the control of time-delay systems.
Some recent comprehensive introductions to the problem
are in Greeki [5], Marshall et al. [12], Kolmanovskii and
Myshkis [7] and Hale and Lunel [6].
In this paper the design task of the stabilizing con-
troller for the closed-loop system is transmuted into LMI
framework and solved. There are two ways to use the
second method of Lyapunov for time-delay systems. The
first is based on the theory of Lyapunov-Krasovskii func-
tionals and the other is based on the theory of Lyapunov-
Razumikhin functions. The paper extends the control
design and stability analyzing methods based on theory
of Lyapunov-Krasovskii functionals with limited speed of
change of time delays on the interval β=h0,1).
2. SYSTEM MODEL
Equations of linear dynamic systems with time delay
are as follows
˙
q(t) = Aq(t) + Adq(tτ(t)) + Bu(t)(1)
y(t) = Cq(t)(2)
0τ(t)τm<, 0 <˙
τ(t)β<1, τm,βIR
with the initial condition
q(ϑ) = ϕ(ϑ),ϑ∈ h−τm,0i(3)
where τ(t)>0 is the time-varying delay of state, q(t)IR n
is a vector of the state, u(t)IR rand y(t)IR mare vectors
of the input and output variables, AIR n×nis the nominal
system matrix, AdIR n×n,BIR n×r,CIR m×n.
Problem of the interest is to design stable close-loop
system with the linear memoryless state feedback controller
of the form
u(t) = Kq(t),(4)
where t0 and KIR r×nis the controller gain matrix.
3. BASIC PRELIMINARIES
Working with the linear matrix inequalities the con-
cept of symmetric positive definite matrix or positive semi-
definite matrix is principally used. The necessary and suf-
ficient conditions for positive definiteness of a symmetric
matrix Pare:
there is a regular matrix Q, such as that
P=QTQ,
all eigenvalues of the matrix Pare positive,
all main subdeterminants of the matrix Pare positive
p11 >0,
p11 p12
p21 p22
>0, . . . , det(P)>0.
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Acta Electrotechnica et Informatica, Vol. 11, No. 3, 2011 71
Working with the norm, in the next k · k denotes the stan-
dard Euclidean norm, and k·kthe Hnorm.
Proposition 3.1. (Schur Complement)
If Q=QT>0,R=RT,Sare real matrices of appro-
priate dimensions, then the next inequalities are equivalent
Q S
STR>0QSR1ST0
0 R >0
m
QSR1ST>0,R>0.
(5)
Proof. (e.g. see [9, 10])
Let the linear matrix inequalities takes form
Q S
STR>0 (6)
than using Gauss elimination it yields
ISR1
0 I  Q S
STR I 0
R1STI=
=QSR1ST0
0 R
(7)
and it is evident that this transformation does not change
positivity (6), and so (7) implies (5).
Definition 3.1. (Lyapunov function [11, 15])
For some τ(t)>0the equilibrium 0of system (1), (2)
is:
uniformly stable if there exists a positive defi-
nite continuous functional v(qt(θ)) whose derivative
˙v(qt(θ)) is negative semi-definite functional;
uniformly asymptotically stable if there exists a pos-
itive definite upper-bounded continuous functional
v(qt(θ)), whose derivative ˙v(qt(θ)) is negative defi-
nite functional;
uniformly exponentially stable if there exist a posi-
tive definite continuous functional v(qt(θ)) and pos-
itive real constants α,β,γ,δsuch that αkqt(θ)k ≤
v(qt(θ)) βkqt(θ)k,˙v(qt(θ)) ≤ −γkqt(θ)k, and
|vqt(θ1)) v(qt(θ2))|δkqt(θ1)qt(θ2)k.
Proposition 3.2. (Lyapunov-Krasovskii Ineqality [3,4])
The linear time delay system of form (1), (2) is sta-
ble with quadratic performance kC(sIA)1Bk=λfor
˙
τ(t)βif exists a symmetric positive definite matrix Pand
Qand a scalar variable γ>0such that
ATP+PA +Q PAdPB CT
∗ −(1β)Q 0 0
∗ −γIr0
∗ −Im
<0
P=PT>0,Q=QT>0,γ>0
(8)
Hereafter, *denotes the symmetric item in a symmetric ma-
trix.
The problem is reduced to solving LMI i.e. to find
matrices Pand Q,PIR n×n,QIR n×n, and a scalar vari-
able γ>0, which simultaneously satisfies the boundary (8).
Proof. Defining Lyapunov-Krasovskii functional candidate
as follows
v(q(t),t) = qT(t)Pq(t) +
t
R
tτ(t)
qT(r)Qq(r)dr+
+
t
R
0
(yT(r)y(r)γuT(r)u(r))dr>0
P>0,Q>0,γ>0
(9)
Using Laplace transform, let
˜
y(s)= G(s)˜
u(s),kG(s)k=kC(sIA)1Bk=λ
(10)
when ˜
y(s),˜
u(s)stand for Laplace transform of mdimen-
sional output vector and rdimensional input vector signal,
respectively and G(s)is the transfer function matrix.
Then Parcevall’s theorem implies
"
R
0
yT(t)y(t)dt#1
2
"
R
0
uT(t)u(t)dt#1
2
=ky(t)k
ku(t)kλ.(11)
Since Hnorm can be interpreted as a maximum gain in
any direction and at any frequency then there exists a scalar
γ λ 2>0 such that Lyapunov-Krasovskkii functional candi-
date be positive definite.
The result of the functional derivative of v(q(t),t)is
˙v(q(t),t) =
=˙
qT(t)Pq(t) + qT(t)P˙
q(t) + qT(t)Qq (t)
qT(tτ(t))Qq (tτ(t)) (1˙
τ(t))+
+yT(t)y(t)γuT(t)u(t)<0
(12)
Inserting (1) into (12) gives
(Aq(t) + Adq(tτ(t)) + Bu(t))TPq(t)+
+qT(t)P(Aq(t) + Adq(tτ(t)) + Bq(t))+
+qT(t)Qq(t)(1˙
τ(t))qT(tτ(t))Qq(tτ(t))+
+qT(t)CTCq(t)γuT(t)u(t)<0
(13)
where
yT(t)y(t) = qT(t)CTCq(t).(14)
Creating a composite vector q(t)
qT(t) = qT(t)qT(tτ(t)) uT(t)(15)
the inequality (13) can be rewritten in a form
qT(t)
11 PAdPB
∗ −(1β)Q 0
∗ −γIr
q(t)(16)
where
11 =ATP+PA +Q+CTC.
Using Schur complement property it is possible to write
ATP+PA +Q PAdPB CT
∗ −(1β)Q 0 0
∗ −γIr0
∗ −Im
<0.(17)
It is obvious that (17) implies (8).
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72 Control and Stability Analyzing of the Time-Delay Systems with Time-Varying Delays
4. PARAMETER DESIGN
Theorem 4.1. For system (1), (2) the necessary and suf-
ficient condition for the stable nominal control (4) with
quadratic performance kC(sIA)1Bk=λis that there
exist positive definite symmetric matrices X,Z, matrix Y
and a scalar variable γ>0, such that the following LMIs
are satisfied
11 X AdZ B XCT
∗ −Z 0 0 0
∗ −(1β)Z 0 0
∗ ∗ γIr0
∗ ∗ ∗ −Im
<0
X>0,Z>0,γ>0
(18)
where 11 =XAT+AX BY YBT.
The control law gain matrix is then given as
K=YX1.(19)
Proof. The linear state feedback control law, defined in (4)
gives rise to the closed-loop system as follows
˙
q(t) = (ABK)q(t) + Adq(tτ(t)).(20)
Subsituting in (8) gives for P=PT>0 and Q=QT>0
11 PAdPB CT
∗ −(1β)Q 0 0
∗ −γIr0
∗ −Im
<0 (21)
where
11 = (ABK)TP+P(ABK) + Q.
Defining the congruence transform matrix
T1=T1T=diagP1III(22)
and premultiplying (21) from the left and right side by T1
gives
11 AdB P1CT
∗−(1β)Q 0 0
∗ ∗ γIr0
∗ −Im
<0 (23)
where
11 =P1(ABK)T+ (ABK)P1+P1QP1.
Substituting P1=Xand Y=KP1then (23) can be
rewriten such that
11 AdB XCT
∗ −(1β)Q 0 0
∗ −γIr0
∗ −Im
<0 (24)
where
11 =XAT+AX BY YBT+XQX.
Using the Schur complement (24) can be modified as
11 X AdB XCT
∗ −Q10 0 0
∗ −(1β)Q 0 0
∗ ∗ γIr0
∗ −Im
<0 (25)
where 11 =XAT+AX BY YBT.
Defining the next congruence transform matrix
T2=T2T=diagIIQ1IrIm(26)
and then premultiplying (25) from the right and left side by
(26) modify it in the form
11 X AdQ1B XCT
∗ −Q10 0 0
∗ ∗ 33 0 0
∗ −γIr0
∗ −Im
<0 (27)
where 33 =(1β)Q1. The substitution Q1=Zleads
to inequality
11 X AdZ B XCT
∗ −Z 0 0 0
∗ −(1β)Z 0 0
∗ ∗ γIr0
∗ ∗ ∗ −Im
<0.(28)
It is obvious that (28) implies (18).
5. ILLUSTRATIVE EXAMPLE
To demonstrate the algorithm properties it was assumed
that system is given by (1), (2), where
A=
2.6 0.0 0.8
1.2 0.2 0.0
0.0 0.53.0
,C=
1 1
2 1
1 0
,
Ad=
0.00 0.02 0.00
0.00 0.00 1.00
0.02 0.00 0.00
,B=
4 0
7 1
1 0
.
Solving (18) with respect to the LMI matrix variables
X>0, Z>0, Yand a scalar variable γ>0 using SeDuMi
package for Matlab, given task was feasible with
X=
1.3892 0.5363 0.0136
0.5363 0.9210 0.2723
0.0136 0.2723 1.1760
,
Z=
3.9383 0.0842 0.0013
0.0842 3.8726 0.0286
0.0013 0.0286 3.0627
,
Y=0.1671 2.0496 0.2288
0.1352 0.4042 0.5026 ,γ=4.6235,
where P1=Xand Q1=Z. The controller gain ma-
trix K1can be obtained from the relation (19) for β=0.1,
where
K1=1.0341 2.9774 0.5068
0.1885 0.7253 0.5975 ,
ρ(Au) = ρ(ABK1) = {−11.0867,2.0799,2.7745}
It is evident that the eigenvalues spectrum ρ(Au)of the
closed control loop is stable.
Setting β=0.5 the gain matrix K2takes the form
K2=1.1799 3.3692 0.5428
0.1997 0.8179 0.6076 ,
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Acta Electrotechnica et Informatica, Vol. 11, No. 3, 2011 73
ρ(Au) = ρ(ABK2) = {−12.4348,2.0624,2.7705}
and the spectrum of the eigenvalues ρ(Au)of the closed
control loop is stable, too.
Solving with respect to the controller gain matrix K3
and β=0.99 results
K3=2.3169 5.3545 1.2816
0.3283 1.1451 0.8732 .
Even for this case is the eigenvalues spectrum ρ(Au)of the
closed control loop is stable, since
eig(Au) = {−18.8092,1.4764,4.2613}.
5.1. Simulation
Time-delay linear dynamic system (1), (2) is raised up
to the steady state starting with the nonzero initial condi-
tions [1,0.5,3]T.
In Figure 1 to 3 is shown the time response of the
closed-loop system with respect to the stability region con-
ditioned by the delay time changes interval 0 <˙
τ(t)1.
Simulations were made with the prescribed values β=0.1,
β=0.5, β=0.99 from above defined interval. Note, the
closed-loop system is t∈ h−τ(t),0)unstable, because one’s
eigenvalue of Adpositive. Closed-loop system reaches
the steady state values of output variables approximately
in time instants less then 11s. It can be easily seen that
the criterion proposed in this paper to design the memory-
less feedback controller gives acceptable solutions. More-
over, the limited range of the delay time changes interval
0<˙
τ(t)1 is given by the necessary negativeness of the
Lyapunov-Krasovskii functional derivative.
6. CONCLUDING REMARKS
Time delays can appear as a part of the dynamics in
many technological processes. Presence of delays in the
system is making the analysis and control design more com-
plex because it has a negative impact on the stability of sys-
tem. Tendency to instability generally grows with respect
to the size of the time delay. Therefore, the proposal of en-
suring stability and control design systems with delays is a
very interesting topic in the scientific community.
The method uses the standard LMI numerical optimiza-
tion procedures to manipulate the system feedback gain
matrix as the direct design variable. Since it is necessary
to propose an asymptotically stable closed-loop system, to
solve the control design problem in the sense of delay-
independent criteria, we typically use such modifications
of Lyapunov-Krasovskii functional as shown above. An ex-
ample, illustrating the design method certifies the presented
principle is applicable.
Fig. 1 Response of the closed-loop system for β=0.1
Fig. 2 Response of the closed-loop system for β=0.5
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74 Control and Stability Analyzing of the Time-Delay Systems with Time-Varying Delays
Fig. 3 Response of the closed-loop system for β=0.99
ACKNOWLEDGEMENT
The work presented in this paper was supported by
VEGA, Grant Agency of Ministry of Education and
Academy of Science of Slovak Republic under Grant No.
1/0256/11. This support is very gratefully acknowledged.
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[3] FILASOV ´
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[4] FILASOV ´
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Received March 4, 2011, accepted July 8, 2011
BIOGRAPHIES
Daniel Gontkoviˇ
cwas born in 1984 in Poprad, Slovakia.
He graduated in automation in 2009 from the Faculty of
Electrical Engineering and Informatics, Technical Univer-
sity of Koˇ
sice, Slovakia. Since 2009 he is a PhD. student
with the Department of Cybernetics and Artificial Intelli-
gence, FEI TU Koˇ
sice. His scientific research is focusing
on time-delay systems and system control.
R´
obert F´
onod was born in 1987 in Veˇ
lk´
y Krt´
ıˇ
s, Slovakia.
He received BSc. degree in Cybernetics in 2009 from the
Faculty of Electrical Engineering and Informatics, Techni-
cal University of Koˇ
sice. Since 2009 he is a MSc. student
in the same study branch with the Department of Cybernet-
ics and Artificial Intelligence, FEI TU Koˇ
sice. His interest
is in dynamic system fault diagnostics.
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Article
New frequency-domain ℓ2-stability criteria are derived for nonlinear discrete-time MIMO systems, having a linear time-invariant block with the transfer function Γ(z), in negative feedback with an aperiodic matrix gain A(k), k=0,1,2,…, and a linear combination of a vector of certain classes of (generalised) first-and-third-quadrant non-monotone nonlinearities ϕ_(⋅_), having arguments with constant and time-varying delays, but without restrictions on their slopes. The framework does not employ Lyapunov–Krasovskii functionals involving linear matrix inequalities (LMIs) or their equivalent. The new stability criteria seem to be the most general for nonlinear and time variying time-delay systems, and have the following structure: (1) positive definiteness of the real part (as evaluated on |z|=1) of the product of Γ(z) and an algebraic sum of general causal and anticausal matrix multiplier functions of z. (2) An upper bound on the L1-norm of the inverse Fourier transform of the multiplier function, the L1-norm being weighted by certain novel, quantitative characteristic parameters (CPs) of the nonlinearities ϕ_(⋅_) without the asumptions of monotonicity, slope restrictions and the like. And (3) constraints on certain global averages of the generalised eigenvalues of (A(k+1),A(k)), k=1,2,…, that are expressed in terms of (i) CPs of the nonlinearities, (ii) their coefficients, and, in general, (iii) time-delays in their arguments, a trade-off among all the three being possible. These global averages imply a restriction on the rate of variation of A(k) in a new sense. The literature results turn out to be special cases of the results of the present paper. Examples illustrate the stability theorems.
Article
Full-text available
Exponential stability of networked control systems with network-induced random delays In this paper, the problem of exponential stability for the standard form of the state control, realized in a networked control system structure, is studied. To deal with the problem of stability analysis of the event-time-driven modes in the networked control systems the delayed-dependent exponential stability conditions are reformulated and proven. Based on the delay-time dependent Lyapunov-Krasovskii functional, exponential stability criteria are derived. These criteria are expressed as a set of linear matrix inequalities and their structure can be modified to use the bilinear inequality techniques.
Book
Preface. 1. Models. 2. General Theory. 3. Stability of Retarded Differential Equations. 4. Stability of Neutral Type Functional Differential Equations. 5. Stability of Stochastic Functional Differential Equations. 6. Problems of Control for Deterministic FEDs. 7. Optimal Control of Stochastic Delay Systems. 8. State Estimates of Stochastic Systems with Delay. Bibliography. Index.
Chapter
In this chapter, some recent stability and robust stability results on linear time-delay systems are outlined. The goal of this guided tour is to give (without entering the details) a wide overview of the state of the art of the techniques encountered in time-delay system stability problems. In particular, two specific stability problems with respect to delay (delay-independent and respectively delay-dependent) are analyzed and some references where the reader can find more details and proofs are pointed out. The references list is not intended to give a complete literature survey, but rather to be a source for a more complete bibliography. In order to simplify the presentation several examples have been considered.