Int. J. Contemp. Math. Sci., Vol. 1, 2006, no. 4, 163 - 176
SPECTRUM ASSIGNMENT AND STABILITY PROPERTIES
OF SYSTEMS
WITH POINT TIME LAGS
M. DE LA SEN
Instituto de Investigacion y Desarrollo de Procesos. IIDP
Department of Electricity and Electronics
Facultad de Ciencias, Universidad del País Vasco
Leioa ( Bizkaia). Aptdo. 644 de Bilbao
48080- Bilbao, SPAIN
Keywords:
α-Stability, Spectrum assignment ,
Stable matrices, Time-delay
systems.
Mathematics Subject Classification: 93B15, 93B52, 94B50.
164
M. de la Sen
Abstract. The asymptotic stability with a prescribed degree of time delayed systems
subject to multiple bounded point delays has received important attention in the last
years (see, for instance [1-5]). It is basically proved that the α-stability locally in the
delays (i.e. all the eigenvalues have prefixed strictly negative real parts located in
Re s ≤ - α < 0) may be tested for a set of admissible delays including possible zero
delays either through a set of Lyapunov' s matrix inequalities or, equivalently, by
checking that an identical number of matrices related to the delayed dynamics are
all stability matrices. The result may be easily extended to check the ε-asymptotic
stability independent of the delays, i.e. for all the delays having any values, the
eigenvalues are stable and located in Re s ≤ ε →0 - , [1] , [5]. The above referred
number of stable matrices to be tested is 2 r for a set of distinct r point delays and
includes all possible cases of alternate signs for summations for all the matrices of
delayed dynamics, [5]. The manuscript is completed with a study for prescribed
closed-loop spectrum- assignment (or "pole -placement") under output-feedback.
1.
Stability Results
Consider the time-invariant time-delay system
x& ( t ) =A 0 x ( t ) +
∑ A x (t − h )
r
k
(1)
k
k =1
where x ∈ R n is the state vector, h k ≥ 0 ( k = 1 , 2 , ... , r ) are r point constant
delays. The initial conditions of (1) are given by any absolutely continuous function
ϕ : [− h, 0 ] → R n ,with possibly finite discontinuities on a subset of zero measure of
[ - h , 0 ], where h = max (h k ). The system (1) is said to be α− asymptotically stable
1≤ k ≤ r
locally in the delays (α−ASLD) for all h k ∈ [0, h k ] for some α ∈ R + , h k > 0
⎛
r
(k=1,2,..., r) (i.e. all the roots of Det ⎜⎜ sI − A 0 − ∑ A k e
⎝
k =1
−h
k
s
⎞
⎟ =0
⎟
⎠
Systems with point time lags
165
lie in Re s ≤ -α < 0 ). The following result was proved in [5]:
Result 1: The system (1) is α-ASLD if there is a real n-matrix P =P T > 0 such that
the following Lyapunov' s matrix inequality holds
PA 0 + A
T
0 P
⎡ r
⎢
+
±e h k α PA k + A Tk P
⎢⎣ k = 1
(
∑
[
⎤
⎥ ≤ −2αP
⎥⎦ m
)
(2)
)
for m=1,2,..., 2 r ρ ∈ − ρ 0 ,0 , where [ ±] m denotes all possible 2 r cases of
alternating sign . The system (1) is asymptotically stable independent of the delays
if
⎡ r
PA 0 + A T0 P + ⎢ ± PA k + A Tk P
⎢⎣ k = 1
∑ (
⎤
⎥ <0
⎥⎦ m
)

Result 1 was proved based on the subsequent technical fact also proved in [5].
Fact 1: For any set of symmetric constant n-matrices {T k ; k=0,1,..., r}, the
r
inequality T 0 + ∑ η k T k ≤ −2αP
k =1
holds for some α ∈ R + , a real n-matrix P =P T > 0 and all real η k ∈[− η k M ,η k M
]
(k=1,2,..., r) iff it holds at the 2 r vertices of the hyper-rectangle:
{
H : = η= (η 1 , η 2 , ..., η r ) T ∈ R r

η k ∈ [−η k M , η k M ]; k = 1, 2, ..., r
}
The following technical lemma will be then used to prove the main results.
Lemma 1. A set of r real matrices A i ( i=1,2,..., r ) are all stability matrices with
stability abscissas ( - α i ) < 0 iff the set of r Lyapunov' s matrix inequalities
166
M. de la Sen
A Ti P + PA i ≤ −2αP ; i = 1, 2, ..., r
( α i ) ⎤⎥ provided that Min (α i ) is sufficiently
hold for any real constant α ∈⎛⎜ 0,Min
≤ ≤
⎝
1 i r
⎦
1≤ i ≤ r
large.
Proof: Assume that all the A i are stability matrices with stability abscissas (-α i ) <
0 then (A i +α i I ) are stable matrices and
(A
T
i
)
(
)
+ α iI P + P A i + α iI < 0
(3)
for i=1,2,..., r , all real n-matrix P = P T > 0 then
A Ti P + PA i ≤ −2α i P ≤ −2αP < 0
(4)
for i=1,2,..., r and all α>0 as specified. To prove the converse, consider three cases
for (3) to fail and then proceed by contradiction
(
)
(
)
T
Case a: Assume A i + α i I P + P A i + α i I > 0 for at least one i ∈ {1,2,..., r}.
Thus, (A i +α i I ) is unstable from Lyapunov' s instability theorem and one of its
eigenvalues has positive real part. Thus, the stability abscissa of A i exceeds ( - α i )
which leads to a contradiction.
(
)
(
)
T
Case b: Assume A i + α i I P + P A i + α i I = 0 for at least one i ∈ {1,2,..., r}.
Consider the linear and time-invariant system x& (t) = (A i +α i I ) x(t) for any
bounded x (0)= x 0 ∈R n with a Lyapunov'- Razumikhin function candidate V(x) =
x T (t) P x (t) , some real matrix P = P T > 0. It turns out that
& ( x ) ≡0⇒V ( x ) =V ( 0 ) <∞
V
Systems with point time lags
2
2
⇒ x (t )
167
( )
≥ λ min P −1 V(0) > 0
for any x 0 ≠ 0 Thus, it turns out that x(t) cannot tend to zero as t →∞ if x 0 ≠ 0 and
then
(A i +α i I ) is not a stability matrix and thus the stability abscissa of A i is less
than (- α i ) what again leads to a contradiction.
Case c: Assume that
(A
T
i
)
(
)
+ α iI P + P A i + α iI
is indefinite. Decompose
A k = A i + ΔA k i for some 1 ≤ k ≤ r and all 1 ≤ i ≠ k ≤ r . Thus for any positive
definite symmetric square n-matrix Q, there exists a positive definite matrix P such
that
( A Tk P + α k I) P + P (A k + α k I)=
( A Ti P + α i I) P + P (A i + α i I)
(
)
+ ΔA Tk i P + PΔA k i + 2 (α k − α i )P
(
)
= − Q + ΔA Tk i P + PΔA k i + 2 (α k − α i )P
with P =
∫
∞
0
e
(A
T
i
) Qe (A
+α i I τ
Note that λ max ( P ) ≤
T
i
) dτ satisfying Q = - (A T P +α I)P − P A +α I
( i i)
i
i
+α i I τ
K2
2 (ρ i + α
i
)
for some real constant K ≥ 1 with (−ρ i )< 0 being
the stability abscissa of A i . Thus,
( A Tk P + α k I) P + P (A k + α k I)< 0 if λ
what is guaranteed if
min
(
( Q )>2 Δ A
ki
2
(
+ α
k
−αi
) )λ
max
(P)
168
1≥
M. de la Sen
λ min (Q )
λ max (Q )
≥
K2 ⎡
α i − α k + Max ΔA k i
1≤ i ≤r
ρ i + α i ⎢⎣
(
2
)⎤⎥⎦
which always holds for sufficiently large ρ i (i.e. for sufficiently stable A i ) for
given Δ A k i
2
; i = 1 , 2, … , r and , thus, for sufficiently large Min (α i ).

1≤ i ≤ r
The main result of this section is now stated:
Theorem 1. The subsequent items hold:
(i) The system (1) is α '-SLD for all h k ∈ [0, hk ] if the 2 r-matrices
⎡ r
⎤
A m = A 0 + ⎢ ±e h k α A k ⎥ m + αI are all stability matrices for m=1, 2 , ..., 2 r and
⎢⎣ k = 1
⎥⎦
∑
some real α >0.
⎡
r
⎤
⎢⎣ k = 1
⎥⎦
(ii) Assume that A mρ = A 0 + ⎢ ∑ ±ρ k A k ⎥ m + αI are all stability matrices for ρ T=
(ρ 1, ρ 2, ., ρ r ) with ρ k ≥1 ; k=1,2,.., r, and some real α > 0. Then, the system (1)
is α - ASLD for all delays h k ∈ [0, h k ] with h k =
1
ln ρ k for all k= 1, 2,..., r.
α
(iii) Assume that
⎡ r
⎤
A mρ = A 0 + ⎢ ±ρ k A k ⎥ m + αI
⎢⎣ k = 1
⎥⎦
∑
are all stability matrices for any real constants ρ k > 0 ; k=1,2,.., r, and some real α >
0. Thus, all the systems of the form
x& ( t ) = A 0 x ( t ) +
r
ρk
∑±β
k =1
(
A kx t − h
k
)
(5)
k
are α-SLD for any prefixed set of real scalars β k >1 ( k=1,2,..., r) and all delays
Systems with point time lags
169
⎡ 1
⎤
h k ∈ ⎢ 0, ln β k ⎥. If β k = ρ k > 1 for all k=1,2,..., r then (1) is α− ASLD.
⎦
⎣ α
⎡ r
'
A
=
A
+
±ρ k A
⎢
(iv) If
0
m
⎢⎣ k = 1
∑
k
⎤
⎥
⎥⎦
m
are all stability matrices with ρ k =1 for all
m=1,2, ..., 2 r ; k=1 ,2 ,.., r then all the delay systems (5) are asymptotically stable
independent of the delays for any set β k > 1 (k=1,2,..., r). If β k = ρ k = 1 (k=1,2,..., r)
then (1) is asymptotically stable independent of delays.
Proof: (i) Consider 2 r Lyapunov' s matrix equations
A Tm P m + P m A m = − Q m = − Q mT < 0
for m= 1, 2 , ..., 2 r . Since A m are stability matrices then the unique solutions P m
to the Lyapunov' s equation are P m= P Tm =
∫
∞
0
T
e A m τ Q m e A m τ dτ ; m=1,2, ..., 2 r .
On the other hand, (−Q m ) ≤ −2αP m (or, equivalently, Q m ≥ 2 α P m ) for all
m=1,2,..., 2 r if
0<α≤
[
[
]
]
Min λ min (Q m )
1 1≤ i ≤ 2 r
2 Max λ max (P m )
r
1≤ i ≤ 2
(6)
Note also that for any symmetric positive definite matrices P m and all P ≥ P m
T
A Tm P + P A m = A m
P m + P mA m
T
+A m
ΔP m + ΔP m A m
(
)
≤ −Q Tm + A Tm ΔP m + ΔP m A m < 0
with ΔP m = P − P m ; m=1,2,..., 2 r satisfying
(7)
170
(A
M. de la Sen
T
m ΔP m
)
+ ΔP m A m = −Q m < 0
since A m is a stability matrix (see the proof of Lemma 1). Thus, for any P
≥ P m (m=1,2, ..., 2 r)
A Tm P + P A m ≤ −2αP
holds so that P is non unique and thus the system (1) is α−SLD from Result 1 and all
the set of delays h k (k=1,2,..., r) satisfying ρ k = e
h
k
α
≥e
h
k
α
≥ 1 , since h k ≥ 0 ;
k=1, 2, .., 2 r equivalently is guaranteed if
0 ≤ τ k ≤ τ kM ≤ 2
1≤ m ≤ 2 r
Max [λ
max
(P m ) ]
Min r [λ
1≤ m ≤ 2
max
(Q m ) ]
ln ρ k
and the proof of (i) has been completed.
(ii) It follows directly from (i) with ρ k = h k α for k=1,2, ..., r and α >0.
(iii) Consider the non unique factorizations ρ k= γ k β k, for any sequences {γ k ;
k=1,2,..., r}, { β k ; k=1,2,..., r}, being only subject to the constraints β k>1 for all
k=1,2,..., r. Thus, it follows from (ii) for ρ k = γ k β k ≡ h k α (k=1,2,..., r) that, if the
matrices
⎡ r
⎛ ρ
⎞⎤
A m β = A 0 + ⎢ ± β k ⎜⎜ ± k A k ⎟⎟ ⎥
⎢⎣ k = 1
⎝ βk
⎠ ⎥⎦
∑
m
+ αI
are all stability matrices for m=1, 2,... ,2 r, then all the systems (5) are α-ASLD.
(iv) It follows from (iii) since the asymptotic stability of the systems (5) for all
possible values of the delays from zero to infinity, with ρ k = γ k β k ≡ h k α =1,
Systems with point time lags
α → 0 + and h
k
(
=o α
−1
171
) → ∞ for all k=1,2,..., r , is guaranteed by testing the 2 r
given n-matrices for all m=1, 2,..., 2 r.

Remarks: 1. Note that the property of α-asymptotic stability locally in the delays
of the system (1), which specifies admissibility domains [0, h
k
] for the delays for
k=1,2,..., r, may be tested by checking if 2 r matrices
⎡ r
A m = A 0 +⎢ ± e h k α A
⎢⎣ k = 1
∑
k
⎤
⎥
⎥⎦
m
+ αI
are all stability matrices for m=1, 2..., 2 r. This property is equivalent to all the
matrices
A 'm
⎡ r
⎤
h kα
⎢
⎥
= A0 +
±e
Ak
⎢k = 1
⎥
⎣
⎦
∑
m
to be stability matrices with stability abscissas of at least (- α) < 0.
2. Note that A m being a stability matrix implies that
⎡ r
⎤
h kα
ˆ
⎢
⎥
=
A
+
±e
A
A
m0
0
k m + α 0I
⎢⎣ k = 1
⎥⎦
∑
are all stable for α 0 ∈ (0 , α ] and h
'
k
=
⎡
h kα
for k=1,2,.., r. Thus, the system (1) is
α0
also α 0 − ASLD for all delays h k ∈ ⎢ 0,
⎣
h kα⎤
⎥ or k=1,2,.., r. As a result, if the
α0 ⎦
system (1) is α − ASLD for h k ∈ [0, h k ] then it is also α 0 − ASLD for all
⎡ h α⎤
α 0 ∈ (0 , α ] and delays h k ∈ ⎢ 0, k ⎥for k=1, 2, ..., r.
⎣ α0 ⎦
172
M. de la Sen
3. Note that all A m being stability matrices for any α≥0 implies that A 0
⎛
⎞
r
and ⎜⎜ ∑ A k ⎟⎟ are both stability matrices. In other words, the delayed dynamics-free
⎝ k= 0
⎠
⎛ r
⎜
&
&
auxiliary system z ( t ) = A 0 z(t) and the delay-free system z ( t ) = ⎜ A k
⎝ k= 0
∑
⎞
⎟ z(t) are
⎟
⎠
both globally exponentially stable. Both conditions are known to be necessary for
stability independent of the delays (see [2-3] and they are obtained in this context as

a direct consequence of Theorem 1.
2. Output-feedback stabilization with prescribed pole-placement
Now system (1) is considered as forced and with a measurable output:
x& ( t ) = A 0 x ( t ) +
r
∑ A x ( t − ih ) + bu ( t )
i
i =1
y( t ) = c T x ( t ) + du( t )
(7)
where h ≥ 0 is now the base delay and h i = ih (i= 1, r ). The change of notation and
specification of delays related to a base one h is made by description simplicity
reasons. The transfer function of (1) is defined in a standard way by using Laplace
transforms of the output and input as P(s) = Y(s) / U(s)
r
⎛
⎞
B(s)
T⎜
P (s) =
= c sI−
A i e − ih s ⎟ −1 b + d
⎜
⎟
A(s)
i= 0
⎝
⎠
∑
where A(s) and B(s) are quasi-polynomials defined by:
] ϕ ≡ 0 leading to:
(8)
Systems with point time lags
r
⎛
A( s) = det ⎜ sI −
A i e − ih s
⎜
i= 0
⎝
∑
⎞
⎟=
⎟
⎠
173
q
∑
A i (s) e − ih s =
n
∑
A * (e − h s )s i =
i
i= 0
i= 0
q
n
∑ ∑a
ik s
k
e − ih s
i= 0 k = 0
(9.a)
r
⎛
⎞
− ihs ⎟
⎜
B(s) = c Adj sI −
Aie
b + d A(s)
⎜
⎟
i= 0
⎝
⎠
∑
T
q
=
,
∑
B i (s)e
i= 0
− ih s
q
m
=
∑
*
B (e
i
i= 0
with q and
q
,
− hs
i
)s =
,
m
∑ ∑b
ik s
k
e − ih s
(9.b)
i= 0 k = 0
being integers satisfying q ,≤ q ≤ rn . For exposition simplicity, it is
assumed with no loss in generality that q , =q. Otherwise, (9.b) still applies by
zeroing the necessary polynomials B ( . ) .
m
B i ( s) =
i
∑
n
b ik s k : A i ( s) =
i= 0
i
∑a
ik s
k
i= 0
are polynomials of respective degrees m i and n i ( i = 0, q ) with m i ≤ m 0= m ≤ n
and n i ≤ n 0= n for i = 0, q with m =n if and only if d ≠ 0 in (1.b), i.e., the plant is
not
strictly
proper
plant
and
m ≤ n −1
,
otherwise.
Note
that
⎛
⎞
n = n 0 ≥ Max ⎜ m , Max ( n i , m i ) ⎟ since the transfer function (8)-(9) obtained from
1≤ i ≤ q
⎝
⎠
(1) is realizable. Alternative polynomials B *i (e − h s ) and A *i (e − h s ) are defined in
the same way leading to an equivalent description of (7). The following results is the
main one of this section
Theorem 2 (Spectrum assignment and closed-loop stability). Assume that the
transfer function (8)-(9) has no pole-zero cancellation and that the property is not
lost under zero delayed dynamics. Thus, the following items hold:
174
M. de la Sen
(i) There exist infinitely many polynomial pairs ( R i (s ) ,S i (s) ) which satisfy the
υ nested diophantine equations of polynomials:
ˆ (s) A 0 (s) R i (s) + B 0 (s) S i (s) = A
mi
i
∑ (A
l
(s) R
i −l
(s) + B l (s) S i − l (s) )
l =1
for i= 0, υ − 1
(10)
υ ≥ 1.
for any integer
(R i (s) ,S i (s) )
Furthermore, if n m 0 ≥ 2 n −1 then there is at least a solution
; i = 0 , υ − 1 which satisfies the following degree constraints:
n ′0 = n m 0 − n ; m ′i ( s) = n −1 for i= 0, υ − 1 ; and
⎛
Max n ′i , m −1 = Max ⎜ n m i , Max n k + n ′i − k
1≤ k ≤ i
⎝
(
)
(
) ⎟⎠⎞ − n
(ii) If Assumptions 1-2 hold and n m 0 ≥ 2 n then it is possible to build infinitely
many proper rational functions of the form
υ−1
∑ [S
Q(s) =
(s) − Λ 0 (s) A 0 (s) ] e − l h s
l
l=0
υ −1
∑ [R
(11)
l
(s) + Λ 0 (s) B 0 (s) ] e
−lhs
l=k
withexisting
polynomial
( R i (s) − Λ 0 (s) A 0 (s), S i (s) + Λ 0 (s) B 0 (s) )
( R i (s) , S i (s) ) are also solutions to (10) where
solution
pairs
verifying (10) provided that
Λ 0 (s) = λ 0 is any real scalar (i.e.
any polynomial of zero degree) if n>m and Λ 0 (s) is any arbitrary polynomial of
arbitrary degree otherwise. If n m 0 = 2 n −1 then (10) is realizable for Λ 0 (s) = 0 if
n>m and with arbitrary Λ 0 (s) if n=m.
Systems with point time lags
175
(iii) Assume that the controller transfer function K υ (s ) =
S(s)
takes the
R (s)
subsequent specific form :
υ−1
∑ [S
l
(s) − Λ 0 ( s) A 0 (s) ] e − l h s
l=0
υ −1
∑ [R
l
(12)
(s) + Λ 0 (s) B 0 (s) ] e
−lhs
+ R υ (s)
l=k
where
(R i (s) ,S i (s) )
are pairs of polynomials being any solutions
to
(10) ; i= 0, υ − 1 , Λ 0 ( s) is chosen according to item (ii) , S υ (s ) is an arbitrary
polynomial of degree not exceeding (n-1), and
R υ ( s) =
N υ (s)
1
=
D υ (s) A(s)
e
−( l-υ) h s
υ+ q
× ∑
l= υ
Min (l , q )
⎡ˆ
⎛ Min (l , q )
⎞⎤
⎢ A m l ( s )−⎜ i = Max ( ∑υ.l − υ + 1) A i ( s )R l −i ( s )+i =Max∑( ν.l − υ ) B i ( s )S l −i ( s ) ⎟ ⎥
⎝
⎠⎦
⎣
(13)
Then, the closed- loop spectrum satisfies:
ˆ * (s) =
A(s) R ( s) + B( s) S(s) = A
m
υ −1
∑ Aˆ
m i (s)
e − ih s
i= 0
with the closed-loop being stable with poles in Aˆ *m (s) = 0 and a closed-loop stable
υ −1
cancellation of the plant poles provided that Aˆ *m ( s) = ∑ A m l ( s) e − l h s is a strictly
l= 0
Hurwitzian quasi-polynomial satisfying n m 0 ≥ 2 n −1.
(iv) If the suited spectrum satisfies n m 0 ≥ 2 n − 1 and the controller is simplified to
have a transfer function K *υ (s) = Q(s) (i.e. R υ (s) and S υ (s) are zeroed) then the
closed-loop spectrum is set to the zeros of
176
{Aˆ
*
m
M. de la Sen
⎧⎪ υ + q
(s) + ⎨
⎪⎩ l = υ
}
∑
Min ( l ,q
)
∑ [A
i = Max ( 0, l − υ
i
)
( s )R
l −i
⎫⎪
( s )+B i (s )S l − i ( s ) e − l h s ⎬
⎪⎭
]
without cancellations of the plant poles.

Acknowledgements
The authors are very grateful to UPV / EHU by its partial support of this work
through Project 9 / UPV / EHU 00I06.I06-1526 /2003 and to the Spanish Ministry
of Education and Technology by its support through Project DPI 2003-0164.
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[3] M. De la Sen: 'Allocation of poles of delayed systems retarded to those
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373-374.
[4] S.I. Niculescu, Delay Effects on Stability. A Rrobust Control Approach. Lecture
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Editors, Springer-Verlag, Berlin, 2001.
[5] B. Xu : 'Stability criteria for linear time-invariant systems with multiple delays',
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Received: July 20, 2005