Stability Analysis for Time-Delay Systems using Rekasius’s Substitution

and Sum of Squares

Christian Ebenbauer and Frank Allgöwer

Abstract— In this paper, a new delay-dependent stability anal- inequalities are very popular and successful. This approach
ysis for time-delay linear time-invariant (TDLTI) systems is allows to circumvent the problems induced by the infinite
derived. In contrast to many recent approaches, which often dimensional character of time-delay systems and allows to
utilize Lyapunov-Krasovskii functionals and linear matrix in-
equalities, an alternative approach is proposed in this paper. use semidefinite programming algorithms. In particular, due
The proposed stability analysis is formulated in the frequency to semidefinite programming, there has been a substantial
domain and investigates the characteristic equation by using the increase of work in this area of research, e.g., [8], [12], [11],
socalled Rekasius substitution and recently established sum of [3], [7], [25], [23], [4], [1].
squares techniques from computational semialgebraic geometry.
The advantages of the proposed approach are that the stability
analysis is often less conservative than many approaches based In this paper, a new stability analysis for TDLTI systems
on Lyapunov-Krasovskii functionals, as demonstrated on a well- based on a frequency domain formulation is proposed. The
known benchmark example, and that the stability analysis is objective is to obtain a lower bound for the maximal tol-
very flexible with respect to additional analysis objectives. erated time-delay of a TDLTI system without jeopardiz-
ing stability. The proposed stability analysis exploits two
I. INTRODUCTION concepts, namely the socalled Rekasius substitution, i.e.,
Stability analysis of time-delays systems has been system- a bilinear transformation, and recently established sum of
atically studied in systems and control theory since many squares techniques from computational semialgebraic geom-
decades [13], [21], [6], [5]. There are at least two main etry. By using the Rekasius substitution, the transcendental
reasons for this continuously strong interest. The first reason characteristic equation can be transformed into an algebraic
is simply due to the fact that time-delay systems appear in equation, i.e., into a polynomial equation. Based on this
a wide range of applications. For example, the range of ap- polynomial equation, a new lower bound for the maximal
plications spreads from traditional engineering applications, tolerated time-delay is derived in this paper. Furthermore,
like conveyor belts, to modern engineering applications, like it is shown that sum of squares techniques can be used
computational delays in real-time computation. Especially to compute this lower bound efficiently. In simple terms,
interesting applications are currently challenging areas like sum of squares techniques allow to check if a polynomial in
communication and information theory, where time-delays several variables is positive semidefinite. Analogous to linear
play an important role, for example when studying stability matrix inequalities, sum of squares techniques are based on
of controlled networks, internet, or guaranteed high-speed efficient and reliable semidefinite programming algorithms.
transmission protocols. The second reason for this contin- Hence, the proposed stability analysis approach is based on
uously strong interest stems from the fact that there are the same efficient numerical tools as Lyapunov-Krasovskii
still many unsolved problems in the stability analysis of functional approaches. One advantage of the new proposed
time-delay systems. Even for time-delay linear time-invariant stability analysis is that one obtains often better lower bounds
(TDLTI) systems, a complete stability analysis is still a than for example by using Lyapunov-Krasovskii functional
difficult problem. One of the main reasons is that time-delay approaches, which is demonstrated on a benchmark example
systems are infinite dimensional systems. This is especially in this paper. Another advantage is the gained flexibility due
difficult, if the stability analysis should be performed in a sum of squares techniques in order to include additional
computationally efficient way. The objective of this paper is analysis objectives.
to establish such a computationally efficient delay-dependent
stability analysis for TDLTI systems. In general, there are The remainder of the paper is structured as follows: In
two basic methodologies to perform a stability analysis for Section II, some preliminary facts on TDLTI systems and
TDLTI systems, namely frequency domain methods and time on the socalled Rekasius substitution are summarized. The
domain methods. Time domain methods are often used in the basic idea of the new proposed stability analysis is motivated
literature to perform a delay-dependent stability analysis. In and explained in Section III. Section IV provides the math-
particular Lyapunov-Krasovskii functionals and linear matrix ematical justification of the proposed stability analysis and
an efficient algorithm based on sum of squares techniques.
Christian Ebenbauer is with the Laboratory for Information and Numerical examples are given in Section V and the results
Decision Systems, Massachusetts Institute of Technology, USA, are compared with methods based on Lyapunov-Krasovskii
ebenbauer@mit.edu. Frank Allgöwer is with the Institute for
Systems Theory and Automatic Control, University of Stuttgart, Germany, functionals. Finally, a summary and an outlook is given in
allgower@ist.uni-stuttgart.de. Section VI.
 II. PRELIMINARIES questions concerning the crossing properties of the roots
In this paper, the following problem is studied. Given the of the characteristic equation with respect to the imaginary
class of time-delay linear time-invariant (TDLTI) systems, axis. However, in comparison to Lyapunov-Krasovskii
i.e., functionals, rather less attention has been paid to this
important transformation. Summarizing, the bottom line of
ẋ(t) = A0 x(t) + A1 x(t − τ ), (1) Rekasius’s substitution [19] is the fact that
where x ∈ Rn is the state, τ ≥ 0 is the time delay, and
A0 , A1 ∈ Rn×n are given constant matrices. Further, it is Lemma 1: s = iω (ω ≥ 0) is a root of the characteristic
assumed that equation

det iωI − A0 − A1 e−τ iω = 0

A0 + A1 is Hurwitz, (2) (7)

i.e., system (1) is asymptotically stable for τ = 0. The for a τ ≥ 0 if and only if
objective is to find the maximal time delay τmax such that  
1 − T iω
the system (1) is asymptotically stable for all τ ∈ [0, τmax ]. det iωI − A0 − A1 =0 (8)
1 + T iω
The basic idea of the proposed approach is to study the for a T ∈ [0, ∞]. The relation between τ and T is given by
location of the roots of the characteristic equation using the (6).
socalled Rekasius substitution [19] and recently established
sum of squares techniques [2], [17], [15] from computational In other words, on the imaginary axis (and only on the
semialgebraic geometry. The characteristic equation of the imaginary axis), (7) has a root on it if and only if (8) has a
system (1) is given by root on it. Hence, (8) instead of (7) can be used to obtain the
∆(s, τ ) = det sI − A0 − A1 e−τ s , stability bound τmax for the TDLTI system (1). Therefore,


(3)
by using for example the Routh-Hurwitz criterion, the roots
or equivalently by of the polynomial
n
X
∆(s, τ ) = ak (s)e−kτ s , (4) D(s, T ) = det ((sI − A0 )(1 + T s) − A1 (1 − T s)) (9)
k=0
on the imaginary axis can be located. Note that the
where ak are polynomials in s of degree n − k with real polynomial equation (9) is obtained by applying Rekasius’s
coefficients. System (1) is asymptotically stable if and only substitution to (4) and by multiplying the resulting equation
if all roots of (3) respectively (4) lie in the open left half with (1 + T s)n . The Routh-Hurwitz type approach was for
complex plane [5]. Unfortunately the characteristic equation example exploited in [24], [14]. Furthermore, note that the
is transcendental due to e−kτ s and has in general infinitely Routh-Hurwitz approach is exact, however, a disadvantage
many roots. This is the main reason why a stability analysis of the Routh-Hurwitz approach is that conditions (e.g.,
of TDLTI system is a nontrivial task. The first central build- socalled Routh’s array) depend now on T and hence
ing block in the proposed approach is the socalled Rekasius one has essentially to compute all roots of (9), i.e., a
substitution [19], [6]. Rekasius introduced the following polynomial in two variables, to finally obtain τmax , which
substitution in (3) respectively (4): is computationally hard.
1 − Ts
e−τ s = , (5)
1 + Ts
T ≥ 0, which is defined only on the imaginary axis in the III. BASIC IDEA
complex plane, i.e., s = iω. However, equation (5) holds for The idea of this paper is to circumvent the computation of
s = iω if and only if all roots of (9), which is shown in the following. Remember,
2  −1 π it is assumed that for τ = 0, system (1) is asymptotic stable.
τ= tan (ωT ) + k , (6) Hence for τ = 0, the characteristic equation (3) has no roots
ω 2
on the imaginary axis. Therefore, by Lemma 1, (8) has no
k = 0, 1, 2, . . . Note that this substitution is a transformation
real positive solutions ω, T . In other words,
(one-to-one) for s = iω, τ ≥ 0, T ≥ 0. This fact is
very well known and is studied in the theory of complex |D(iω, T = 0)| > 0 (10)
functions under the name bilinear or linear fractional or
Möbius transformation. However, the crucial property for all positive ω. Now by increasing T , which corresponds
of this substitution is that it turns the transcendental to increasing τ due to (6), one expects that there exists a
characteristic equation (3) respectively (4) into an algebraic maximal T , i.e., Tmax such that the roots of (3) cross the
(rational) equation. Interesting results based on the Rekasius imaginary axis, and hence there exists a maximal Tmax such
substitution (also known as the socalled pseudodelay that
technique [13]) has been established for example in [14].
There, results have been derived which give answers to |D(iω = iω0 , T = Tmax )| = 0 (11)
for a certain ω0 , where |.| denotes the absolute value. From ω
this, one could suggest the following approach. In a first step,
find the maximal T , i.e., Tmax such that
|D(iω, T )| > 0 (12)
holds for (T̃max , ω̃0)

∀T ∈ [0, Tmax ], ∀ω ∈ [0, ∞), (13)

(Tmax , ω0)
where Tmax is found for example by a line search. In a
second step, compute ω0 in (11) and by using the function
(0, 0) T
(6), compute a lower bound of τmax , i.e., τ0 given by (a)
Tmax
2
tan−1 (ω0 Tmax ). (14)
ω0
ω
Let’s discuss the first step in the above approach. The
main idea is to replace the computation of the roots by a (Tmax , ωmin )
positivity condition. This is an advantage because condition 2 −1
ω tan (ωT ) = τ1
(12) can be checked via efficient numerical methods, in ωmin
particular by using sum of squares techniques. Hence, one (T0, ωmin ) (T̃max , ω̃0)
can avoid to compute roots of polynomials. Now, let’s 2 −1
ω tan (ωT ) = τ 2 > τ1
discuss the second step in the above approach. The idea is
(Tmax , ω0)
rather obvious but unfortunately not correct. The fault in
2 −1
the arguments above is that there may exists another pair ω tan (ωT ) = τ 3 > τ2

(ω̃0 , T̃max ) with T̃max > Tmax but with τ̃0 < τ0 , since the (0, 0) T
(b)
relation between T and τ in function (6) depends also on Tmax
ω and is therefore not monotonic. Summarizing, the first
step bridges the connection to sum of squares techniques. Fig. 1. Schematic representation of the T − ω plane.
The second step, however, does not lead to a lower bound
for τmax . Nevertheless, it is possible to modify the above
approach, by keeping the positivity conditions, which finally
leads to the main results of this paper in Section IV. The
modified approach is as follows:
4
Stability Analysis Test: Step 1: Compute the maximum
(supremum) Tmax and the minimal (infimum) ωmin such that 2
τ

|D(iω, T )| > 0 (15) 0

4 2
holds for 3
∀T ∈ [0, Tmax ], ∀ω ∈ [0, ∞), 2 1
(16) 1
∀T ∈ [0, ∞), ∀ω ∈ [ωmin , ∞). ω T
0 0
Step 2: Compute a lower bound of τmax by
2
τmax = tan−1 (ωmin Tmax ). (17)
ωmin

2
The only modification made is to replace (13) by a (16).
0
τ

Note that the strict inequality in (15) is crucial since

|D(iω, T )| ≥ 0. Furthermore, for the strict inequality −2
|D(iω, T )| > 0, one will not achieve a maximum or
minimum but a supremum or infimum. For computational 5
2
purposes, however, |D(iω, T )| ≥ ε, ε > 0 a small constant, 0 0
ω −5 T
will be used instead of (15). To see that this is indeed −2
sufficient, i.e., to see that τmax is a lower bound of τmax , Fig. 2. Function (6) for k = 0.
one has to investigate the function (6). In the following, this
is shown graphically and a proof is given in Section IV. Figure 1(a) shows basically the situation as outlined at the
beginning of this section. One can see the T − ω plane for
T ≥ 0, ω ≥ 0. The black three points represent roots of (Tmax , ωmin ) defines a lower bound for τmax via (20). In
the (absolute values) characteristic equation (9). The white other words,
domain represents the set (13). Hence all roots of of (9) 2  −1 π 2
must lie in the gray domain. Figure 1(b) shows basically tan (ω̃ T̃ ) + k ≥ tan−1 (ωmin Tmax ) (21)
ω̃ 2 ωmin
the situation as outlined in the proposed stability analysis
test. The white domain represents the set (16). To see that holds for all k = 0, 1, 2, . . . and all (T̃ , ω̃) ∈ (Tmax , ∞) ×
τmax defined by (17) is a lower bound of τmax , observe [0, ωmin ). This hypothesis is true, if one shows, for example,
now that the level sets of the function (6), which are given that the function
by the dashed-dotted lines in Figure 1(b) reveal the crucial 1
τ (T, ω) = tan−1 (ωT ), (22)
observation that the minimum of the function (6) subject ω
to the domain (16) is attained at (Tmax , ωmin ). Hence, for satisfies
all roots of the characteristic equation (9), which must lie ∂ ∂
τ (T, ω) ≥ 0, τ (T, ω) ≤ 0 (23)
in the gray domain, τ is larger or equal to the value of ∂T ∂w
characteristic equation (9) evaluated at (Tmax , ωmin ). Hence, for all T ≥ 0, ω ≥ 0. Namely, if τ = τ (T, ω) is
(Tmax , ωmin ) is a lower bound of the τmax . Finally, notice monotonically increasing in direction T and monotonically
that it makes only sense to study the function (6) for k = 0 decreasing in direction ω and since possible roots satisfy
is obvious, since the objective is to find a lower bound (T̃ , ω̃) ∈ (Tmax , ∞) × [0, ωmin ), (21) obviously holds, due
of (Tmax , ωmin ). Figure 2 shows the function (6) for the to T̃ ≥ Tmax and ω̃ ≤ ωmin . Hence, it has to be shown
domain of interest, i.e, T ≥ 0, ω ≥ 0 and, to give a better that (23) holds. By differentiation of (22) w.r.t. to T follows
global picture, also for negative arguments. Summarizing, immediately
the crucial observation is that the level sets of the function
∂ 1
(6) have a particular shape (convex set). From this, the τ (T, ω) = ≥ 0. (24)
proposed stability analysis test is easy to understand and ∂T 1 + (ωT )2
justifies graphically the proposed test. That this graphical By differentiation of (22) w.r.t. ω follows
discussion is indeed airtight, is proved in the next section. ∂ 1 T
τ (T, ω) = − 2 tan−1 (ωT ) + . (25)
∂ω ω ω(1 + (ωT )2 )
IV. MAIN RESULTS
Hence, it must hold
In this section, the main results of this paper, namely the 1 T
justification of the stability analysis test in Section III and tan−1 (ωT ) ≥ (26)
ω2 ω(1 + (ωT )2 )
a sum of squares formulation of the stability analysis test
are derived. Collecting the ideas in Section III together, the or equivalently,
following theorem justifies the stability analysis test: ωT
tan−1 (ωT ) ≥ . (27)
1 + (ωT )2
Theorem 1: Suppose that the matrix A0 − A1 has no purely
imaginary eigenvalues and suppose that But this is indeed true, since
d d
a(x) ≥ b(x) (28)
|D(iω, T )| > 0 (18) dx dx
x
holds for with a(x) = tan−1 (x) and b(x) = 1+x 2 holds, which

follows by simple algebraic manipulations and (28) implies

∀T ∈ [0, Tmax ], ∀ω ∈ [0, ∞),
(19) a(x) ≥ b(x), (29)
∀T ∈ [0, ∞), ∀ω ∈ [ωmin , ∞).
i.e., (27), due to a(0) = b(0) = 0. Therefore
Then
∂
2 τ (T, ω) ≤ 0. (30)
τmax = −1
tan (ωmin Tmax ) (20) ∂ω
ωmin
It is assumed that det(sI − A0 + A1 ) has no zeros on
is a lower bound of τmax , i.e., the TDLTI system (1) under the imaginary axis. This is because the case T = ∞,
1−T iω
the Assumption (2) is stable for all τ ∈ [0, τmax ]. 1+T iω = −1, in (8) is not appropriately reflected in (9),
(18). In case A0 − A1 has purely imaginary eigenvalues,
Proof. Due to Lemma 1, |D(iω, T )| = 0 (D(iω, T ) = 0) the stability test is still applicable. More details on this
if and only if |∆(iω, τ )| = 0 (∆(iω, τ ) = 0), where T special case and how to proceed then can be found in [5], p.
and τ are related via equation (6). Further, Assumption 36f. Alternatively, one can use instead of (5) the following
s)2
(2) guarantees that |D(iω, T = 0)| > 0 for all ω ∈ substitution: e−τ s 7→ (1−T
(1+T s)2 . In this case, the 2/ω in
[0, ∞). Moreover, (18) and (19) guarantees that all roots relation (6) has to be replaced by 4/ω and the assumption
(T̃ , ω̃), T̃ ≥ 0, ω̃ ≥ 0, of D on the iω-axis must lie in made on A0 − A1 can be skipped, cf. [24], [10]. 
domain (Tmax , ∞) × [0, ωmin ). The hypothesis is now that
Theorem 1 justifies the stability analysis test proposed in certain cases which includes the case of two variables, it
Section III. In a next step, it is shown how the proposed has been shown by Hilbert (cf. e.g. [20]), that a polynomial
stability analysis test can be implemented on a computer is nonnegative (positive) if and only if the polynomial can
very efficiently using sum of squares techniques. Some be written as a sum of squares of polynomials. Hence to
background material on sum of squares techniques can be check (32), (33) by using sum of squares techniques does
found in [16], [20]. Furthermore, the Matlab toolboxes [18] not introduce any conservatism, since polynomials in the
and [9] are available to solve sum of squares problems. two variables u, T respectively in ω, v are investigated.
Without going into details, the bottom line of sum of squares In particular, in the delay-independent stability case the
techniques is that one can check very efficiently whether a proposed stability test, i.e., |D(iv 2 , T 2 )|2 > 0(ε) ∀v, T , is
multivariable polynomial is globally positive or not by using necessary and sufficient, neglecting the constant ε (cf. also
efficient and reliable convex optimization, namely semidefi- [22]).
nite programming algorithms. For the stability analysis test,
this means one has to write down (15) together with (16) in Summarizing, a new theorem for the stability analysis of
terms of a polynomial inequality. Obviously, |D(iω, T )|2 is TDLTI systems was proved which justifies the stability anal-
a polynomial. To ensure strict positivity of |D(iω, T )|2 , the ysis test in Section III. Further, a numerical stability analysis
first step is to replace (15) with |D(iω, T )|2 ≥ ε, where ε algorithm was derived which is based on efficient sum
is a very small constant, i.e., 10−4 . The second step is to of squares techniques. Theorem 1 is a sufficient condition
incorporate the constraints (16), where the positivity test has and delivers a lower bound τmax for τmax . However, the
to take place. For this, the following substitutions in (9), are following examples demonstrate that this lower bound is less
used: conservative then many stability tests based on Lyapunov-
Tmax Krasovskii functionals and linear matrix inequalities.
T = , ω = ωmin + v 2 , (31)
1 + u2
which maps from u 7→ T : (−∞, ∞) → [0, Tmax ),
v 7→ ω : (−∞, ∞) → [ωmin , ∞) respectively. Note that V. EXAMPLES
the first substitution turns the characteristic equation (9)
into a rational function, however, by multiplying again
Example 1. Consider the TDLTI system (1) with
with (1 + u2 )n , one obtains again a polynomial. Hence,
one arrives at the following numerical stability analysis    
algorithm: −2 0 −1 0
A0 = , A1 = . (35)
0 −0.9 −1 −1
Stability Analysis Algorithm using Rekasius’s Substitu-
tion and Sum of Squares (RSOS): Step 1: Choose a small This is a well known benchmark example from the
ε > 0. Check if literature. Table I, which is taken from [4], summarizes
 2 several attempts from the literature over the past 10 years to
Tmax
(1 + u2 )n D iω, ≥ε (32) obtain a best possible lower bound for τmax . In particular,
1 + u2
the paper from [4] comes already very close to the exact
holds for all ω, u ∈ R using sum of squares techniques. bound. However, the proposed stability analysis algorithm
Check if RSOS in Section IV obtains the best bound in comparison

2 to all other methods in the table. Even more, the exact
D i(ωmin + v 2 ), T 2 ≥ε (33)
bound is practically achieved. The difference is caused due
holds for all T, v ∈ R using sum of squares techniques. to ε. To check the inequalities (32) and (33), the Matlab
Step 2: Compute a lower bound of τmax by toolbox SOSTOOLS [18] was used. The parameters are:
2 ε = 10−5 , Tmax = 9.985, and ωmin = 0.436. Tmax and
τmax = tan−1 (ωmin Tmax ). (34) ωmin were obtained via a line search.
ωmin

Some additional remarks are at this place necessary. First, Method Lower bound τmax
one would like to have the lower bound τmax as large as [8] 0.8571
[12] 0.99
possible. One way to achieve is via a line search with respect [11] 4.3588
to Tmax , ωmin . An alternative is to maximize Tmax and to [3] 4.47
minimize ωmin . This is also possible by sum of squares [7], [25], [23] 4.4721
[4] 4.4721 - 6.09
techniques. Second, note that (32) is satisfied for negative RSOS (Theorem 1) 6.1698
ω ′ s if it is satisfied for positive ω ′ s, since the absolute value Exact bound τmax 6.17
of D is symmetric with respect to ω. It is, of course, also TABLE I
possible to replace in (32), ω by ω 2 . Notice also that the R ESULTS FOR E XAMPLE 1.
sum of squares decomposition is a sufficient condition to
check positivity of multivariable polynomials. However, for
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