Finite-time Stability Analysis for Nonlinear Descriptor Systems

N. Shopa

, D. Konovalov

, A. Kremlev

and K. Zimenko

Faculty of Control Systems and Robotics, ITMO University, Russia Federation

Keywords:

Descriptor Systems, Finite-time Stability, Nonlinear Systems, Stability Analysis.

Abstract:

Sufﬁcient conditions of ﬁnite-time stability are presented for the class of nonlinear descriptor systems. Both,

explicit and implicit Lyapunov function methods, are extended for ﬁnite-time stability analysis of descriptor

systems and the corresponding settling time estimates are obtained. The theoretical results are supported by

numerical examples.

1 INTRODUCTION

Frequently, in control practice there are nonlinear sys-

tems for which it is hard to derive useful representa-

tion in the form of Ordinary Differential Equations

(ODEs). Compared to ODEs, descriptor (singular)

systems in addition to the dynamic part also include

a static (uncausal) one (e.g., algebraic constraints). In

this way, descriptor models are more ﬂexible for sys-

tem description. Additionally, descriptor models al-

low preserving physical meaning of variables. There-

fore, descriptor systems have often been a subject of

research (see, for example, (Sun et al., 2014; Zheng

and Cao, 2013; Mo et al., 2017; Ikeda et al., 2004;

Yang et al., 2012; Wu and Mizukami, 1994)).

Stability (stabilizability) analysis based on the

Lyapunov function method for nonlinear descriptor

systems was considered in (Ikeda et al., 2004; Hill and

Mareels, 1990; Wu and Mizukami, 1994; Yang et al.,

2012; Chen and Yang, 2016). Finite-time stability

analysis (stabilization) is important if all transitions

of the system has to be terminated in a ﬁnite (spec-

iﬁed in advance) time (see, for example, (Bhat and

Bernstein, 2000; Roxin, 1966; Polyakov et al., 2015;

Bacciotti and Rosier, 2005; Moulay and Perruquetti,

2006; Orlov, 2004), etc.). The papers (Sun et al.,

2014; Zheng and Cao, 2013; Konovalov et al., 2021)

are devoted to the ﬁnite-time control design problem

for descriptor systems. In these papers an analysis

of ﬁnite-time stability is based on preliminary trans-

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formation to the canonical semi-explicit form (i.e.,

ODEs with constraints) and subsequent ﬁnite-time

stability analysis for ODEs. A ﬁnite-time stability

analysis based on the Lyapunov function method was

proposed in the paper (Chen and Yang, 2016) for non-

linear descriptor systems. However, stability condi-

tions proposed in (Chen and Yang, 2016) are too con-

servative and can be applied only for rather speciﬁc

examples.

In this paper, sufﬁcient Lyapunov-based condi-

tions of ﬁnite-time stability are presented for the class

of nonlinear descriptor systems. The corresponding

estimates of settling time functions are also derived.

Compared to the paper (Chen and Yang, 2016), stabil-

ity conditions are signiﬁcantly relaxed. The proposed

conditions are applicable for descriptor systems in not

necessarily semi-explicit form.

The paper is organized in the following way. Sec-

tion II introduces notation used in the paper. Sec-

tion III recalls some basics on descriptor systems and

ﬁnite-time stability. Section IV presents the main re-

sult on ﬁnite-time stability conditions for nonlinear

descriptor systems. Some numerical examples are

also presented there. Finally, concluding remarks are

given in Section V.

2 NOTATION

Through the paper the following notation will be

used:

• R

= {x ∈ R : x > 0}, where R is the ﬁeld of real

numbers;

• R

denotes the n dimensional Euclidean space

Shopa, N., Konovalov, D., Kremlev, A. and Zimenko, K.

Finite-time Stability Analysis for Nonlinear Descriptor Systems.

DOI: 10.5220/0011347500003271

In Proceedings of the 19th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2022), pages 711-716

ISBN: 978-989-758-585-2; ISSN: 2184-2809

711

with vector norm k·k;

• O

m×n

denotes zero matrix with dimension m ×n;

• I

∈ R

n×n

is the identity matrix;

• the order relation P > 0 (< 0; ≥ 0; ≤ 0) for

P ∈ R

n×n

means that P is symmetric and positive

(negative) deﬁnite (semideﬁnite);

• A continuous function σ : R

∪{0} → R

∪{0}

belongs to class K if it is strictly increasing and

σ(0) = 0. It belongs to class K

∞

if it is also un-

bounded.

3 PRELIMINARIES

Let us consider a nonlinear descriptor system

E ˙x(t) = f (x(t)), x(0) = x

, t ≥ 0, (1)

where x ∈R

is the state vector, f : R

→R

, f (0) = 0

and E ∈ R

n×n

is a constant matrix, which is singular

in general (rankE = r < n).

Deﬁnition 1 (Newcomb, 1981). Initial conditions x

are consistent at t = 0 if there exists a solution Φ(t,x

)

to the system (1), such that x

= lim

t→0

Φ(t, x

Assumption 1. The system (1) has a unique solution

Φ(t, x

), t ≥ 0, and initial value x

∈ R

is consistent.

Deﬁnition 2 (Wu and Mizukami, 1994). The sys-

tem (1) is said to be

• globally Lyapunov stable if for any ε > 0 there ex-

ists δ(ε) > 0 such that kx

k< δ, then kΦ(t,x

)k<

ε for all t ≥ 0.

• globally asymptotically stable if it is Lyapunov

stable and there exists δ(ε) > 0 such that kx

k< δ

implies lim

t→+∞

Φ(t, x

) = 0.

Similar to systems presented by ODEs (see (Bhat

and Bernstein, 2000), (Polyakov, 2011)) let us give

ﬁnite-time stability deﬁnitions for the descriptor sys-

tem (1).

Deﬁnition 3. The origin of (1) is said to be glob-

ally ﬁnite-time stable if it is globally asymptotically

stable and any solution Φ(t, x

) of the system (1)

reaches the equilibrium point at some ﬁnite time mo-

ment, i.e., Φ(t, x

) = 0, ∀t ≥ T (x

) and Φ(t, x

) 6= 0,

∀t ∈ [0,T(x

)), x

6= 0, where T : R

→ R

∪{0},

T (0) = 0 is a settling-time function.

Deﬁnition 4. The set M is said to be ﬁnite-time at-

tractive for (1) if any solution Φ(t,x

) of (1) reaches

M in a ﬁnite instant of time t = T

) and remains

there ∀t ≥ T

). As before, T

: R

→ R

∪{0} is

a settling-time function.

Deﬁne a full column rank matrix U ∈ R

n×(n−r)

whose column vectors consist of the bases of NullE

Deﬁne the set M = {x ∈R

: Ex = 0}.

The following theorem gives a sufﬁcient condition

for asymptotic stability analysis of nonlinear descrip-

tor systems.

Theorem 1 (Ikeda et al., 2004). Let there exist a

continuously differentiable function V : R

→ R

, a

continuous function W : R

×R

n−r

→ R, functions a,

b ∈K

∞

and c ∈K satisfying the following conditions:

1) a(kExk) ≤V (Ex) ≤ b(kExk) for ∀x ∈ R

;

V (Ex) + W (x,U

f (x)) ≤ −c(kxk) for ∀x ∈ R

where

V (Ex) = gradV (z)



z=Ex

· f (x);

3) W(x, 0) ≡ 0 for ∀x ∈ R

Then the zero solution x ≡ 0 of the descriptor system

(1) is globally asymptotically stable.

In (Chen and Yang, 2016) is proposed a sufﬁcient

condition for ﬁnite-time stability analysis of descrip-

tor systems.

Theorem 2 (Chen and Yang, 2016). Let there exist

a continuous function V : R

→R

and two functions

a,b ∈ K

∞

such that the following conditions hold:

1) a(kEkkxk) ≤V (x) ≤b(kEkkxk);

V (x) ≤ −βV (x)

, where σ ∈ (0,1), β ∈ R

;

Then the system (1) is ﬁnite-time stable with a settling

time satisfying the inequality

T (x

) ≤

1−σ

)

β(1 −σ)

Remark 1. The dynamical part of the system (1) is

represented by state-space equations whose state vari-

able corresponds to the variable Ex, i.e., Ex represents

the dynamical behavior of the systems by itself (Ikeda

et al., 2004). Thus, it is ’natural’ to consider a Lya-

punov function as a positive deﬁnite function of the

variable Ex. In this sense, the condition 1) in Theo-

rem 2 is too restrictive and it signiﬁcantly narrows the

class of descriptor systems, where Theorem 2 can be

useful.

4 MAIN RESULT

The next theorem extends the result of Theorem 1 for

ﬁnite-time stability analysis.

Theorem 3. Suppose there exist continuously differ-

entiable functions V

: R

→R

, continuous func-

tions W

: R

×R

n−r

→ R, functions a,b ∈ K

∞

c ∈ K and real numbers β ∈ R

and σ ∈ [0, 1), such

that:

1) a(kExk) ≤V

(Ex) ≤b(kExk) for ∀x ∈R

and i =

1,2;

(Ex) +W

(x,U

f (x)) ≤−c(kxk) for ∀x ∈ R

;

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

712

3) W

(x,0) ≡0 for i = 1,2 and ∀x ∈ R

;

(Ex) + W

(x,U

f (x)) ≤ −βV

(Ex) for ∀x ∈

\M ;

where

(Ex) = gradV

(Ex) · f (x) for i = 1, 2. Then

the system (1) is globally ﬁnite-time stable and

T (x

) ≤

1−σ

(Ex

)

β(1 −σ)

Proof. Conditions 1) −3) provide asymptotic sta-

bility of the system (1) according to Theorem 1.

Since U satisﬁes U

E = O

(n−r)×n

, the term W

(as

well as W

in the condition 2)) becomes zero along the

solutions of the system (1), and 4) provides

(Ex) ≤ −βV

(Ex), ∀x ∈R

\M . (2)

Then, 1) and (2) provide the set M is ﬁnite-time at-

tractive with the settling-time estimate

) ≤

1−σ

(Ex

)

β(1 −σ)

that can be checked by integration of (2) (see (Lopez-

Ramirez et al., 2018), (Zimenko et al., 2021) for more

details):

(x) =

(Ex)

−

(EΦ(θ

(s),x))

≤

(Ex)

βV

(EΦ(θ

(s),x))

(Ex)

βs

β(1−σ)

(Ex)

1−σ

< +∞,

(3)

where θ

is the inverse of t → V

(EΦ(t, x)). More-

over,

(0) = 0. On the other hand, due to Ex(t) = 0

for ∀t ≥ T

), then from 1) and 2) we have that

kxk= 0 for ∀t ≥ T

), i.e. the system is ﬁnite-time

stable with T (x

) ≡ T

). 

Note that Theorem 2 is a particular case of Theo-

rem 3.

Example 1. Consider the following nonlinear de-

scriptor system:

˙x

= −x

1/3

+ 2x

0 = x

+ x

sin(x

(4)

where E =



1 0

0 0



, x =





Let us choose

V (Ex) = V

(Ex) = V

(Ex) =

, (5)

and

W (x,U

f (x)) = W

(x,U

f (x)) = W

(x,U

f (x))

= −(x

+ x

sin(x

))

(6)

for which the conditions 1) and 3) of Theorem 3 are

satisﬁed. Then, we obtain

V (Ex) +W (x,U

f (x))

= −x

4/3

+ 2x

−(x

+ x

sin(x

))

= −x

4/3

−x

−

+ x

)sin(x

)

−

sin

)

≤ −x

4/3

−

sin

(7)

From (7) we have

V (Ex) +W (x,U

f (x)) ≤−

kxk

and, on the other hand,

V (Ex) +W (x,U

f (x)) ≤ −x

4/3

= −2V

2/3

(Ex),

i.e., the conditions 2) and 4) of Theorem 3 are also

satisﬁed. Thus, all conditions of Theorem 3 are satis-

ﬁed, and the system is ﬁnite-time stable with the fol-

lowing settling time estimate T (x

) ≤

4/3

2/3

, where

= x

(0). The results of simulation with using

the logarithmic scale are shown in Fig. 1 in order to

demonstrate ﬁnite-time convergence rate of the Eu-

clidean norm kxk.

Figure 1: The results of simulation for different initial con-

ditions.

The advantages of the proposed result are based

on the following observations:

• under the condition 2) the ﬁnite-time attractive-

ness of the set M is equivalent to ﬁnite-time sta-

bility of the origin;

• the condition 1) allows to choose a Lyapunov

function depending only on the variable Ex (e.g.,

= x

PEx, P > 0 and its nonlinear varia-

tions);

• the terms W

and W

in the conditions 2) and 4)

become zero along the solutions of the system (1).

In spite of this, these terms are crucial for the

satisfaction of the conditions 2) and 4) in prac-

tice (see, for example, the results of (Ikeda et al.,

2004), (Uezato and Ikeda, 1999) on asymptopic

stability analysis).

Finite-time Stability Analysis for Nonlinear Descriptor Systems

713

If f (x) = 0 only at the origin for x ∈ M then the

following result one can obtain:

Corollary 1. Suppose there exist a continuously dif-

ferentiable function V : R

→ R

, a continuous func-

tion W : R

×R

n−r

→R, functions a,b ∈ K

∞

and real

numbers β ∈ R

and σ ∈ [0,1), such that:

1) a(kExk) ≤V (Ex) ≤ b(kExk) for ∀x ∈ R

;

V (Ex) +W (x,U

f (x)) ≤−βV

(Ex)

for ∀x ∈ R

\M ;

3) W(x, 0) ≡ 0 for ∀x ∈ R

;

4) {x ∈M : f (x) = 0} ≡{0}.

Then the system (1) is globally ﬁnite-time stable and

T (x

) ≤

1−σ

(Ex

)

β(1 −σ)

Proof. The proof is straightforward. Accord-

ing to the proof of Theorem 3 the conditions 1) −

3) provide ﬁnite-time attractiveness of the set M .

Due to Ex(t) = 0 for ∀t ≥ T

) =

1−σ

(Ex

)

β(1−σ)

, then

f (x(t)) = 0 for ∀t ≥ T

) and by the condition 4)

we have that kxk= 0 for ∀t ≥ T

), i.e. the system

is ﬁnite-time stable with T (x

) ≡ T

). 

Example 2. Consider the three-tank hydraulic system

(Fig. 2) from (Duro et al., 2008). The system consists

of three cylinders T

, T

, and T

with the same cross-

section A. These cylinders are connected serially to

each other by cross-section S

pipes.

Figure 2: Schematic diagram of the three-tank system.

The mathematical model of the plant can be rep-

resented by the following descriptor system:

= −Q

= Q

−Q

= Q

−Q

= S

√

2gbh

−h

0.5

= S

√

2gbh

−h

0.5

= S

√

2gbh

0.5

(8)

where h

, h

and h

are the liquid levels in each

tank, their derivatives represent the balance equations;

and Q

are the ﬂow rates between tanks, Q

is the rate of ﬂow exiting the system; g is the ac-

celeration due to gravity; b·e

= |·|

sign(·). The

system is in the form (1) with E =



3×3







. It is easy to

check that the condition 4) of Corollary 1 is satisﬁed.

Now let us choose the following candidate Lya-

punov function

V (Ex) = |x

−x

1.5

+ |x

−x

1.5

+ |x

1.5

(9)

Recall that by Minkowski inequality for any z

∈R

and p ≥ 1, the inequality

+ z

≤ 2

p−1

(|z

+ |z

)

hold. Applying this inequality one can obtain

(|x

1.5

+|x

1.5

+ |x

1.5

) ≤V (Ex)

≤ 2

3/2

(|x

1.5

+ |x

1.5

+ |x

1.5

i.e., the condition 1) of Corollary 1 is satisﬁed.

Let

W(x,U

f (x)) =

1.5

−x

0.5

(2x

−2 f

(x)−x

+ f

(x))

1.5

−x

0.5

( f

(x)

−x

+2x

−2 f

(x)−x

+ f

(x))

1.5

0.5

(−x

+ f

(x) + x

− f

(x)),

(10)

where f

(x) = S

√

2gbx

−x

0.5

, f

(x) =

√

2gbx

−x

0.5

and f

(x) = S

√

2gbx

0.5

According to (8) the function W become zero along

the solutions of the system (the condition 3) of

Corollary 1 hold).

Since

V =

1.5

−x

0.5

(−2x

+ x

)

1.5

−x

0.5

−2x

+ x

)

1.5

0.5

−x

(11)

then, taking into account the equivalence of norms k·

1.5

≤ k·k

, we obtain

V (Ex) +W (x,U

f (x))

√

−x

0.5

−x

0.5

(H + H

)

−x

0.5

−x

0.5

≤ −

1.5S

√

(|x

−x

|+ |x

−x

|+ |x

≤ −

1.5S

√

2/3

(Ex),

(12)

where H =

−3 3 0

0 −3 3

0 0 −1.5

Thus, by Corollary 1 the system (8) is ﬁnite-time

stable.

Remark 2. Consider the system (1) in the canonical

semi-explicit form

˙x

= f

0 = f

)

, (13)

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

714

where x =





, E =



r×(n−r)

(n−r)×r

(n−r)×(n−r)



f (x) =



)



. Then under the assumption the

function f

(x) satisﬁes the condition that there ex-

ists a continuous function h such that x

= h(x

) and

h(0) = 0, the system (13) can be reduced to an ODE

˙x

= f

,h(x

)),

and ﬁnite-time stability analysis methods correspond-

ing to ODEs can be applied (for example, (Bhat

and Bernstein, 2000), (Roxin, 1966), (Bacciotti and

Rosier, 2005), (Moulay and Perruquetti, 2006), etc.).

Note, the representation in the canonical form (13)

and the subsequent transition to ODEs in some cases

can be accompanied by computational complexity

and errors. It is also worth noting that for the sys-

tem in the form (13) the condition 4) of Corollary 1 is

satisﬁed and the principal conditions are 1) −3).

The next theorem presents the Implicit Lyapunov

Function method (see (Korobov, 1979), (Adamy and

Flemming, 2004)) for ﬁnite-time stability analysis of

descriptor systems (1).

Theorem 4. Suppose that there exists a continuous

function

Q: R

×R

→ R

(V,z) 7→ Q(V,z)

such that

1) Q(V,z) is continuously differentiable ∀z ∈

\{0} and ∀V ∈ R

;

2) for any z ∈ R

\{0} there exist V

−

∈ R

and

∈ R

Q(V

−

,z) < 0 < Q(V

,z); (14)

3) for Ω = {(V,z) ∈ R

n+1

: Q(V,z) = 0}

lim

z→0

(V,z)∈Ω

V = 0, lim

V →0

(V,z)∈Ω

kzk = 0, lim

kzk→∞

(V,z)∈Ω

V = +∞;

4) the inequality

−∞ <

∂Q(V,z)

∂V

< 0

holds ∀V ∈ R

and ∀z ∈ R

\{0};

5) the inequality

∂Q(V,z)

∂z

f (x) +W (V,x,U

f (x)) ≤βV

∂Q(V,z)

∂V

hold ∀x ∈ R

; ∀(V,x) ∈ R

n+1

: Q(V,Ex) = 0, where

z = Ex, W : R

× R

n−r

→ R is such that

W (V,x,0) ≡ 0, and σ ∈ [0,1), β ∈ R

are some con-

stants;

6) {x ∈M : f (x) = 0}≡ {0}.

Then the origin of the system (1) is globally ﬁnite-

time stable and

T (x

) ≤

1−σ

β(1 −σ)

where V

∈ R

: Q(V

,Ex

) = 0.

Proof. The proof follows the same arguments as

one of Corollary 1 and the Implicit Lyapunov Func-

tion method for ﬁnite-time stability analysis of ODE

systems (Polyakov et al., 2015, Theorem 4). The

conditions 1), 2), 4) and the implicit function theo-

rem (Courant and John, 2000) imply that the equa-

tion Q(V,z) = 0 implicitly deﬁnes a unique function

V : R

\{0} → R

such that Q(V (z), z) = 0 for all

z ∈ R

\{0}. Due to the condition 3) the function V

can be continuously prolonged at the origin by setting

V (0) = 0. In addition, V is radially unbounded and

positive deﬁnite. Then there exist functions a,b ∈ K

∞

such that a(kzk) ≤V (z) ≤ b(kzk) (Khalil, 1992), i.e.,

the condition 1) of Corollary 1 is satisﬁed with z =

Ex.

Since by means of Implicit Function Theorem

(Courant and John, 2000)

V = −

∂Q

∂V

−1

∂Q

∂z

˙z, then

with z = Ex the condition 5) repeats the conditions 2)

and 3) of Corollary 1. Finally, the condition 6) repeats

the condition 4) of Corollary 1. Thus, all conditions

of Corollary 1 are satisﬁed and the system (1) is ﬁnite-

time stable. 

Remark 3. In (Konovalov et al., 2021) a ﬁnite-time

homogeneous control is proposed for linear descrip-

tor systems, and the following implicitly deﬁned Lya-

punov function is considered

Q(V,x) = x

−G

lnV

−G

lnV

x −1,

where G

∈R

n×n

is an anti-Hurwitz matrix (i.e., −G

is Hurwitz); X ∈ R

n×n

is such that X

E = E

X ≥ 0

and x

Ex = 0 iff Ex = 0. In view of Theorem 4,

the result of (Konovalov et al., 2021) can be revisited

in order to consider an implicitly deﬁned Lyapunov

function in the form

Q(V,x) = z

−L

lnV

−L lnV

z −1,

P > 0, L is anti-Hurwitz, z = Ex,

(15)

which can be considered as homogeneous generaliza-

tion (see (Konovalov et al., 2021) for more details) of

the quadratic Lyapunov function V (Ex) = x

PEx.

Based on the stability analysis of linear descriptor

systems proposed in (Xu and Lam, 2006), it is ex-

pected that basing on Theorem 4 the choice of a Lya-

punov function in the form (15) may allow one to

obtain more reliable LMI-based stability conditions

and investigate the robustness properties of the con-

trol scheme given in (Konovalov et al., 2021). This is

one of the main directions for future research.

Finite-time Stability Analysis for Nonlinear Descriptor Systems

715

5 CONCLUSION

In this paper, the sufﬁcient conditions of ﬁnite-time

stability are proposed for the class of nonlinear de-

scriptor systems. The settling time estimates are ob-

tained. Both, explicit and implicit Lyapunov func-

tion methods are considered. The conditions are suf-

ﬁciently less restrictive than those proposed in (Chen

and Yang, 2016). The presented ﬁnite-time stability

analysis opens a lot of topics for future research. For

example, control and observer design for descriptor

systems based on the proposed stability conditions.

ACKNOWLEDGEMENTS

This work is supported by RSF under grant 22-29-

00344 in ITMO University.

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