# Finite-Time Boundedness Control of Time-Varying Descriptor Systems

Authors: Xiaoming Su; Yali Zhi; Qingling Zhang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper mainly studies a control problem of finite-time boundedness of time-varying descriptor systems. Firstly, a sufficient and necessary condition of finite-time stability is given, then a sufficient condition of finite-time boundedness for time-varying descriptor systems is given. Secondly, we analyze the finite-time boundedness control problem and design the finite-time state feedback controller; the controller is given based on LMIs for time-varying descriptor systems and time-varying uncertain descriptor systems, respectively. Finally, a numerical example is given to prove the effectiveness of the method.


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## Body

## 1. Introduction

Recently, descriptor systems (which are also known assingular systems, semistate systems, systems of differential-algebraic equations, or generalized state space systems.) theory has been well studied since they are very important from the engineering point of view. Let the time-varying descriptor systems be described as E(t)x˙(t)=A(t)x(t)+B(t)u(t)+G(t)ω(t). They arise naturally in many physical applications such as electrical networks, aircraft and robot dynamics, neutral delay and large-scale systems, economics and optimization problem, biology, constrained mechanics, as result of partial discretization of partial differential equation, and so forth. The time-varying descriptor system has the common properties of general descriptor system and some special properties. In control theory, the time-varying descriptor system stability is mainly Lyapunov stability; however, Lyapunov stability deals with the whole state performance of the system, but it does not reflect a transient performance of the system. The transient performance refers to the system stability in a short period, but it is different from Lyapunov stability. In engineering, a whole stability of the system may have a very bad transient performance, which would cause a bad effect in the engineering. Therefore, people tend to be more concerned about the transient performance of the system [1–5] than the overall steady state performance of the system.In order to study the transient performance of the system, scholars have given the concept of finite-time stability. Finite-time stability of time-varying descriptor system is a new field; scholars have mainly studied the finite-time stability of linear system for years and also made some corresponding results. For example, [6–10] proposed the definition of finite-time stability and finite-time boundedness; [5–8] proposed the sufficient conditions for the finite-time stability of general linear system; [1, 11] studied the input and output finite-time stability for time-varying descriptor system.In addition, a finite-time stability problem with external disturbance is called finite-time boundedness; the concept of finite-time boundedness came from finite-time stability. We have some preliminary research results about the finite-time boundedness. For example, [12–14] studied the finite-time bounded problem of the linear time-varying system with impulse; [15–18] proposed the design methods of dynamic compensators; [19–21]  discussed the finite-time control problems of uncertain system with disturbance. Scholars introduced the definition of finite-time stability and have given the necessary and sufficient conditions for finite-time stability of the descriptor system, but the finite-time stability problems of time-varying descriptor system have no available research results, especially when E(t) is time-variant. So, it is necessary to study finite-time stability of the time-varying uncertain descriptor system.The paper is divided into two parts: firstly, we deals with the problem of finite-time stability of time-varying descriptor system. We give the necessary and sufficient condition of finite-time stability and finite-time boundedness of system with time-varying function matrixE(t); the unstable system can be controlled by the state feedback controllers. We also make the corresponding research for time-varying uncertain descriptor system. Secondly, we design a finite-time boundedness state feedback controller for time-varying uncertain descriptor system to make the close-loop system finite-time boundedness.The paper is organized as follows. In Section2, we give some results of finite-time stability and boundedness for time-varying descriptor system. In Section 3, some results of time-varying uncertain descriptor system are provided. In Section 4, a numerical example is presented to illustrate the efficiency of the proposed result.


## 2. Finite-Time Control of Time-Varying Descriptor Systems

### 2.1. Finite-Time Stable of Time-Varying Descriptor Systems

Definition 1.
Time-varying matrixE(t) is singular on time interval [0,T], if there exists a t∈[0,T] such that rank E(t)<n.
Consider the following time-varying descriptor systems(1)E(t)x˙(t)=A(t)x(t),
where x(t)∈Rn is the state. A(·) is given continuous matrix-valued function. E(t)∈Rn is singular on time interval [0,T].Definition 2.
The time-varying descriptor system (1) is said to be finite-time stable with respect to (c1,c2,T,R(t)) with positive definite matrix R(t), and given three positive scalars c1,c2,T, with c1<c2, if xT(0)ET(0)R(0)E(0)x(0)≤c1, then xT(t)ET(t)R(t)E(t)x(t)<c2, for all t∈[0,T].Theorem 3.
The following statements are equivalent:(i)
system (1) is FTS respect to (c1,c2,T,R(t)).(ii)
For allt∈[0,T], ΦT(t,0)ET(t)R(t)E(t)<(c2/c1)ET(0)R(0)E(0), where Φ(t,0) is the state transition matrix and R(t) is positive definite matrix.(iii)
For allt∈[0,T], the differential Lyapunov inequality, with terminal and initial conditions
(a)
M(t)<0, where
(2)M(t)=AT(t)P(t)E(t)+ET(t)P(t)A(t)+E˙T(t)P(t)E(t)+ET(t)P˙(t)E(t)+ET(t)P(t)E˙(t),(b)
R(t)≤P(t)≤P(0)<(c2/c1)R(0), R(t) is positive definite matrix, admits a piecewise continuously differentiable symmetric solution p(·).Proof.
(
ii
)
⇒
(
i
) Let xT(0)ET(0)R(0)E(0)x(0)≤c1; assume x(t)=Φ(t,0)x(0), have
(3)xT(t)ET(t)R(t)E(t)x(t)=xT(0)ΦT(t,0)ET(t)R(t)E(t)x(0)<c2c1ET(0)R(0)E(0)<c2.
Therefore, system (1) is FTS.
(
i
)
⇒
(
ii
) by contradiction. Let us assume ∃t¯,x¯(4)x¯T(t¯)ΦT(t¯,0)ET(t¯)R(t¯)E(t¯)Φ(t¯,0)x¯(t¯)≥c2c1x¯T(0)ET(0)R(0)E(0)x¯(0).
Let xT(0)ET(0)R(0)E(0)x(0)=c1, ∃λ, such that x¯(t)=λx(0); then (4) implies that
(5)xT(0)ΦT(t¯,0)ET(t¯)R(t¯)E(t¯)Φ(t¯,0)x(0)≥c2;
therefore
(6)xT(t¯)ET(t¯)R(t¯)E(t¯)x(t¯)=xT(0)ΦT(t¯,0)ET(t¯)R(t¯)E(t¯)Φ(t¯,0)x(0)≥c2.
Obviously, it contradicts the initial assumption that system (1) is FTS.
(
iii
)
⇒
(
i
) Let V(t,x)=xT(t)ET(t)P(t)E(t)x(t), V˙(t,x)=xT(t)M(t)x(t).
Then (a) implies thatV˙(t,x) is negative definite along the trajectories of system (1).
NowxT(0)ET(0)R(0)E(0)x(0)≤c1, then for a generic t, such that
(7)xT(t)ET(t)R(t)E(t)x(t)≤xT(t)ET(t)P(t)E(t)x(t)<xT(0)ET(0)P(0)E(0)x(0)≤c2c1xT(0)ET(0)R(0)E(0)x(0)≤c2.(i)⇒(iii) By contradiction. Because system (1) is FTS. Let z=ϵx for a small ϵ>0, for all t∈[0,T], such that
(8)xT(0)ET(0)P(0)E(0)x(0)≤c1⟹xT(t)ET(t)R(t)E(t)x(t)+∥z∥22<c2.
Let P(·) be the solution of
(9)M(t)=ϵ2I,(10)R(t)=P(t).
And assume that ∃x¯, such that
(11)x¯T(t)ET(t)P(0)E(t)x¯(t)≥c2c1x¯T(t)ET(t)R(0)E(t)x¯(t).
Now let xT(0)ET(0)R(0)E(0)x(0)=c1, ∃λ, such that x¯(t)=λx(0).
Then (11) implies
(12)xT(0)ET(0)R(0)E(0)x(0)≥c2;
from (9) we obtain that
(13)ddtxT(t)ET(t)P(t)E(t)x(t)=-ϵ2xT(t)x(t).
Integrating (13) from 0 to t we have
(14)xT(t)ET(t)P(t)E(t)x(t)-xT(0)ET(0)R(0)E(0)x(0)=-ϵ2∥x∥22.
Therefore
(15)xT(t)ET(t)R(t)E(t)x(t)≥xT(t)ET(t)P(t)E(t)x(t)=xT(0)ET(0)P(0)E(0)x(0)-ϵ2∥x∥22≥c2-∥z∥22,
which contradicts (8).


### 2.2. Finite-Time Boundedness of Time-Varying Descriptor Systems

Consider the following time-varying descriptor systems(16)E(t)x˙(t)=A(t)x(t)+G(t)ω(t),
where x(t)∈Rn is the state and ω(t)∈Rl is exogenous input. A(t)∈Rn×n, G(t)∈Rn×l are given constant matrices. E(t)∈Rn×n is a singular function matrix, and rankE(t)=q<n.The exogenous disturbanceω(t) is time varying and satisfies the constraint
(17)∫0T‍ωT(t)ω(t)dt≤d,d≥0.Definition 4.
The system (16) subject to an exogenous disturbance ω(t) satisfies (17) and is said to be finite-time bounded with respect to (c1,c2,T,R(t),d) with positive definite matrices R(t), and given three positive scalars c1,c2,T, with c1<c2, if xT(0)ET(0)R(0)E(0)x(0)≤c1, then xT(t)ET(t)R(t)E(t)x(t)<c2.Theorem 5.
The time-varying descriptor system (16) is finite-time bounded, if there exist two positive definite nonsingular matrices P(t)∈Rn×n, Q∈Rl×l, such that
(18)[M(t)ET(t)P(t)G(t)GT(t)PT(t)E(t)Q]<0,(19)R(t)≤P(t)≤P(0)≤R(0),(20)c1+dλmax(Q)<c2
hold, where λmax(·) denote the maximum eigenvalue of the argument,
(21)M(t)=AT(t)P(t)E(t)+ET(t)P(t)A(t)+E˙T(t)P(t)E(t)+ET(t)P˙(t)E(t)+ET(t)P(t)E˙(t).Proof.
LetV(t,x)=xT(t)ET(t)P(t)E(t)x(t).
We have(22)V˙(t,x)=xT(t)M(t)x(t)+ωT(t)GT(t)PT(t)E(t)x(t)+xT(t)ET(t)P(t)G(t)ω(t)=(xT(t),ωT(t))×[M(t)ET(t)P(t)G(t)GT(t)PT(t)E(t)Q]×(x(t)ω(t))-ωT(t)Qω(t)<0.
Therefore,
(23)V˙(t,x)<ωT(t)Qω(t);
integrating from 0 to t, with t∈[0,T], we have
(24)V(t,x)<V0+dλmax(Q).
From (19), we have
(25)xT(t)ET(t)R(t)E(t)x(t)≤V(t,x)<V(0,x)+dλmax(Q)<xT(0)ET(0)R(0)E(0)x(0)+dλmax(Q)<c1+dλmax(Q)<c2.
Therefore, system (16) is FTB. The proof is completed.


### 2.3. Finite-Time Bounded Control of Time-Varying Descriptor Systems

Consider the following time-varying descriptor systems(26)E(t)x˙(t)=A(t)x(t)+B(t)u(t)+G(t)ω(t),
where x(t)∈Rn is the state; u(t)∈Rm is control input; ω(t)∈Rl is exogenous input. A(t)∈Rn×n,B(t))∈Rn×mG(t)∈Rn×l are given continuous matrix-valued functions. E(t)∈Rn×n is a singular function matrix, and rankE(t)<n.Find a state feedback control law(27)u(t)=K(t)x(t),
where K(t)∈Rm×n. Let Ac(t)=(A(t)+B(t)K(t)). Then the closed-loop system is given by
(28)E(t)x˙(t)=Ac(t)x(t)+G(t)ω(t).Theorem 6.
The time-varying descriptor system (19) is finite-time bounded, if there exist a symmetric positive definite matrix P(t)∈Rn×n, and a nonsingular matrix Q∈Rl×l, such that inequality (19), (20) and the following condition hold:
(29)[M(t)ET(t)P(t)G(t)GT(t)PT(t)E(t)-QT]<0.
Moreover, the state feedback control law is given by
(30)K(t)=B-1(t)G(t)Q-1(t)GT(t)PT(t)E(t),
where λmax(·) denote the maximum eigenvalue of the argument,
(31)M(t)=AT(t)P(t)E(t)+ET(t)P(t)A(t)+E˙T(t)P(t)E(t)+ET(t)P˙(t)E(t)+ET(t)P(t)E˙(t).Proof.
From Theorem5, the conditions for FTB is that there exist a nonsingular matrix Q such that (19), (20) and the following matrix inequality hold:
(32)[M¯(t)ET(t)P(t)G(t)GT(t)PT(t)E(t)Q]<0,
where
(33)M¯(t)=(A(t)+B(t)K(t))TP(t)E(t)+ET(t)P(t)(A(t)+B(t)K(t))+E˙T(t)P(t)E(t)+ET(t)P˙(t)E(t)+ET(t)P(t)E˙(t).
According to Schur’s theorem, it is easy to see that (32) can be rewritten as
(34)M¯(t)-ET(t)P(t)G(t)Q-1(t)GT(t)PT(t)E(t)<0.
Using (30), (34), we have
(35)M(t)+ET(t)P(t)G(t)Q-T(t)GT(t)P(t)E(t)<0.
Since P(t) is symmetric, then
(36)M(t)+ET(t)P(t)G(t)Q-T(t)GT(t)PT(t)E(t)<0.
By Schur’s theorem, (36) is equivalent to (29). The proof is completed.


### 2.4. Finite-Time Bounded Control of Time-Varying Uncertain Descriptor Systems

Consider the following time-varying descriptor systems(37)E(t)x˙(t)=(A(t)+ΔA(t)x(t))+(B(t)+ΔB(t))u(t)+G(t)ω(t),
where x(t)∈Rn is the state; u(t)∈Rm is control input; ω(t)∈Rl is exogenous input. A(t)∈Rn×n, B(t)∈Rn×m, G(t)∈Rn×l are given continuous matrix-valued functions. E(t)∈Rn×n is a singular function matrix, and rankE(t)<n.
(38)[ΔA(t)ΔB(t)]=HF[E1(t)E2(t)],
where F∈Rq×s is unknown and satisfies
(39)FTF≤I.Theorem 7.
The time-varying descriptor system (37) is finite-time bounded, if there exist a symmetric positive definite matrix P(t)∈Rn×n, and a nonsingular matrix Q∈Rl×l, such that inequality (19), (20), and (32) hold, and the state feedback control law is given by
(40)K(t)=(B(t)+HFE2(t))-1×(G(t)Q-1(t)GT(t)PT(t)E(t)-HFE1(t)),
where
(41)M¯(t)=[A(t)+ΔA(t)+(B(t)+ΔB(t))K(t)]T×P(t)E(t)+ET(t)P(t)×[A(t)+ΔA(t)+(B(t)+ΔB(t))K(t)]+E˙T(t)P(t)E(t)+ET(t)P˙(t)E(t)+ET(t)P(t)E˙(t).Proof.
Applying (40) to (34), and P(t) is symmetric, then the proof is completed.


## 2.1. Finite-Time Stable of Time-Varying Descriptor Systems

Definition 1.
Time-varying matrixE(t) is singular on time interval [0,T], if there exists a t∈[0,T] such that rank E(t)<n.
Consider the following time-varying descriptor systems(1)E(t)x˙(t)=A(t)x(t),
where x(t)∈Rn is the state. A(·) is given continuous matrix-valued function. E(t)∈Rn is singular on time interval [0,T].Definition 2.
The time-varying descriptor system (1) is said to be finite-time stable with respect to (c1,c2,T,R(t)) with positive definite matrix R(t), and given three positive scalars c1,c2,T, with c1<c2, if xT(0)ET(0)R(0)E(0)x(0)≤c1, then xT(t)ET(t)R(t)E(t)x(t)<c2, for all t∈[0,T].Theorem 3.
The following statements are equivalent:(i)
system (1) is FTS respect to (c1,c2,T,R(t)).(ii)
For allt∈[0,T], ΦT(t,0)ET(t)R(t)E(t)<(c2/c1)ET(0)R(0)E(0), where Φ(t,0) is the state transition matrix and R(t) is positive definite matrix.(iii)
For allt∈[0,T], the differential Lyapunov inequality, with terminal and initial conditions
(a)
M(t)<0, where
(2)M(t)=AT(t)P(t)E(t)+ET(t)P(t)A(t)+E˙T(t)P(t)E(t)+ET(t)P˙(t)E(t)+ET(t)P(t)E˙(t),(b)
R(t)≤P(t)≤P(0)<(c2/c1)R(0), R(t) is positive definite matrix, admits a piecewise continuously differentiable symmetric solution p(·).Proof.
(
ii
)
⇒
(
i
) Let xT(0)ET(0)R(0)E(0)x(0)≤c1; assume x(t)=Φ(t,0)x(0), have
(3)xT(t)ET(t)R(t)E(t)x(t)=xT(0)ΦT(t,0)ET(t)R(t)E(t)x(0)<c2c1ET(0)R(0)E(0)<c2.
Therefore, system (1) is FTS.
(
i
)
⇒
(
ii
) by contradiction. Let us assume ∃t¯,x¯(4)x¯T(t¯)ΦT(t¯,0)ET(t¯)R(t¯)E(t¯)Φ(t¯,0)x¯(t¯)≥c2c1x¯T(0)ET(0)R(0)E(0)x¯(0).
Let xT(0)ET(0)R(0)E(0)x(0)=c1, ∃λ, such that x¯(t)=λx(0); then (4) implies that
(5)xT(0)ΦT(t¯,0)ET(t¯)R(t¯)E(t¯)Φ(t¯,0)x(0)≥c2;
therefore
(6)xT(t¯)ET(t¯)R(t¯)E(t¯)x(t¯)=xT(0)ΦT(t¯,0)ET(t¯)R(t¯)E(t¯)Φ(t¯,0)x(0)≥c2.
Obviously, it contradicts the initial assumption that system (1) is FTS.
(
iii
)
⇒
(
i
) Let V(t,x)=xT(t)ET(t)P(t)E(t)x(t), V˙(t,x)=xT(t)M(t)x(t).
Then (a) implies thatV˙(t,x) is negative definite along the trajectories of system (1).
NowxT(0)ET(0)R(0)E(0)x(0)≤c1, then for a generic t, such that
(7)xT(t)ET(t)R(t)E(t)x(t)≤xT(t)ET(t)P(t)E(t)x(t)<xT(0)ET(0)P(0)E(0)x(0)≤c2c1xT(0)ET(0)R(0)E(0)x(0)≤c2.(i)⇒(iii) By contradiction. Because system (1) is FTS. Let z=ϵx for a small ϵ>0, for all t∈[0,T], such that
(8)xT(0)ET(0)P(0)E(0)x(0)≤c1⟹xT(t)ET(t)R(t)E(t)x(t)+∥z∥22<c2.
Let P(·) be the solution of
(9)M(t)=ϵ2I,(10)R(t)=P(t).
And assume that ∃x¯, such that
(11)x¯T(t)ET(t)P(0)E(t)x¯(t)≥c2c1x¯T(t)ET(t)R(0)E(t)x¯(t).
Now let xT(0)ET(0)R(0)E(0)x(0)=c1, ∃λ, such that x¯(t)=λx(0).
Then (11) implies
(12)xT(0)ET(0)R(0)E(0)x(0)≥c2;
from (9) we obtain that
(13)ddtxT(t)ET(t)P(t)E(t)x(t)=-ϵ2xT(t)x(t).
Integrating (13) from 0 to t we have
(14)xT(t)ET(t)P(t)E(t)x(t)-xT(0)ET(0)R(0)E(0)x(0)=-ϵ2∥x∥22.
Therefore
(15)xT(t)ET(t)R(t)E(t)x(t)≥xT(t)ET(t)P(t)E(t)x(t)=xT(0)ET(0)P(0)E(0)x(0)-ϵ2∥x∥22≥c2-∥z∥22,
which contradicts (8).


## 2.2. Finite-Time Boundedness of Time-Varying Descriptor Systems

Consider the following time-varying descriptor systems(16)E(t)x˙(t)=A(t)x(t)+G(t)ω(t),
where x(t)∈Rn is the state and ω(t)∈Rl is exogenous input. A(t)∈Rn×n, G(t)∈Rn×l are given constant matrices. E(t)∈Rn×n is a singular function matrix, and rankE(t)=q<n.The exogenous disturbanceω(t) is time varying and satisfies the constraint
(17)∫0T‍ωT(t)ω(t)dt≤d,d≥0.Definition 4.
The system (16) subject to an exogenous disturbance ω(t) satisfies (17) and is said to be finite-time bounded with respect to (c1,c2,T,R(t),d) with positive definite matrices R(t), and given three positive scalars c1,c2,T, with c1<c2, if xT(0)ET(0)R(0)E(0)x(0)≤c1, then xT(t)ET(t)R(t)E(t)x(t)<c2.Theorem 5.
The time-varying descriptor system (16) is finite-time bounded, if there exist two positive definite nonsingular matrices P(t)∈Rn×n, Q∈Rl×l, such that
(18)[M(t)ET(t)P(t)G(t)GT(t)PT(t)E(t)Q]<0,(19)R(t)≤P(t)≤P(0)≤R(0),(20)c1+dλmax(Q)<c2
hold, where λmax(·) denote the maximum eigenvalue of the argument,
(21)M(t)=AT(t)P(t)E(t)+ET(t)P(t)A(t)+E˙T(t)P(t)E(t)+ET(t)P˙(t)E(t)+ET(t)P(t)E˙(t).Proof.
LetV(t,x)=xT(t)ET(t)P(t)E(t)x(t).
We have(22)V˙(t,x)=xT(t)M(t)x(t)+ωT(t)GT(t)PT(t)E(t)x(t)+xT(t)ET(t)P(t)G(t)ω(t)=(xT(t),ωT(t))×[M(t)ET(t)P(t)G(t)GT(t)PT(t)E(t)Q]×(x(t)ω(t))-ωT(t)Qω(t)<0.
Therefore,
(23)V˙(t,x)<ωT(t)Qω(t);
integrating from 0 to t, with t∈[0,T], we have
(24)V(t,x)<V0+dλmax(Q).
From (19), we have
(25)xT(t)ET(t)R(t)E(t)x(t)≤V(t,x)<V(0,x)+dλmax(Q)<xT(0)ET(0)R(0)E(0)x(0)+dλmax(Q)<c1+dλmax(Q)<c2.
Therefore, system (16) is FTB. The proof is completed.


## 2.3. Finite-Time Bounded Control of Time-Varying Descriptor Systems

Consider the following time-varying descriptor systems(26)E(t)x˙(t)=A(t)x(t)+B(t)u(t)+G(t)ω(t),
where x(t)∈Rn is the state; u(t)∈Rm is control input; ω(t)∈Rl is exogenous input. A(t)∈Rn×n,B(t))∈Rn×mG(t)∈Rn×l are given continuous matrix-valued functions. E(t)∈Rn×n is a singular function matrix, and rankE(t)<n.Find a state feedback control law(27)u(t)=K(t)x(t),
where K(t)∈Rm×n. Let Ac(t)=(A(t)+B(t)K(t)). Then the closed-loop system is given by
(28)E(t)x˙(t)=Ac(t)x(t)+G(t)ω(t).Theorem 6.
The time-varying descriptor system (19) is finite-time bounded, if there exist a symmetric positive definite matrix P(t)∈Rn×n, and a nonsingular matrix Q∈Rl×l, such that inequality (19), (20) and the following condition hold:
(29)[M(t)ET(t)P(t)G(t)GT(t)PT(t)E(t)-QT]<0.
Moreover, the state feedback control law is given by
(30)K(t)=B-1(t)G(t)Q-1(t)GT(t)PT(t)E(t),
where λmax(·) denote the maximum eigenvalue of the argument,
(31)M(t)=AT(t)P(t)E(t)+ET(t)P(t)A(t)+E˙T(t)P(t)E(t)+ET(t)P˙(t)E(t)+ET(t)P(t)E˙(t).Proof.
From Theorem5, the conditions for FTB is that there exist a nonsingular matrix Q such that (19), (20) and the following matrix inequality hold:
(32)[M¯(t)ET(t)P(t)G(t)GT(t)PT(t)E(t)Q]<0,
where
(33)M¯(t)=(A(t)+B(t)K(t))TP(t)E(t)+ET(t)P(t)(A(t)+B(t)K(t))+E˙T(t)P(t)E(t)+ET(t)P˙(t)E(t)+ET(t)P(t)E˙(t).
According to Schur’s theorem, it is easy to see that (32) can be rewritten as
(34)M¯(t)-ET(t)P(t)G(t)Q-1(t)GT(t)PT(t)E(t)<0.
Using (30), (34), we have
(35)M(t)+ET(t)P(t)G(t)Q-T(t)GT(t)P(t)E(t)<0.
Since P(t) is symmetric, then
(36)M(t)+ET(t)P(t)G(t)Q-T(t)GT(t)PT(t)E(t)<0.
By Schur’s theorem, (36) is equivalent to (29). The proof is completed.


## 2.4. Finite-Time Bounded Control of Time-Varying Uncertain Descriptor Systems

Consider the following time-varying descriptor systems(37)E(t)x˙(t)=(A(t)+ΔA(t)x(t))+(B(t)+ΔB(t))u(t)+G(t)ω(t),
where x(t)∈Rn is the state; u(t)∈Rm is control input; ω(t)∈Rl is exogenous input. A(t)∈Rn×n, B(t)∈Rn×m, G(t)∈Rn×l are given continuous matrix-valued functions. E(t)∈Rn×n is a singular function matrix, and rankE(t)<n.
(38)[ΔA(t)ΔB(t)]=HF[E1(t)E2(t)],
where F∈Rq×s is unknown and satisfies
(39)FTF≤I.Theorem 7.
The time-varying descriptor system (37) is finite-time bounded, if there exist a symmetric positive definite matrix P(t)∈Rn×n, and a nonsingular matrix Q∈Rl×l, such that inequality (19), (20), and (32) hold, and the state feedback control law is given by
(40)K(t)=(B(t)+HFE2(t))-1×(G(t)Q-1(t)GT(t)PT(t)E(t)-HFE1(t)),
where
(41)M¯(t)=[A(t)+ΔA(t)+(B(t)+ΔB(t))K(t)]T×P(t)E(t)+ET(t)P(t)×[A(t)+ΔA(t)+(B(t)+ΔB(t))K(t)]+E˙T(t)P(t)E(t)+ET(t)P˙(t)E(t)+ET(t)P(t)E˙(t).Proof.
Applying (40) to (34), and P(t) is symmetric, then the proof is completed.


## 3. Numerical Example

Consider a time-varying descriptor system (26) with
(42)E=[t000],A=[t00-1],B=[100t],G=[1-t].
Let R(t)=P(t), Q=t, T=1. Exist P(t)=[-t00t], such that
(43)M(t)=[-2t3-3t2000],ET(t)P(t)G(t)=[-t20],GT(t)PT(t)E(t)=[-t20].
So, condition (29), (19), (20) hold. the system (32) is finite-time bounded and the state feedback controller with
(44)K(t)=B-1(t)G(t)Q-1(t)GT(t)PT(t)E(t)=[-t010].


## 4. Conclusions

In this paper, we have studied the finite-time stability and given a sufficient and necessary condition of the finite-time stability; the state feedback controller was designed, then we studied the finite-time boundedness of time-varying descriptor systems, and a sufficient and necessary condition of the finite-time boundedness is given and a state feedback controller was designed. In the end, a numerical example is given to prove the effectiveness of the method.



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*Source: 754268-2013-12-25.xml*