# Finite-Time Simultaneous Stabilization of a Set of Nonlinear Singular Systems

Authors: Liying Sun; Meiqing Li
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This study addresses the problems of finite-time simultaneous stabilization for two nonlinear singular systems and more than two nonlinear singular systems. First, we design a suitable output feedback controller and combine the two nonlinear singular systems to generate an augmented system by using an augmented technique. Based on a sufficient condition of the augmented system impulse-free, an important result, that is, the augmented system is finite-time stabilization is presented. Then, the finite-time stabilization problem of more than two singular systems is investigated by dividing theN systems into two sets. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed finite-time stabilization controller for two nonlinear singular systems.


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## Body

## 1. Introduction

In recent years, the singular system has attracted extensive attention of researchers, which is also called the descriptor system, generalized state-space system, differential-algebraic system, and so on. Due to its unique properties and structure, the singular system is more complex than the normal state-space system. It needs to consider impulse-free (continue-time system) or causality (discrete-time system), regularity, and stable [1].It is worth pointing out that the simultaneous stabilization problem of the normal state-space system is introduced in 1982. Abundant results of simultaneous stabilization have been obtained in recent years [2–6]. In [2], several simultaneous external and internal stabilization problems have been considered under appropriate adaptive low-and-high gain feedback controllers. Simultaneous stabilization of linear systems has been solved by the static output feedback in [3]. In [4], the simultaneous stabilization of the multiinput-multioutput system has been investigated to develop a stabilizing controller. A necessary and sufficient condition for the existence of simultaneously quadratically stabilizing state feedback laws is derived for a collection of single-input discrete-time nonlinear systems in [5]. In [6], simultaneous stabilization of a set of nonlinear port-controlled Hamiltonian systems is investigated by using the dissipative Hamiltonian structural properties. Subsequently, some results of simultaneous stabilization are obtained for singular systems [7, 8]. Simultaneous stabilization problems are investigated for nonlinear singular systems with actuator saturation by using a suitable output feedback in [7] and an uncertain singular system with input saturation via using linear matrix inequalities in [8], respectively.By the virtue of the method of simultaneous stabilization, the finite-time simultaneous stabilization problem is considered for the nonlinear singular system. Definition of finite-time stabilization is proposed for the linear system [9, 10] and then extended to the linear singular system [11]. Furthermore, the finite-time stabilization is investigated for the nonlinear discrete-time Hamiltonian singular system [12]. Finite-time stabilization, which is different from asymptotic stability [13–17], is used to describe that the state does not exceed a certain bound during a fixed finite-time interval. The finite-time stabilization of the nonlinear continue-time singular system and nonlinear discrete-time singular system is investigated via the state-undecomposed method in [18, 19], respectively. In [20], the stochastic finite-time H∞ filtering issue for a class of nonlinear continuous-time singular semi-Markov jump systems is discussed in the forms of strict LMIs. While, stability of the robust finite-time for linear singular Markovian jump systems with impulsive effects and time-varying norm-bounded disturbance is considered under the designed state feedback controller and estimation of domain of attraction in [21]. To our best knowledge, a few results are presented for finite-time simultaneous stabilization of the singular system. In [22], finite-time control and fault detection are studied simultaneously for the singular system via the average dwell time approach and using some novel integral inequalities. When it comes to the nonlinear singular system, the finite-time simultaneous stabilization of two nonlinear singular systems and more than two nonlinear singular systems is considered in this study. It is worth noticing that it is also called finite-time stability, i.e., states of the system reach the equilibrium point within a fixed time T and stay at the equilibrium point when t > T [23–29]. The finite-time robust stabilization problem of general nonlinear time-delay systems is studied based on the Hamiltonian function method and observer design in [26]. In [27], the finite-time stabilization is investigated for a class of singular systems by the constructed new Lyapunov functional, while the finite-time robust simultaneous stabilization and adaptive robust simultaneous stabilization have been investigated for nonlinear systems with time delay in [28, 29], respectively.In this study, our goal is to discuss the simultaneous stabilization problem of two nonlinear singular systems or more than two nonlinear singular systems and propose the designed method of the output feedback controller. The study is divided into three parts. (i) Two nonlinear singular systems discussed about state bound in finite-time interval under the nonlinear function need not to satisfy the Lipschitz condition, and the suitable output feedback law is constructed. (ii) Based on the above method, finite-time simultaneous stabilization of more than two nonlinear singular systems is studied by dividing theN systems into two sets. (iii) The numerical example of two nonlinear singular systems is used to illustrate the validity of the proposed finite-time stable controller.


## 2. Finite-Time Simultaneous Stabilization of Two Nonlinear Singular Systems

Consider the following two nonlinear singular systems:(1)E1x˙=f1x+g1xu,E1x0=E1x0,f10=0,y=b1xx,(2)E2ξ˙=f2ξ+g2ξu,E2ξ0=E2ξ0,f20=0,η=b2ξξ,where x∈ℝn1 and ξ∈ℝn2 are the states of the two systems, u∈ℝq is the control input, y,η∈ℝm are the outputs, Ei∈ℝni×ni is the singular matrix, and 0< rank Ei=ri<n; bix∈ℝm×ni, gi.∈ℝni×q, fi. is a sufficiently smooth vector field with proper dimensions, i=1,2.To facilitate the analysis of the two systems (1) and (2), we show the following definition and lemma.Definition 1.
(see [30]). A control law ux is called an admissible control law, if for any initial condition Ex0, the resulted closed-loop descriptor system has no impulsive solution, and accordingly, the original system is called impulse controllable.Lemma 1.
(see [31]). If a scalar function hx with h0=0x∈ℝn has continuous nth order partial derivatives, then hx can be expressed as(3)hx=a1xx1+⋯+anxxn=Axx,where aix,i=1,2,…,n are the scalar functions.
As mentioned above, the two systems can be described as follows:(4)E1x˙=A1xx+g1xu,E1x0=E1x0,f10=0,y=b1xx,(5)E2ξ˙=A2ξξ+g2ξu,E2ξ0=E2ξ0,f20=0,η=b2ξξ.
We can design an output feedback controller:(6)u=−Ky−η,where K∈ℝm×m is a symmetric matrix. Substituting (6) into systems (4) and (5), the systems can be given as follows:(7)E1x˙=A1x−g1xKb1xx+g1xKb2ξξ,y=b1xx,(8)E2ξ˙=A2ξ+g2ξKb2ξξ−g2ξKb1xx,η=b2ξξ.Combine the two closed-loop singular systems (7) and (8) into an augmented system:(9)EX˙=AXX,Y=BXX,where(10)X=xξ,Y=yη,E=E100E2,BX=b1x00b2ξ,AX=A1x−g1xKb1xg1xKb2ξ−g2ξKb1xA2ξ+g2ξKb2x.Lemma 2.
Ifrank0EEAX=n1+n2+r1+r2, then system (9) is impulse-free.Proof.
FromE=diagE1,E2, we know rankE=r1+r2. So, there exist two nonsingular matrices M,N∈ℝn1+n2×n1+n2, such that(11)MEN=Ir1+r2000.
Denote(12)X¯=N−1X=X1X2,MAXN=A˜11XA˜12XA˜21XA˜22X=A11XA12XA21XA22X,MBX=B˜1XB˜2X=B1X¯B2X¯,where X¯∈ℝn1+n2, X1∈ℝr1+r2, and X2∈ℝn1+n2−r1−r2. Then, system (9) can be expressed as follows:(13)X˙1=A11X¯X1+A12X¯X2,0=A21X¯X1+A22X¯X2,Y=B1X¯X1+B2X¯X2.
Noticing that(14)n1+n2+r1+r2=rank0EEAX=rankMM0EEAXNN=rankA22X¯+2r1+2r2,we can obtain that rankA22X¯=n1+n2−r1−r2, which implies that the index of equivalent closed-loop singular system (9) is one at the equilibrium point 0, that is, system (9) is impulse-free.
Next, we present the definition of the finite-time simultaneous stable.Definition 2.
The two nonlinear singular systems (4) and (5) are said to be the finite-time simultaneous stable (FTSS) with respect to c1,c2,T,R, with 0<c1<c2 and R=diagR1,R2>0, if XT0ETREX0≤c1, such that XTtETREXt<c2,∀t∈0,T, where xT0E1TR1E1x0≤c11, ξT0E2TR2E2ξ0≤c21, and c1≥c11+c21, c11,c12>0.
Based on augmented system (9), we give the following result of the finite-time simultaneous stable of systems (4) and (5).Theorem 1.
If there exist symmetric positive definite matricesQ∈ℝn1+n2×n1+n2, P1∈ℝn1×n1, and P2∈ℝn2×n2, such that(15)Ω1Ω2Ω2TΩ3≤0,(16)λmaxQc1eαT<λminQc2,(17)rank0EEAX=n1+n2+r1+r2.
Then, under the output feedback control law (6), the nonlinear singular systems (4) and (5) are finite-time simultaneous stable with respect to c1,c2,T,R, where(18)Ω1=A1x−g1xKb1xTP1E1+E1TP1A1x−g1xKb1x−αE1TP1E1,Ω2=g1xKb2ξTP1E1−E2TP2g2ξKb1x,Ω3=A2ξ+g2ξKb2ξTP2E2+E2TP2A2ξ+g2ξKb2ξ−αE2TP2E2,R=R100R2>0, Q=Q100Q2>0, P1=R11/2Q1R11/2, P2=R21/2Q2R21/2, α>0, and λmax. and λmin. indicate the maximum and minimum eigenvalues of the argument, respectively.Proof.
According to Lemma2 and condition (17), it is clear that augmented system (9) is impulse-free. Choose the Lyapunov function(19)VXt=XTtETPEXt≥0,∀t∈0,T,i.e., VXt=V1xt+V2ξt=xTtE1TP1E1xt+ξTtE2TP2E2ξt≥0, where P=P100P2=R1/2QR1/2.
Based on system (9),(20)V˙Xt−αVXt=XTATXPE+ETPAX−αETPEX=xξTA1x−g1xKb1xTP1E1+E1TP1A1x−g1xKb1x−αE1TP1E1−g2ξKb1xTP2E2+E1TP1g1xKb2ξg1xKb2ξTP1E1−E2TP2g2ξKb1xA2ξ+g2ξKb2ξTP2E2+E2TP2A2ξ+g2ξKb2ξ−αE2TP2E2xξ=xξTΩ1Ω2Ω2TΩ3xξ.
Under condition (15), there is(21)V˙Xt≤αVXt,∀t∈0,T.
Furthermore, by integrating the inequality (21) between 0 and t, it is clear that(22)VXt≤eαtVX0,∀t∈0,T.
Consider the following chain of inequalities:(23)VXt=XTtETPEXt=XTtETR1/2QR1/2EXt≥λminQXTtETREXt,(24)VX0eαt=XT0ETPEX0=XT0ETR1/2QR1/2EX0eαt≤λmaxQXT0ETREX0eαT,where λminQ=minλminQ1,λminQ2, and λmaxQ=maxλmaxQ1,λmaxQ2.
IfXT0ETREX0≤c1, taking account of (16), (22)–(24), it can be deduced that(25)XTtETREXt≤λmaxQXT0ETREX0λminQeαT≤λmaxQc1λminQeαT<c2.
Hence, system (9) is finite-time stable with respect to c1,c2,T,R.
To proceed further, systems (4) and (5) can be finite-time simultaneous stabilization with respect to c1,c2,T,R.


## 3. Finite-Time Simultaneous Stabilization of More than Two Nonlinear Singular Systems

ConsiderN nonlinear singular systems:(26)Eix˙i=fixi+gixiu,Eixi0=Eix0i,fi0=0,yi=bixixi,i=1,2,…,N,where xi∈ℝni, yi∈ℝm, and u∈ℝq are the states, the outputs, and the inputs of the N systems, respectively; gixi∈ℝni×q, Ei∈ℝni×ni,0<rankEi=ri<ni, and bix∈ℝm×ni,i=1,2,…,N.Based on the method given in Section2, we can represent system (26) into the following forms:(27)Eix˙i=Aixixi+gixiu,yi=bixixi,i=1,2,…,N.Suppose thati1,i2,…,iN is an arbitrary permutation of 1,2,…,N and that L is a position integer satisfying 1≤L≤N−1. Denote T1=ni1+⋯+niL and T2=niL+1+⋯+niN. We divide the N systems into two sets:(28)EaX˙a=AaXaXa+GaXau,Ya=BaXaXa,(29)EbX˙b=AbXbXb+GbXbu,Yb=BbXbXb,where(30)Xa=xi1T,…,xiLTT∈ℝT1,Xb=xiL+1T,…,xiNTT∈ℝT2,Ea=diagEi1,…,EiL,Eb=diagEiL+1,…,EiN,AaXa=diagAi1xi1,…,AiLxiL,AbXb=diagAiL+1xiL+1,…,AiLxiN,Ya=yi1+⋯+yiL,Yb=yiL+1+⋯+yiN,GaXa=gi1Txi1,…,giLTxiLT,GbXb=giL+1TxiL+1,…,giNTxiNT,BaXa=bi1xi1,…,biLxiL,BbXb=biL+1xiL+1,…,biNxiN.Design an output feedback controller:(31)u=−KYa−Yb≔−KBaXaXa−BbXbXb,where matrix K∈ℝm×m is the symmetric. Substitute (31) into systems (28) and (29), respectively. Then, the closed-loop systems can be given as follows:(32)EaX˙a=AaXa−GaXaKBaXaXa+GaXaKBbXbXb,Ya=BaXaXa,(33)EbX˙b=AbXb+GbXbKBbXbXb−GbXbKBaXaXa,Yb=BbXbXb.By the system-augmentation technique, systems (32) and (33) can be transformed as(34)E¯X¯˙=A¯X¯X¯,Y¯=B¯X¯X¯,where(35)X¯=XaXb,E¯=Ea00Eb,A¯=Aa−GaKBaGaKBb−GbKBaAb+GbKBb,Y¯=YaYb,B¯=Ba00Bb.Based on Theorem1, we can obtain the following result.Theorem 2.
If there exists a positive scalarα and symmetric positive definite matrices Q¯∈ℝT1+T2×T1+T2, Pa∈ℝT1×T1,andPb∈ℝT2×T2, such that(36)Ξ1Ξ2Ξ2TΞ3≤0,(37)λmaxQ¯c¯1eαT<λminQ¯c¯2,(38)Rank0E¯E¯A¯X¯=∑i=1Nni+ri,then, under the output feedback control law (31), system (26) is finite-time simultaneous stabilization with respect to c¯1,c¯2,T¯,R¯, where c¯2>c¯1>0, R¯>0, K,λmax., and λmin. are the same as those in Theorem 1. Noted that,(39)Ξ1=AaXa−GaXaKBaXaTPaEa+EaTPaAaXa−GaXaKBaXa−αEaTPaEa,Ξ2=GaXaKBbXbTPaEa−EbTPbGbXbKBaXa,Ξ3=AbXb+GbXbKBbXbTPbEb+EbTPbAbXb+GbXbKBbXb−αEbTPbEb.Proof.
Taking the same as Theorem1, system (34) is impulse-free depending on Lemma 2 and condition (38). We choose a Lyapunov function VX¯t=X¯TtE¯TP¯E¯X¯t≥0,∀t∈0,T, where P¯=Pa00Pb=Ra1/200Rb1/2Qa00QbRa1/200Rb1/2=R¯1/2Q¯R¯1/2, and Q¯,P¯,R¯ are the symmetric positive definite matrices.
Based on conditions (36) and (37), we have(40)V˙X¯−αVX¯=X¯TE¯TP¯A¯+A¯TP¯E¯−αE¯P¯E¯X¯=XaXbTAaXa−GaXaKBaXaTPaEa+EaTPaAaXa−GaXaKBaXa−αEaTPaEaEaTPaGaXaKBbXb−GbXbKBaXaTPbEbGaXaKBbXbTPaEa−EbTPbGbXbKBaXaAbXb+GbXbKBbXbTPbEb+EbTPbAbXb+GbXbKBbXb−αEbTPbEbXaXb=XaXbTΞ1Ξ2Ξ2TΞ3XaXb≤0.
To sum up,(41)V˙X¯t≤αVX¯t.By integrating inequality (41) between 0 and T with t∈0,T, the following result can be presented:(42)VX¯t≤eαtVX¯0.
Next, we will prove the finite-time simultaneous stabilization for more than two nonlinear singular systems.
It is straightforward to see(43)VX¯t=X¯TtE¯TP¯E¯X¯t=X¯TtE¯TR¯1/2Q¯R¯1/2E¯X¯t≥λminQ¯X¯TtE¯TR¯E¯X¯t,VX¯0eαt=X¯T0E¯TP¯E¯X¯0eαt=X¯T0E¯TR¯1/2Q¯R¯1/2E¯X¯0eαt≤λmaxQ¯X¯T0E¯TR¯E¯X¯0eαT,where λmaxQ¯=maxλmaxQa,λmaxQb, and λminQ¯=minλminQa,λminQb. According to the concept of finite-time simultaneous stable, from X¯T0E¯TR¯E¯X¯0≤c¯1 and (37), we can obtain(44)X¯TtE¯TR¯E¯X¯t≤λmaxQ¯X¯T0E¯TR¯E¯X¯0λminQ¯eαT≤λmaxQ¯c¯1λminQ¯eαT<c¯2.
By using the similar method as that in Theorem1, system (34) is finite-time stable with respect to c¯1,c¯2,T¯,R¯, and it can be shown that system (26) is finite-time simultaneous stabilization with respect to c¯1,c¯2,T¯,R¯.


## 4. Illustrative Example

In this section, we present an example to illustrate the effectiveness of the proposed method which addresses the finite-time simultaneous stabilization problem of the two nonlinear singular systems.Example 1.
Consider the two nonlinear singular systems:(45)E1x˙=f1x+g1xu,E1x0=E1x0,f10=0,y=b1xx,(46)E2ξ˙=f2ξ+g2ξu,E2ξ0=E2ξ0,f20=0,η=b2ξξ,where x=x1,x2T∈ℝ2,ξ=ξ1,ξ2T∈ℝ2,u∈ℝ2, we give the following parameters:(47)E1=1212,E2=1100,f1x=−2x1x22−4x1−5x2−5x1−2x23−1.5x2,f2ξ=−0.55ξ1ξ22−2.3ξ1+5ξ25ξ1−2.2ξ23−2ξ2,g1x=−100−1,g2ξ=−100−1,b1x=1001,b2ξ=1001.
Based on Lemma1, it can deduce that(48)A1x=−2x22−4−5−5−2x22−1.5,A2ξ=−0.55ξ22−2.355−2.2ξ22−2.
Design an output feedback controller(49)u=−Ky−η=−3550.5x1−ξ1x2−ξ2=−3x1−5x2+3ξ1+5ξ2−5x1−0.5x2+5ξ1+0.5ξ2.
Substituting (49) into systems (45) and (46), respectively, we obtain(50)E1x˙=A1x−g1xKb1xx+g1xKb2ξ,y=b1xx,(51)E2ξ˙=A2ξ+g2ξKb2ξξ−g2ξKb1x,η=b2ξξ.
Based on form (9), we have(52)EX˙=AXX,Y=BXX,where(53)X=xξT=x1x2ξ1ξ2,E=1200120000110000,B=1000010000100001,AX=−2x22−10−3−50−2x22−1−5−0.535−0.55ξ22−5.3050.50−2.2ξ22−2.5.
Choose(54)M=10000010−11000001,N=100−20001001001−10.
According to Lemma2, we can easy check that the augmented system is impulse-free.
Next, we give Lyapunov functionVXt=XTtETPEXt, where P=1000020000200004. Let α=0.2,Q=P,R=I,c1=1.34,c2=8.75,andT=2. By the virtue of conditions (15) and (16) in Theorem 1, we have(55)Ω1Ω2Ω2TΩ3=−4x22−2.6−8x22−5.2−7−16−8x22−5.2−16x22−10.40−2−70−2.2ξ22−21.2−1.1ξ22−10.6−16−2−1.1ξ22−10.6−0.4≤0,λmaxQc1eαTλminQ=4c11e0.2T<c2.
Choose the initial stateE1x0=0.7,0.7T and E2ξ0=0.6,0T, then X0TETREX0≤c1, the trajectory of Ψ=XtTETREXt<c2,∀t∈0,2. Denote ψ1=xtTE1TR1E1xt and ψ2=ξtTE2TR2E2ξt. The trajectories of Ψ, ψ1 and ψ2, state response, and control signal are shown in Figures 1–3, respectively. Obviously, the augmented nonlinear singular system (52) is finite-time stable with respect to 1.34,8.75,2,I. Hence, under the output feedback controller (49), systems (45) and (46) are finite-time simultaneous stabilization.Figure 1
Responses of the valuesψ, ψ1, and ψ2.Figure 2
Responses of the statex and ξ.Figure 3
Responses of the control signalu.


## 5. Conclusion

The finite-time simultaneous stabilization problem is considered for the two nonlinear singular systems or more than two nonlinear singular systems in this study. Under a suitable output feedback controller, the augmented singular system is first proved to be impulse-free and then is proved to be finite-time simultaneous stable by using the system-augmented technique. Finally, the finite-time simultaneous stabilization problem of more than two nonlinear singular systems is investigated with the same method. Moreover, the proposed method in this study can be extended to handle finite-timeH∞ simultaneous stabilization of the two or more than two nonlinear singular systems.



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