Bingxin LI,Xiangfei ZHAO,Xuefeng ZHANG,Xin ZHAO,4

1Institute of Robotics and Automatic Information System,Nankai University,Tianjin 300071,China

2Tianjin Key Laboratory of Intelligent Robotics,Nankai University,Tianjin 300071,China

3College of Sciences,Northeastern University,Shenyang 110819,China

4Shenzhen Research Institute of Nankai University,Shenzhen 518083,China

Abstract:In this paper,observer-based control for fractional-order singular systems with order α(0<α<1)and input delay is studied.On the basis of the Smith predictor and approximation error,the system with input delay is approximately equivalent to the system without input delay.Furthermore,based on the linear matrix inequality(LMI)technique,the necessary and sufficient condition of observer-based control is proposed.Since the condition is a nonstrict LMI,including the equality constraint,it will lead to some trouble when solving problems using toolbox.Thus,the strict LMI-based condition is improved in the paper.Finally,a numerical example and a direct current motor example are given to illustrate the effectiveness of the strict LMI-based condition.

Key words:Observer-based control;Singular systems;Fractional order;Input delay;Linear matrix inequality

1 Introduction

Recently,a large number of researchers have focused on fractional-order systems,which describe the model of the real-world phenomena with memory more concisely and precisely(Du and Lu,2021;Hua et al.,2021;Saffarian and Mohebbi,2021;Jiang et al.,2022).In general,stability analysis is fundamental for control systems,including fractionalorder systems.The first criterion has been proposed by Matignon(1998)for fractional-order systems with 0<α<2.Although stability can be judged through the location of poles,it is still difficult to design controllers in practice.The linear matrix inequality(LMI)technique can be better used to solve control theory,which can be expressed by convex optimization problems.Thus,to perform the stability analysis for fractional-order systems with orderα(0<α<1 or 1<α<2),the LMI-based conditions have been derived(Lu and Chen,2009,2010;Sabatier et al.,2010).

Singular systems,also referred to as descriptor systems,are a class of widespread systems(Xu et al.,2002).Many real-world systems,such as economic systems,circuit systems,and viscoelastic systems,can be more accurately described by singular systems(Xu and Lam,2006;Guerrero et al.,2021;Li YC and Ma,2021;Zhang L et al.,2021).Different from the normal system,admissibility analysis for singular systems,including regular,impulse-free,and stable ones,is the fundamental problem.Recently,we have noticed that,by introducing the fractional calculus,singular systems can describe phenomena more accurately.Therefore,on the basis of fractionalorder and singular system results,a large number of papers have been published(N’Doye et al.,2013;Jiand Qiu,2015;Marir et al.,2017;Zhang XF and Chen,2018;Marir and Chadli,2019;Wei et al.,2019;Wu et al.,2020).In these works,stability and stabilization have been widely studied(N’Doye et al.,2013;Wu et al.,2020),since they can be directly derived.In Marir et al.(2017)and Marir and Chadli(2019),the admissibility conditions were derived,and the strict LMI-based conditions were discussed further for fractional-order singular systems with 1<α<2.Moreover,the admissibility conditions were derived for 0<α<1 in Zhang XF and Chen(2018),and to improve the nonstrict LMI-based conditions,two kinds of strict LMI-based conditions were proposed.Lin et al.(2018)studied observer-based control for fractional-order singular systems with orderα(1≤α<2).It is noteworthy that the stability region ofα(0<α<1)is nonconvex.Thus,the study ofα(0<α<1)is more difficult and interesting(Zhang XF and Chen,2018).In addition,various researchers(Li RC and Zhang,2020;Udhayakumar et al.,2020)have reported the conditions of polytopic uncertainties and Takagi–Sugeno(T-S)fuzzy control for fractional-order singular systems.Delayed neutral networks have also been studied(Aghayan et al.,2021).

In practice,state estimation and observer-based controller should be required for fractional-order systems.Many efforts have been made for observerbased control(Lan et al.,2012;Lan and Zhou,2013;Li C et al.,2014;Ibrir and Bettayeb,2015;Li BX and Zhang,2016;Geng et al.,2020).In Lan et al.(2012)and Lan and Zhou(2013),the conditions of observer-based control of 1<α<2 have been proposed.Furthermore,Li C et al.(2014)investigated nonlinear systems with 0<α<2.Observer-based control of 0<α<1 has been developed in Li BX and Zhang(2016)using the singular value decomposition concept.In addition,new sufficient conditions of observer-based control have been proposed for uncertain systems of 1<α<2 and 0<α<1 in Ibrir and Bettayeb(2015).Moreover,observer-based control has been studied for fractional-order systems with input delay(Geng et al.,2020)using the Smith predictor,which is a class of effective dead-time compensators(Nguyen et al.,2021).To overcome the shortcomings,methods that include sensitivity to model errors and unstable processes have been proposed in Stamova(2014).Pu and Wang(2020)proposed the conditions to deal with state and input delay for fractional-order systems.Marir et al.(2022a)studied the bounded real lemma for fractional-order singular systems with orderα(1≤α<2),and Marir et al.(2022b)derived the conditions ofH∞static output-feedback control.Even so,H∞control of orderα(0<α<1)was rarely studied.

Although a lot of papers have been published for input delay,it is worth mentioning that most research is mainly in the field of nonsingular fractionalorder systems with input delay.For general singular systems with input delay,these were rarely studied.In short,the contributions are as follows:

1.Using the Smith predictor,observer-based control is first studied for fractional-order singular systems with orderα(0<α<1)and input delay.

2.The necessary and sufficient condition based on nonstrict LMI is presented.Then,the condition based on strict LMI is improved.

The notations used throughout the paper are as follows:R denotes the real set,ATdenotes the transpose ofA,sym(X)=X+XT,and spec(X,Y,α)is the spectrum of det(sαX?Y)=0.Moreover,det(·)and deg(·)denote the determinant and degree,respectively.

2 Preliminaries and problem formulation

2.1 Preliminaries

The Caputo definition(Du and Lu,2021)of a fractional derivative operator is given by

whereΓ(·)is the Gamma function andα∈(l?1,l).

Consider the following system:

where 0<α<1,Dαdenotes the Caputo derivative,andx(t)∈Rn,u(t)∈Rm,andy(t)∈Rpdenote the state vector,input,and output,respectively.E∈Rn×n,A∈Rn×n,B1∈Rn×m,B2∈Rn×m,andC∈Rp×nare constant matrices,and rank(E)

Before proceeding,one definition and one lemma for system(1)are introduced as follows:

Definition 1(Zhang XF and Chen,2018) The fractional-order linear singular systemEDαx(t)=Ax(t)with rank(E)πα/2(stable).

Lemma 1(Zhang XF and Chen,2018) The fractional-order linear singular systemEDαx(t)=Ax(t)with rank(E)

wherea=sin(πα/2)andb=cos(πα/2).

2.2 Problem formulation

For system(1),define the following state transformation:

In particular,ifEis the identity matrix,the following equation holds according to the results in Si-Ammour et al.(2009):

According to Geng et al.(2020),we have

From Eqs.(2)and(4),we obtain

According to Eq.(3),system(1)can be rewritten as follows:

Thus,the following observer-based controller is obtained:

We define the errore(t)=z(t)??z(t).Hence,from Eqs.(5)and(6),we have

Then,we obtain

where

Remark 1 Our aim is to design the controller in Eq.(7),and to ensure that system(9)is asymptotically admissible.According to Lemma 2.1 in Geng et al.(2020),LCB2u(s)dsdoes not affect the admissibility for systems(1)and(9).

3 Main results

In this section,the LMI-based necessary and sufficient conditions of system(9)are proposed.First,the nonstrict LMI-based condition is given in the following theorem:

Theorem 1 System(9)with matricesKandLis asymptotically admissible,if and only if there exist matricesX1,X2,Y1,Y2,Z,andRsuch that the following inequalities hold:

wherea=sin(πα/2)andb=cos(πα/2).Furthermore,LandKare given by

Proof(Sufficiency)LetZ=P1LandR=KP2.Then,according to inequalities(13)and(14),we obtain

Pre-and post-multiplying inequality(14)byand its transpose,we obtain

So,we can easily find a scalarμsatisfying

ChooseP==aX+bY;from Lemma 1,system(9)is asymptotically admissible.

(Necessity)Suppose that system(9)is asymptotically admissible.Choose a nonsingular matrix

where“?”is the unused part in the following proof.

Substituting the above equation into Lemma 1,we have

Therefore,we can obtain

Pre-and post-multiplying inequality(19)byand its tr anspose,we obtain

Then,setthus,inequality(11)can be obtained.

Thus,we have

Letting,inequality(12)can be derived.The proof is completed.

Remark 2 Theorem 1 in this study is equivalent to Theorem 3.1 in Geng et al.(2020),whenE=I;i.e.,Theorem 1 can be regarded as the extension of results of normal fractional-order systems.Inequalities(11)and(12)are nonstrict LMIs,which contain equality constraints.Due to round-offerrors in numerical calculations,equality constraints are fragile and usually cannot be well satisfied.As a result,the strict LMI-based condition is proposed in the following theorem:

Theorem 2 System(9)withKandLis asymptotically admissible,if and only if there exist matricesX1,X2,Y1,Y2,Q,Z,andRsuch that the following inequalities hold:

wherea=sin(πα/2),b=cos(πα/2),andS∈Rn×(n?r)satisfyingES=0.Here,LandKare given by

Proof Suppose that inequalities(24)and(25)hold with matricesX1,X2,Y1andY2,and thatQ∈R(n?r)×nshould satisfySQ∈Rn×n.Let

Then,substituting the above equations into inequalities(26)and(27),we can obtain

According to Theorem 1,Theorem 2 can be proved directly.

Remark 3 The condition of observer-based control for system(9)is presented based on the strict LMI in Theorem 2,which eliminates the equality constraints.Thus,the condition is less conservative and easier to solve using the LMI toolbox.

4 Simulations

In this section,two examples are given to illustrate the effectiveness of our condition.The first one is a numerical example,and the second one is a practical system of direct current(DC)motor.

4.1 Example 1

System(9)with parameters as follows is considered:

Choose,which can satisfyES=0.Then,solving inequalities(24)–(27)and Eq.(28)in Theorem 2,we can obtain the feasible solutions as follows:

Fig.1 shows that system(9)with the aboveKandLis asymptotically admissible.Fig.2 shows that the observation errors can converge to zero,and the effectiveness of the observer-based controller is verified.

Fig.1 System states z(t)(a)and observer states?z(t)(b)

Fig.2 Observer errors e(t)

Remark 4 Theorem 2 is based on the strict LMI,which avoids the computational complexity and it can be viewed as a generalization of the results of integer-order systems.Compared with the results inWu et al.(2020),Theorem 2 overcomes the problem of solving different ordersα∈(0,1)and reduces the computational cost.

4.2 Example 2

DC motor,as a kind of actuator,is used widely.In this study,the DC motor model is used as the example shown in Fig.3.

Fig.3 Block diagram of the direct current(DC)motor

The variables and inputs of the system are shown in Tables 1 and 2,respectively.

In fact,the DC motor with delayed inputs actually exists(Léchappéet al.,2016),and thusu(t?τ)is introduced to denote the delay voltage of the source.Based on the mechanical and electrical laws(Li H and Yang,2019;Lee,2020),we can obtain the following system with 0<α<1:

wherey(t)is the output,andJandbare as follows:

The parameters in Eqs.(31)and(32)are given in Table 3.

Table 1 Variables of the system

Table 2 Inputs of the system

Supposei(t)anduL(t)are measurable.Theni(t),w(t),anduL(t)can be expressed asx1(t),x2(t),andx3(t),respectively.From Eq.(31),we have

Choose the parameter settings shown in Table 4.Hence,system(33)can be expressed as follows:

We choosewhich satisfiesES=0;different combinations ofαandτcan beused to verify the applicability of Theorem 2 in this study,as shown in Tables 5 and 6.

Table 3 Parameters of the system

Table 4 Parameter settings

Table 5 Simulation results for K

Table 6 Simulation results for L

The states and observer errors of system(9)are given in Figs.4–6,using three different combinations ofαandτ.The system with the state-feedback controller can be stabilized,and observer errors of the system can fluctuate in a small range,which can illustrate that the controller design and observer design are effective.

Remark 5 According to the results of Example 2,it is shown that we can always find a set ofKandLwhich is suitable to make the system stable,no matter what values ofαandτwe take.The effectiveness of Theorem 2 proposed in this study can be verified.

5 Conclusions

The paper deals with observer-based controller design for fractional-order singular systems with 0<α<1 and input delay.Using the linear matrix inequality(LMI)technique,the necessary andsufficient condition based on the nonstrict LMI is obtained.In the case of random error,the nonstrict LMI-based condition will cause trouble.When we improve the condition based on the strict LMI,the condition is easier to handle.Finally,the numerical example and the DC motor example are given to illustrate the effectiveness of the proposed condition.

Fig.4 States z(t)(a)and errors e(t)(b)with α=0.21 and τ=0.37

Fig.5 States z(t)(a)and errors e(t)(b)with α=0.64 and τ=1.21

Fig.6 States z(t)(a)and errors e(t)(b)with α=0.98 and τ=2.35

In the future,observer-based robust control and observer-basedH∞control for fractional-order singular systems will be studied.

Contributors

Bingxin LI designed the research.Bingxin LI and Xiangfei ZHAO processed the data.Bingxin LI drafted the paper.Xuefeng ZHANG helped organize the paper.Xuefeng ZHANG and Xin ZHAO revised and finalized the paper.

Compliance with ethics guidelines

Bingxin LI,Xiangfei ZHAO,Xuefeng ZHANG,and Xin ZHAO declare that they have no conflict of interest.