Open access peer-reviewed chapter

Stability Conditions for a Class of Nonlinear Systems with Delay

Written By

Sami Elmadssia and Mohamed Benrejeb

Submitted: 05 December 2017 Reviewed: 19 March 2018 Published: 18 July 2018

DOI: 10.5772/intechopen.76600

From the Edited Volume

Nonlinear Systems - Modeling, Estimation, and Stability

Edited by Mahmut Reyhanoglu

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Abstract

This chapter presents an extension and offers a more comprehensive overview of our previous paper entitled “Stability conditions for a class of nonlinear time delay systems” published in “Nonlinear Dynamics and Systems Theory” journal. We first introduce a more complete approach of the nonlinear system stability for the single delay case. Then, we show the application of the obtained results to delayed Lur’e Postnikov systems. A state space representation of the class of system under consideration is used and a new transformation is carried out to represent the system, with delay, by an arrow form matrix. Taking advantage of this representation and applying the Kotelyanski lemma in combination with properties of M-matrices, some new sufficient stability conditions are determined. Finally, illustrative example is provided to show the easiness of using the given stability conditions.

Keywords

  • nonlinear systems
  • time delay
  • arrow matrix
  • M-matrix
  • Lur’e Postnikov
  • stability conditions

1. Introduction

Studying stability of dynamical systems with time delay has received the attention of many researchers from the control community in the past decades, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] and the references therein. Time-varying delay which varies within an interval with nonzero lower bound is encountered in a variety of engineering applications which spreads from recurrent neural networks to chemical reactors and power systems with loss-less transmission lines. It is therefore more appropriate to study stability analysis and control synthesis of these dynamical systems with time-varying delays as these delays are usually time varying in nature. There are mainly two strategies in obtaining stability conditions. We can obtain delay-independent (i.o.d) results [28, 29] and the references therein, which are applicable to delays of arbitrary size or when there is no information about the delay. In general this lack of information about the delay will result in conservative criteria, especially when the delay is relatively small. Whenever it is possible to include information on the size of the delay, we can get delay-dependent (d.d) conditions which are usually less conservative. Most of the systems described above are nonlinear in practical engineering problems. For this reason, the chapter focuses on determining easy to test sufficient stability conditions for nonlinear systems with time-varying delay [30, 31, 32, 33].

New delay dependent stability conditions are derived by employing arrow form state space representation [31, 32, 33, 34], Kotelyanski lemma and using tools from M-matrix theory and Lyapunov functional method.

The obtained results are exploited to design a state feedback controller that stabilizes Lur’e systems with time-varying delay and sector-bounded nonlinearity [26, 28, 34]. In fact, Lur’e control systems is considered as one the most important classes of nonlinear control systems and continue to be one of the important problems in control theory that has been studied widely because it has many practical applications [32, 33, 34, 35, 36].

The chapter is organized as follows: Section 2 presents the notation used throughout the chapter and some facts on M-matrices that will be needed in proving the obtained results. In sections 3 the main results are given. Application of these results to delayed nonlinear nth order all pole plant and the well-known Lur’e systems, is presented in Section 4. Illustrative example is given in Section 5 and some concluding remarks are provided in Section 6.

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2. Notation and facts

Let us fix the notation used. Let Cn=Cτ0Rn be the Banach space of continuous functions mapping the interval τ0 into Rn with the topology of uniform convergence. Let xtCn be defined by xtθ=xt+θ,θτ0 where xt=ytẏtyn1t. For a given φCn, we define φ=supτθ0φθ,φθRn. The functions ai., bi., i=1,,n1 are completely continuous mapping the set Ja×CnH×Sϖ into R, where CnH = φCnφ<H, H>0,Ja=a+,aR and Sϖ=ϖk1ϖk2/k1k2R. In the sequel, we denote txtϖ=..

Now we introduce several useful facts, including some definitions of M-matrices and the Kotelyanski lemma that will be used in subsequent parts of the chapter.

The n×n matrix A=ai,j1i,jn is called an M-matrix if the following conditions are satisfied for i=1,2,,n [34]:

  1. ai,i>0, ai,j0ijj=12n.

  2. Successive principal minors of A are positive, i.e.

deta1,1a1,iai,1ai,i>0

The matrix A is the opposite of an M-matrix if (−A) is an M-matrix. There are many equivalent conditions for characterizing an M matrix. In fact, the following definition is the most appropriate for our purposes [34].

The matrix A=ai,jni,jn is called an M-matrix if ai,i>0i=12n, ai,j0, ij, ij=12n and for any vector σR+n, the algebraic equation A'c=σ has a solution c=A'1σR+n [34].

Kotelyanski Lemma

The real parts of the eigenvalues of a matrix A, with non-negative off diagonal elements, are less than a real number μ if and only if all those of the matrix M, M=InμA, are positive, with In the n×n identity matrix [34, 35].

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3. Sufficient stability conditions

Our work consists of determining stability conditions for systems described by the following equation:

ynt+i=0n1aitxtϖyit+j=0nbjtxtϖyjtτ=utyit=φit,tτ0,i=0,,n1,E1

where τ is a constant delay and ai., bi., i=1,,n1 are nonlinear functions.

We start by representing the system (1), under another form. Using the following notation:

xi+1t=yit,i=0,,n1E2

we get:

ẋit=xi+1ti=1,,n1ẋnt=i=0n1ai.xiti=0n1bi.xitτE3

or under matrix form:

ẋt=A.xt+B.xtτE4

A. and B. are n×n matrices given by:

A.=010001a0.a1.an1.,B.=0000b0.bn1.E5

The regular basis change P transforms the original system to the new one defined by:

xt=Pzt,E6

with:

P=111α1α2αn1α1n1α2n1αn1n1001E7

The new state space representation is:

żt=F.zt+D.ztτE8

with:

F.=P1A.P=α1β1α2β2αn1βn1γ1.γ2.γn1.γn.E9

Elements of the matrix F. are defined in [33] by:

γi.=pAαi.fori=1,,n1,γn.=an1.i=1n1αiE10

where

pAs.=sn+i=0n1ai.siE11

and

βi=λαiQλλ=αifori=1,,n1E12

where

Qλ=j=1n1λαjE13

and the matrix D. is given by:

D.=P1B.P=On1,n1On1,1δ1.δn1.δn.E14

Elements of the matrix D. are defined in [18] by:

δi.=pBαi.,i=1,,n1δn.=bn1.E15

Based on this transformation and the arbitrary choice of parameters αi,i=1,,n1 which play an important role in simplifying the use of aggregate techniques, we give now the main result. Let us start by writing our system in another form. By using the Newton-Leibniz formula

xtτ=tτtẋuduE16

Equation (Eq. 8) becomes

żt=F.+D.ztD.tτtẋθE17

Let Ω be a domain of Rn, containing a neighborhood of the origin, and supJτ,Ω,Sϖ the suprema calculated for tJτi.etτ, for functions x with values in Ω, and for ϖ in Sϖ.

Next, using the special form of system (Eq. (1)) and applying the notation supJτ,Ω,Sϖ=sup., we can announce the following theorem.

The system (Eq. (1)) is asymptotically stable, if there exist distinct parameters αi<0,i=1,,n1, such that the matrix F˜. is the opposite of an M-matrix, where F˜. is given by

F˜.=α1β1α2β2αn1βn1γ˜1.γ˜2.γ˜n1.γ˜n.E18

and the elements γ˜i.,i=1,,n, are given by

γ˜i.=γi.+δi.+ταisup.δi.1τsup.δn.,i=1,,n1γ˜n.=γn.+δn.+τsup.δn.γn.+δn.1τsup.δn.+i=1nτβisup.δi.1τsup.δn.E19

Proof:

We use the following vector norm pz=p1zp2zp3zpnz, where

piz=zi,i=1,,n1pnz=zn+i=1nsup.δi.1τsup.δn.τ0t+θtżiϑdϑdθE20

with the condition

τsup.δn.<1E21

Let Vt be a radially unbounded Lyapunov function given by (Eq. (22)).

Vt=pztw=i=1nwipiztE22

where wR+n,wi>0,i=1,,n. First, note that

Vt0i=1nwizit0+wnznt0+sup.δn.1τsup.δn.supτ0φ̇nτ22r<+

and

Vti=1nwizit

The right Dini derivative of Vt, along the solution of (Eq. (22)), gives

D+Vt=i=1nwid+piztdt+E23

For clarification reasons, each element of d+piztdt+, i = 1, …, n is calculated separately. Let us begin with the first n1 elements. Because zi=zisignzi, we can write, for i=1,,n1,

d+piztdt+=d+zitdt+=d+zitdt+signzit=αizit+βizntsignzitαizit+βizntE24

and

d+pnzdt+=d+zndt++i=1nsup.δi.1τsup.δn.d+dt+τ0t+θtżiυdυE25

because

i=1nsup.δi.1τsup.δn.d+dt+τ0t+θtżiϑdϑdθ=i=1nsup.δi.1τsup.δn.τżittτtżiϑ

and

d+zntdt+γn.+δn.znt+i=1n1γi.+δi.zit+i=1nsup.δi.tτtżiθ

Finally, it is easy to see that equation (Eq. (25)) can be overvalued by the following one

d+pnzdt+i=1nγ˜i.zi

Then we obtain the following inequality

D+Vt<F˜.ztwE26

where zt=z1tznt, and

F˜.=α1β1α2β2αn1βn1γ˜1.γ˜2.γ˜n1.γ˜n.E27

Because the nonlinear elements of F˜. are isolated in the last row, the eigenvector vtxtϖ relative to the eigenvalue λm is constant [34, 35], where λm is such that Reλm=maxiReλiλiλF˜.. Then, in order to have D+Vt<0, it is sufficient to have F˜. as the opposite of an M-matrix. Indeed, according to properties of M-matrices, we have σR+n,wR+n such that F˜.1σ=w. This enables us to write the following equation

D+Vt<F˜.ztw=ztF˜.w=ztσ=i=1nσizit<0E28

This completes the proof of theorem.

The system (Eq. (1)) is asymptotically stable, if there exist distinct parameters αi<0,i=1,,n1, such that the following condition:

μ.+2τν.ξ.<0E29

is satisfied.

where:

μ.=γn.+δn.+τsup.δn.γn.+δn.γn.+δn.ν.=i=1n1βisup.δi.ξ.=i=1n1γi.+δi.βiαi+E30

Proof:

Basing on definition 1 and definition 2, the choice of αk<0, k=1,,n1, αiαj for ij, the condition of signs on the principal minors is as follows

detα100αi>0,i=123n1E31

and

detF˜.=γ˜n.i=1n1γ˜i.βiαii=1n1αi>0E32

which yields to the following condition

γ˜n.i=1n1γ˜i.βiαi<0E33

Replacing each term in (Eq. (33)) of by its expression we get

γ˜n.i=1n1γ˜i.βiαiγn.+δn.+τsup.δn.γn.+δn.1τsup.δn.+τi=1n1βisup.δi.1τsup.δn.i=1n1γi.+δi.+ταisup.δi.βi1τsup.δn.αi=12sup.δn.γn.+δn.+τi=1n1βisup.δi.i=1n1γi.+δi.ταisup.δi.βiαi

which can be re-written as:

μ.+τν.i=1n1γi.+δi.βiαii=1n1ταisup.δi.βiαi=μ.+τν.ξ.+τν.=μ.+2τν.ξ.

where:

μ.=(12τsup.δn.γn.+δn.ν.=i=1n1βisup.δi.ξ.=i=1n1γi.+δi.βiαi

which completes the proof.

If the couple pAs.+pBs.Qs forms a positive pair, then there exist distinct negative parameters αi,i=1,,n1, verifying the condition γi.+δi.βi>0 for i=1,,n1.

Using Theorem 2.1 and Remark 2.1, the obtained supremum of time delay is a function of αi values, i=1,,n1. As a result, a sufficient condition for asymptotic stability of our system is when values of the time delay are less than this supremum.

If the couple Ds.+Ns.Qs forms a positive pair and there exist distinct negative parameters αi,i=1,,n1, such that:

2τγn.+δn.sup.δn.ν.+D0.+N0.Q0>0E34

then the system (Eq. (1)) is asymptotically stable.

Proof.

According to Remark 2.1, we find that

γn.+δn.j=1n1γj.+δj.βjαj=γn.+δn.j=1n1γj.+δj.βjαj=D0.+N0.Q0

The result of Theorem 2.1 becomes

2τγn.+δn.sup.δn.ν.+D0.+N0.Q0>0

This completes the proof of corollary.

  • Theorem 2.1 depends on the new basis change, where parameters αi of the matrix P are arbitrary chosen such that matrix T. is the opposite of an M-matrix. The appropriate choice of the set of free parameters αi makes the given stability conditions satisfied.

  • The theorem takes into account the fact that delayed terms may stabilize our system. Theorem 2.1 can hold even if pAs. is unstable. This is another advantage as the majority of previously published results assume that pAs. is linear and stable.

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4. Application to delayed nonlinear nth order all pole plant

Consider the complex system S given in Figure 1.

Figure 1.

Block diagram of studied system.

Ds=pAs defined by (Eq. (11)) and pBs=1, respectively. In this case f˜i. are constants and g is a function satisfying the finite sector condition.

Let ĝ be a function defined as follows

ĝeθyθ=geθyθeθyθ,eθyθθ[τ+[E35
sup.ĝetyt=g¯R+.

The presence of delay in the system of Figure 1 makes stability study difficult. The following steps show how to represent this system in the form of system (Eq. (1)). Then we can write

ynt+i=0n1aidiytdti=ĝetτytτytτ+ĝetτytτetτ.

Using the following notation ĝ.=ĝetτbxtτ, therefore

ynt+i=0n1aiyit+ĝ.ytτ=ĝ.etτ.E36

It is clear that system (Eq. (36)) is equivalent to system (Eq. (1)) in the special cases eθ=0 and eθ=Kxθ, xt=ytẏtynt', θ[τ+[. We will now consider each case separately.

4.1. Case et=0

In case, et=0t[τ+[, the description of the system becomes

ynt+i=0n1aiyit+ĝ.ytτ=0.

This is a special representation of system (Eq. (1)) where f˜i.=ai, g˜1.=ĝ. g˜i.=0 i=2,,n1, Ds.=Ds, Ns.=ĝ., γn.=γn=an1i=1n1αi and δn.=0.

A sufficient stability condition for this system is given in the following proposition.

If there exist distinct αi<0 i=1,,n1, such that the following conditions

γn<0μ1.+2τν1.ξ1.<0E37

where

μ1.=γnν1.=g¯ξ1.=Dα1+ĝ.β1α1+i=2n1DαiβiαiE38

are satisfied. Then the system S is asymptotically stable.

Suppose that Ds admits n distinct real roots pi,i=1,,n among which there are n1 negative ones. By using the fact that an1=i=1npi, then the choice αi=pi, i=1,..,n2 and αn1=pn1+ε permit us to write γn=an1i=1n1pi=pnε. In this case the last proposition becomes.

If Ds admits n1 distinct real negative roots such that the following conditions

pnε<0μ2.+2τν2.ξ2.<0E39

are satisfied, where

μ2.=pnεν2.=g¯ξ2.=ĝ.β1α1+Dαn1βn1αn1E40

then the system S is asymptotically stable.

4.2. Case et=Kxt

In this case, take et=Kxt with K=k0k1kn1, then the obtained system has the same form as (Eq. (1)), with ĝ1K.=ĝK.k0+1 and ĝiK.=ĝK.ki1, i=2,,n.

The stabilizing values of K can be obtained by making the following changes:

γn=an1i=1n1αi,δnK.=ĝK.kn1,ν1K.=g¯Ki=1n1N˜αiwhereg¯K=sup.ĝK.andN˜α=1+k0+i=1n1bi+kiαi.

If there exist distinct αi<0, i=1,,n1, such that the following conditions

γnĝK.kn1<0τ<12g¯Kkn1μ1K.+2τν1K.ξ1k.<0E41

where

μ1K.=12g¯Kτkn1γn+δnK.ν1K.=g¯Ki=1n1βiN˜αiξ1K.=i=1n1Dαi+ĝK.N˜αiβiαiE42

are satisfied. Then the system S is asymptotically stable.

By a special choice of K the result of proposition 3.3 can be simplified. In fact, if the conditions of this proposition are verified we can choose the vector K such that Dpi=N˜pi. In this case we obtain Dpi=N˜pi=0, i=1,,n1 and ν1.=ξ1.=0 which yields the following new proposition.

If Ds admits n1 distinct real negative roots pi such that the following conditions are satisfied.

γnĝK.kn1<0τ<12g¯Kkn1μ1K.<0E43

Then the system S is asymptotically stable.

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5. Illustrative example

Let us study the same example in [34] defined by Figure 2 which refer to the dynamics of a time-delayed DC motor speed control system with nonlinear gain, Block diagram of time-delayed DC motor speed control system with nonlinear gain.

Figure 2.

Delayed nonlinear model of DC motor speed control.

where:

  • p1=1Te and p2=1Tm where TeandTm are, respectively, electrical constant and mechanical constant.

  • τf presents the feedback delay between the output and the controller. This delay represents the measurement and communication delays (sensor-to-controller delay).

  • τc the controller processing and communication delay (controller-to-actuator delay) is placed in the feedforward part between the controller and the DC motor.

  • g.:RR is a function that represents a nonlinear gain.

The process of Figure 2 can also be modeled by Figure 1, where τ=τf+τc.

It is clear that model of Figure 2 is a particular form of delayed Lurie system in the case where Ds=ss+p1s+p2=s3+p1+p2s2+p1p2s and Ns=1. Thereafter, applying the result of Theorem 2.1, a stability condition of the system is that the matrix T. given by:

T.=α10α1α210α2α2α11t1.t2.t3.

where:

t1.=γ1+ĝ.+τα1g¯,t2.=γ2,t3.=γ3+τβ1g¯

must be the opposite of an M-matrix. By choosing αi, i=1,2, negative real and distinct, we get the following stability condition:

γ3+2τβ1g¯β1γ1+ĝ.α1β2γ2α2<0

For the particular choice of α1=p1 and α2=p2+ε, ε>0.

yields β1=β2=ε+p1p21 and we obtain the new stability condition:

2τg¯+p11ĝ.+α21Dα2<εε+p1p2

Assume that we have this inequality g¯<Dα2, we can find from \ref.{ops} the stabilizing delay given by the following condition:

τ<12εε+p1p2Dα2p11α21E44

By applying the control et=Kxt with K=k0k1k2, we can determine the stabilizing values of K can be obtained by making the following changes:

γ3=p1+p2i=12αi,δ1K.=ĝK.k0+1,δiK.=ĝK.ki1,i=2,3
ν1K.=g¯Ki=12βiN˜αiwhereg¯K=sup.ĝK.andN˜α=1+k0+i=12kiαi

If we choose αi<0, i=1,2, such that the following conditions

Dαi=N˜αi=0,,i=1,2

we get

1+k0k2=p1+p2,k1k2=p1p2

and from proposition 3.3 the stabilizing gain values satisfying the following relations:

0g¯K.k2<0k2<12τg¯KE45

Finally we find the domain of stabilizing k0,k1,k2 as follows:

0<k2<12τg¯Kk1=p1p2k2andk0=p1+p2k21E46
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6. Conclusion

In this chapter, a joined and structured procedure for the analysis of delayed nonlinear systems is proven. A complete structured analysis formulation based on the comparison principle and vector norms for the asymptotic stability is presented. Based on the arrow form matrices, and by taking into account for the system parameters, a new stability conditions are synthesized, leading to a practical estimation of the stability domain. In order to highlight the feasibility and the main capabilities of the proposed approach, the case of nonlinear nth order all pole plant and delayed Lur’e Postnikov systems are presented and discussed. In addition, the simplicity of the application of these criteria is demonstrated on model of time-delayed DC motor speed control.

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Written By

Sami Elmadssia and Mohamed Benrejeb

Submitted: 05 December 2017 Reviewed: 19 March 2018 Published: 18 July 2018