# On the Delay Interval in Which the Control Delay Systems are Stabilizable

Authors: Jiang Wei
Journal: Complexity
 (2020)

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## Abstract

In this paper, we firstly derive some criteria for considered control delay systems to be stabilizable and then get the interval of the control delay, in which the stabilizability of the systems can be kept.


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## Body

## 1. Introduction

Recently, the phenomenon of time delay has aroused many people’s attention [1–22]. In some practical systems, such as economic systems, biological systems, and space-light industry systems, due to the transmission of the signal or the mechanical transmission, the control variables must have time delay. From reference [1, 2], we can see that time delay usually destroys the stability. So, in the study of stabilizability of the control system, we must consider control delay seriously.References [1–4] give the basic theory of functional differential equations that give us a lot of useful method and tools to discuss time delay systems. In [7], Jean-Pierre Richard gives an overview of some recent advances and open problems on time delay systems.As we know, we can use the controllability to determine the stabilizability. Some scholars [2–4] and [8–11, 15, 22] have given many results on the controllability of the systems with delay.System stability theory is the foundation of stabilizability of the control system. In papers [5, 6, 15–20, 22], the authors have given some excellent results for the stability of delay systems.In this paper, we will point out that we can use the controllability of the systems with control delay to determine the stabilizability of such systems and give the method to do so. Therefore, we can use the results of references [2, 4] for the study of stabilizability of systems with control delay.In reference [5, 6, 15], John Chiasson, Jie Chen, and Jiang Wei give some results for computing the interval of delay values for stability of delay systems. These results will be useful in the study of the range of the control delay in which stabilizability of the systems can be kept.The systems we consider in this paper will be the control systems with control delay:(1)x˙t=Axt+But−h,t≥t0,xt0=x0,ut=φt,t0−h≤t≤t0,where xt∈Rn is a state vector; ut∈Rm is a control vector; u⋅ an admissible control, that is, it is contained in the square integrable functions L2 on every finite interval; A ∈Rn×n and B ∈Rn×m are constant matrices; h>0 is time delay; and φt is the initial control function.In this paper, we will consider the stabilizability for the control systems with control delay (1). Firstly, we derive some criterions to criticize the stabilization for the control delay systems and then get the interval of the control delay, in which the stabilizability of the systems can be kept.


## 2. Preliminaries

For the control systems with control delay (1), we give some preliminaries involved in this paper.Definition 1.
System (1) is said to be controllable at t1≥t0 if one can reach any state at t1 from any admissible initial state x0 and initial control φt.Lemma 1.
System (1) is controllable if and only if(2)RankB,AB,…,An−1B=n.
The proof of this Lemma can be easily gotten from Theorem1 in [4], but we must assume n1=n and n2=0 there.
If there exist a matrixK∈Rn×m, such that(3)ut=Kxt,system (1) will become(4)x˙t=Axt+BKxt−h.Definition 2.
The differential systems with delay (4) are called asymptotic stable, if there exist scalars α<0,β>0 such that, for t>t0, xt satisfy(5)xt≤βeαtx0,t≥t0.Definition 3.
The control systems with control delay (1) are said to be stabilizable, if there is a matrix K such that system (4) is asymptotically stable. The matrix K is said to be a feedback stabilization matrix.
For an entire functionGs, we define set(6)JG=△s∈CGs=0,where C is the complex plane.
For system (4), we define the entire function:(7)ps;A,BK;h=△detsI−A−BKe−hs.
From [1], we have the following.Lemma 2.
System (4) is asymptotically stable if and only if(8)Jps;A,BK;h⊆C−,where C− is the left complex plane.If system (1) is controllable, that is, (2) is true, matrices A and B are equivalent to their controllable normalized forms. In the following, if (1) are controllable, we might as well let AandB be normalized forms:(9)A=A1A12⋯A1l0A2⋯A2l⋯⋯⋯⋯00⋯Aln×n,B=b10⋯0∗0b2⋯0∗⋯⋯⋯⋯⋯00⋯bl∗n×m,where Ai,bii=1,2,…,l are the normalized forms as(10)Ai=010⋯00001⋯00⋯⋯⋯⋯⋯⋯000⋯01−ai,0−ai,1−ai,2⋯−ai,ni−2−ai,ni−1,bi=00⋮01.Let(11)K=K^0,where(12)K^=k¯1T0⋯000k¯2T⋯00⋯⋯⋯⋯⋯00⋯0k¯lT,k¯i=k1,k2,⋯,kni.Then,(13)ps;A,BK;h=∏i=1lps;Ai,bikiT;h.From (13) and Lemma 2, we have the following.Lemma 3.
If the control systems with control delay (1) are controllable, they are stabilizable if and only if(14)Jps;Ai,bikiT;h⊆C−,for every i=1,2,…,l, is true.Letd=e−sh and(15)αs,d=sn+an−1dsn−1+⋯+a0d,where aidi=1,2,⋯,n−1 are the polynomials of d.Definition 4.
Letαs,d be a two-variable polynomial of form (15) and r=degdαs,d, and we call the two-variable polynomial,(16)α˜s,d=△drα−s,1/d=−1ndrsn+dran−11/d−sn−1+⋯+dra01/d,the auxiliary polynomial of αs,d.
Supposesi,di,i=1,2,…,k be the common zeros ofαs,d,α˜s,d for which Resi=0,si≠0 and di=1,di≠1. For each such pair si,di, let hi=minh>0h∈R|di=e−hsi. Since si≠0 and di=1, such an hi exists. Define(17)h∗=min1≤i≤khi>0.
From [5], we have the following.Lemma 4.
(18)
α
s
,
e
−
h
s
≠
0
,
R
s
≥
0
,
0
≤
h
<
h
∗
,if and only if(19)αs,1≠0,Rs≥0.
Furthermore, there exists ans∗=jω∗ for which αs∗,e−h∗s∗=0.


## 3. Main Results

In this section, we will give the main results of this paper. Firstly, we give a general theorem, then forn=2, discuss in two cases, and give some results.Theorem 1.
For system (1), if they are controllable, there exists a real number h∗>0 such that they are stabilizable with delay h∈0,h∗.Proof.
From Lemma1, system (1) is controllable if and only if (2) is true. It is obvious that if (2) is true, there exists a matrix K such that(20)JdetsI−A−BK⊆C−.
Letαs,d=detsI−A−BKd and take h∗ as that in (17); then, from Lemma 4, we have that Theorem 1 is true.
From Lemma3, to discuss the stabilizability of system (1) in detail, if they are controllable, we might as well let(21)A=010⋯00001⋯00⋯⋯⋯⋯⋯⋯000⋯01−a0−a1−a2⋯−an−2−an−1,B=00⋮01.
TakeK=−k0,−k1,…,−kn−1T, where k0,k1,…,kn−1 are undetermined real numbers, and we have(22)ps;A,BKT;h=detsI−A−BKTe−sh=sn+an−1sn−1+⋯+a0+e−shkn−1sn−1+⋯+k0.
If there exist some eigenvalues ofA with the negative real part, let λ1,λ2,…,λp be all the eigenvalues of A with Reλi<0,i=1,2,…,p, then there exist real numbers a0′,a1′,…,an−p−1′ such that(23)ps;A=sn+an−1sn−1+⋯+a0=s−λ1s−λ2⋯s−λpsn−p+an−p−1′sn−p−1+⋯+a0′.
Here,sn−p+an−p−1′sn−p−1+⋯+a0′=0 and there exists only nonnegative real part roots.
Obviously, for any leading coefficient which is 1 andn−p order polynomial gs, we can set(24)sn+an−1sn−1+⋯+a0+kn−1sn−1+⋯+k0=s−λ1s−λ2⋯s−λpgs,that is,(25)s−λ1s−λ2⋯s−λpsn−p+an−p−1′sn−p−1+⋯+a0′+kn−1sn−1+⋯+k0=s−λ1s−λ2⋯s−λpgs.
Let(26)gs=sn−p+bn−p−1sn−p−1+⋯+b0,there bn−p−1,…,b0 are arbitrary constants. Then, we have(27)kn−1sn−1+⋯+k0=s−λ1s−λ2⋯s−λpgs−s−λ1s−λ2⋯s−λpsn−p+an−p−1′sn−p−1+⋯+a0′=s−λ1s−λ2⋯s−λpsn−p+bn−p−1sn−p−1+⋯+b0−s−λ1s−λ2⋯s−λpsn−p+an−p−1′sn−p−1+⋯+a0′=s−λ1s−λ2⋯s−λpbn−p−1−an−p−1′sn−p−1+⋯+b0−a0′.
Letk0′=b0−a0′,k1′=b1−a1′,…,kn−p−1′=bn−p−1−an−p−1′, then(28)kn−1sn−1+⋯+k0=s−λ1s−λ2⋯s−λpkn−p−1′sn−p−1+⋯+k0′.
From (28), we can determine real numbers k0,k1,…,kn−1.
Take (28) into (22), we have(29)ps;A,BKT;h=detsI−A−BKTe−sh=sn+an−1sn−1+⋯+a0+e−shkn−1sn−1+⋯+k0=s−λ1s−λ2⋯s−λpsn−p+an−p−1′sn−p−1+⋯+a0′+e−shs−λ1s−λ2⋯s−λpkn−p−1′sn−p−1+⋯+k0′=s−λ1s−λ2⋯s−λpsn−p+an−p−1′sn−p−1+⋯+a0′+e−shkn−p−1′sn−p−1+⋯+k0′.
Let(30)ps;A′,B′K′T;h=sn−p+an−p−1′sn−p−1+⋯+a0′+e−shkn−p−1′sn−p−1+⋯+k0′.
We can see that, if we want(31)Jps;A,BKT;h⊆C−,it is enough to make(32)Jps;A′,B′K′T;h⊆C−.
So, in the following discussion, we might as well assume thatA have only nonnegative real part eigenvalues, that is,(33)ps;A=sn+an−1sn−1+⋯+a0=0,there exists only nonnegative real part roots.
Now we discuss the stabilizability of system (1) with n=2. We discuss that in two cases.Case I.
n
=
2 and Ps,A and there exists only positive real part zeros.
If the quadratic polynomial equation(34)ps;A=s2+a1s+a0=0,there exists only positive real part roots, and we can rewrite Ps,A as(35)ps;A=s−a2+b=0,a>0,b≥0.Theorem 2.
For system (1), when n=2 and ps;A=s−a2+b=0,a>0,b≥0, if they are controllable, then they are stabilizable with delay h∈0,π/34a2+b+3a.Proof.
Forps;A=s−a2+b=0,a>0,b≥0, we have that(36)p−s;A=s+a2+b=0,and there exists only negative real part roots.
Let(37)s−a2+b+k1s+k0=s+a2+b,then k1=4aandk0=0.(38)αs,d=s−a2+b+d4as=0,(39)α˜s,d=dα−s,1/d=ds+a2+b−4as=0.
From (38), we have(40)d=−s−a2+b4as.
Taking it into (39), we have(41)s4+14a2+2bs2+a2+b2=0.
The solutions of (41) are(42)s1=7a2+b+234a2+baj=4a2+b+3aj,s2=7a2+b−234a2+baj=4a2+b−3aj,s3=−7a2+b+234a2+baj=−4a2+b+3aj,s4=−7a2+b−234a2+baj=−4a2+b−3aj.
Here,j2=−1. Taking s1,s2,s3,ands4 into (40), we have(43)d1=−4a2+b+3aj−a2+b4a4a2+b+3aj=12−32j,d2=−4a2+b−3aj−a2+b4a4a2+b−3aj=12+32j,d3=−4a2+b+3aj+a2+b−4a4a2+b+3aj=12+32j,d4=−4a2+b−3aj+a2+b−4a4a2+b−3aj=12−32j.
For(44)d1=e−hs1⟺12−32j=e−h4a2+b+3aj,⟺cosh4a2+b+3a=12,sinh4a2+b+3a=32,⟺h4a2+b+3a=2kπ+π3,k=0,1,2,….
Then, we have(45)h1=minh>0h∈Rd1=e−hs1=π34a2+b+3a.
Using the same method, we have(46)h2=minh>0h∈R∣d2=e−hs2=5π34a2+b−3a,h3=minh>0h∈R∣d3=e−hs3=π34a2+b+3a,h4=minh>0h∈R∣d3=e−hs3=5π34a2+b−3a.
So,(47)h∗=minh1,h2,h3,h4=π34a2+b+3a.
Then, we have that Theorem2 is true.Case II.
n
=
2and Ps,A has only pure imaginary zeros.
If the quadratic polynomial equation(48)ps;A=s2+a1s+a0=0,there exists only pure imaginary roots, and we can rewrite Ps,A as(49)ps;A=s2+b=0,b>0.Theorem 3.
For system (1), when n=2 and ps;A=s2+b=0,b>0, if they are controllable, then they are stabilizable with delay h∈0,π/2b+2b.Proof.
Let(50)s2+b+k1s+k0=s+b2=s2+2bs+b,then k1=2bandk0=0.(51)αs,d=s2+b+d2bs=0,(52)α˜s,d=dα−s,1/d=ds2+b−2bs=0.
From (51), we have(53)d=−s2+b2bs.
Taking it into (52), we have(54)s2+b2+4bs2=0.
The solutions of (54) are(55)s1=b+2bj,s2=−2b−bj,s3=−b+2bj,s4=2b−bj.
Taking them into (53), we have(56)d1=d2=−j,d3=d4=j.
For(57)d1=e−hs1⟺−j=e−hb+2bj,⟺coshb+2b=0,sinhb+2b=1,⟺hb+2b=2kπ+π2,k=0,1,2,….
Then, we have(58)h1=minh>0h∈Rd1=e−hs1=π2b+2b.
Using the same method, we have(59)h2=minh>0h∈R∣d2=e−hs2=3π22b−b,h3=minh>0h∈R∣d3=e−hs3=3π2b+2b,h4=minh>0h∈R∣d3=e−hs3=π22b−b.
So,(60)h∗=minh1,h2,h3,h4=π2b+2b.
Then, we have Theorem3 is true.
Now, we discuss the general case: for any positive integern, the polynomial(61)ps;A=sn+an−1sn−1+⋯+a1s+a0=0,and there exists only positive real part roots.
It is obvious that the polynomial equation(62)p^s;A=sn−an−1sn−1+⋯+−1n−1a1s+−1na0=0,and there exists only negative real part roots.
Let(63)sn+an−1sn−1+⋯+a1s+a0+kn−1sn−1+⋯+k1s+k0=sn−an−1sn−1+⋯+−1n−1a1s+−1na0,then ki=−1n−i−1aii=0,1,…,n−1.
Whenn is an even number, let n=2r, and we have(64)αs,d=s2r+a2r−1s2r−1+⋯+a1s+a0−2da2r−1s2r−1+a2r−3s2r−3+⋯+a1s=0,(65)α˜s,d=s2r−a2r−1s2r−1+⋯−a1s+a0d+2a2r−1s2r−1+a2r−3s2r−3+⋯+a1s=0.
From (64), we have(66)d=s2r+a2r−1s2r−1+⋯+a1s+a02a2r−1s2r−1+a2r−3s2r−3+⋯+a1s.
Taking it into (65), we obtain(67)s2r−a2r−1s2r−1+⋯−a1s+a0s2r+a2r−1s2r−1+⋯+a1s+a0+4a2r−1s2r−1+a2r−3s2r−3+⋯+a1s2=0.
We can prove that all the roots of (67) are pure imaginary roots, let them be si=jyi,i=1,2,…,2r; then (67) will become(68)jyi2r−a2r−1jyi2r−1+⋯−a1jyi+a0×jyi2r+a2r−1jyi2r−1+⋯+a1jyi+a0+4a2r−1jyi2r−1+a2r−3jyi2r−3+⋯+a1jyi2=0.
For(69)jyi2r−a2r−1jyi2r−1+⋯−a1jyi+a0×jyi2r+a2r−1jyi2r−1+⋯+a1jyi+a0=jyi2r+a2r−2jyi2r−2+⋯+a2jyi2+a0−a2r−1jyi2r−1+a2r−3jyi2r−3+⋯+a1jyi×jyi2r+a2r−2jyi2r−2+⋯+a2jyi2+a0+a2r−1jyi2r−1+a2r−3jyi2r−3+⋯+a1jyi.
We obtain(70)jyi2r+a2r−2jyi2r−2+⋯+a2jyi2+a02−a2r−1jyi2r−1+a2r−3jyi2r−3+⋯+a1jyi2+4a2r−1jyi2r−1+a2r−3jyi2r−3+⋯+a1jyi2=0,then(71)jyi2r+a2r−2jyi2r−2+⋯+a2jyi2+a02=−3a2r−1jyi2r−1+a2r−3jyi2r−3+⋯+a1jyi2.
So,(72)jyi2r+a2r−2jyi2r−2+⋯+a2jyi2+a0=±3ja2r−1jyi2r−1+a2r−3jyi2r−3+⋯+a1jyi.
From (66), we have(73)di=jyi2r+a2r−1jyi2r−1+⋯+a1jyi+a02a2r−1jyi2r−1+a2r−3jyi2r−3+⋯+a1jyi=12a2r−1jyi2r−1+a2r−3jyi2r−3+⋯+a1jyi×a2r−1jyi2r−1+a2r−3jyi2r−3+⋯+a1jyi+jyi2r+a2r−2jyi2r−2+⋯+a2jyi2+a0=12a2r−1jyi2r−1+a2r−3jyi2r−3+⋯+a1jyi×a2r−1jyi2r−1+a2r−3jyi2r−3+⋯+a1jyi±3ja2r−1jyi2r−1+a2r−3jyi2r−3+⋯+a1jyi=12±32j,where i=1,2,…,n.
In the same way, we can also prove that whenn is an odd number, we still have di=1/2±3/2j,i=1,2,…,n.
Fromdi=e−hsi=e−hyij and hi=minh>0h∈R∣di=e−hyij, we have hiyj=π/3 or hiyj=5π/3.
Lety∗=maxy1,y2,…,yn. It is obvious that y∗>0. Then, we have(74)h∗=minh1,h2,…,hn=π3y∗.
From the discussion above, we obtain the following.Theorem 4.
For system (1), if they are controllable and(75)ps;A=sn+an−1sn−1+⋯+a1s+a0=0,there exists only positive real part roots, and they are stabilizable with control delay h∈0,π/3y∗.Now, we give an example to illustrate the use of results.Example 1.
Consider systems(76)x1˙t=x2t,t≥0,x2˙t=−x1t+2x2t,t≥0,x10=1,x20=1,where x1t,x2t∈R.
The solutions of (76) are(77)x1t=et,t≥0,x2t=et,t≥0.
They are obviously unstable.
Adding a control with the delay term in (76), we have control systems(78)x1˙t=x2t,t≥0,x2˙t=−x1t+2x2t+ut−h,t≥0,x10=1,x20=1,ut=φt,−h≤t≤0,where x1t,x2t∈R, ut∈R, and φt∈R are the initial control function.
Comparing (78) with (1), we have(79)A=01−12,B=01.
For(80)RankB,AB=Rank0112=2,from Lemma 1, we have that system (78) is controllable.
For(81)ps;A=s1001−01−12=s−11s−2=s−12,use Theorem 2, for a=1andb=0, we have that system (78) is stabilizable with delay h∈0,π/32+3.
We get an interval0,π/32+3 of the control delay, in which the stabilizability of system (78) can be kept.
We must take notice that if the control delay is outside of interval0,π/32+3, system (78) is perhaps unstable. In fact, if we take the feedback matrix as K=0,4, the control delay h=6ln2 and system (78) is unstable. For h=6ln2=ln32>lne=1>2/3>4/31/2+3>π/3×2+3, h=6ln2 is outside of interval 0,π/32+3.
For(82)ps;A,BK;h=△detsI−A−BKe−hs=s1001−01−12−0004e−hs=s−11s−2−4e−hs=s−12−4se−hs,s=1/2 satisfies equation(83)ps;A,BK;6ln2=0.
System (78) has solutions(84)x1t=e1/2t,t≥0,x2t=e1/2t,t≥0.
They are obviously unstable.


## 4. Concluding Remarks

In practice, we often meet many unstable systems such as(85)x˙t=Axt,xt0=x0,t≥t0,where xt∈Rn is a state vector. We usually add control terms in them. But unfortunately, due to the transmission of the signal or the mechanical transmission takes a period of time, time delay may appear in control variables, and the control systems will become as system (1). The results of this paper give the methods to find the interval of the control delay, in which the stabilizability of the systems can be kept. From Example 1, we can see that, on the outside of this interval, the systems may be unstable. So, the method given in this paper has practical significance.Now, we give some Remarks to finish this paper.Remark 1.
By Lemmas1–4 and Theorem 1, system (1) is controllable and(86)ps;A=sn+an−1sn−1+⋯+a1s+a0=0.
There exists some negative real part roots, and we can also get analogous results and find the interval of control delay with which the systems are stabilizable.Remark 2.
Although Theorem4 requires system (1) to be controllable, if it is uncontrollable but the uncontrollable subsystems are stable, we can also get analogous results and find the interval of control delay with which the systems are stabilizable.Remark 3.
In Theorem4, interval 0,π/3y∗ of control delay h is obtained, with which the systems are stabilizable. But whether 0,π/3y∗ includes all delay values for which system (1) is stabilizable can be left as an open problem.Remark 4.
In papers [5, 6, 15–20], the authors have given some excellent results for the stability of delay systems. But all systems, they study, are not control systems, we cannot use their results directly to judge the stabilizability for the control systems with control delay. In comparison with these, we can see that the results of this paper are very useful.Remark 5.
We note that, in recent years, many scholars researched delayed systems with fast time-varying delay [20, 21] and gave many wonderful consequences. Usually the methods they used to deal with stability are linear matrix inequalities (LMIs). In this paper, we use the characteristic root methods to deal with stability (or the stabilizability) for the systems with time-invariant delay. If we want to use the characteristic root methods to deal with the stability (or the stabilizability) for the delayed systems with fast time-varying delay, or to use the results of this paper to the delayed systems with fast time-varying delay, we will meet more difficulties.



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