Stabilization of neutral type indirect control systems to absolute stability state

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R E S E A R C H

Open Access

Stabilization of neutral-type indirect control

systems to absolute stability state

Andriy Shatyrko

1

_{, Ronald RP van Nooijen}

2*

_{, Alla Kolechkina}

2

_{and Denys Khusainov}

1

*_{Correspondence:}

[email protected]

2_{Faculty of Civil Engineering and}

Geo Sciences, Delft University of Technology, Stevinweg 1, Delft, 2628 CN, Netherlands Full list of author information is available at the end of the article

Abstract

This paper provides suﬃcient conditions for absolute stability of an indirect control Lur’e problem of neutral type. The conditions are derived using a Lyapunov-Krasovskii functional and are given in terms of a system of matrix algebraic inequalities. From these matrix inequalities a suﬃcient condition for linear state feedback stabilizability follows.

MSC: 34H15; 34K20; 93C10; 93D05

Keywords: Lyapunov-Krasovskii functional; stabilization; absolute stability; neutral-type time-delay argument

1 Introduction

The problem of absolute stability is often encountered in engineering practice. One spe-cific form of this problem is the indirect control Lur’e problem, where the system to be controlled is linear, but the control action is the output of a nonlinear scalar system that itself receives output feedback. The special case where the output of the controller is a nonlinear function of one variable whose graph lies between two lines in the first and third quadrants of the coordinate plane is usually studied. Initially only systems of ordi-nary differential equations were considered; see for example [–]. A historical overview of the absolute stability problem can be found in [] or in the introduction of [].

In practical control processes time delays are common and they often cause instabilities, as a result, the absolute stability problem of nonlinear control systems with delay has at-tracted a lot of interest [, , –]. Nonlinear systems of neutral type with indirect control are considered in [–]. Sufficient conditions for absolute stability for such systems are derived in [, ] by the direct Lyapunov method using Lyapunov-Krasovskii functionals, these conditions are given in Theorem  of this paper. The functionals are constructed by taking the sum of a quadratic form of the current coordinates, integrals over the delay of quadratic forms of the state and its derivative, and the integral of the nonlinear compo-nents of the considered system [, , –]. All results from [, , ] can be put into a unified form in terms of matrix algebraic inequalities. A very different approach is given in [, ] or [, ], where integral operators are used.

In this paper we also consider what to do if absolute stability of the system under inves-tigation cannot be established using the result given in Theorem . There are two obvious options: either change the method of investigation or change the Lyapunov function or functional. But there is a third option: we can try to add a linear state feedback to stabilize

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the closed loop system for the previously chosen Lyapunov function or functional. There are some interesting papers devoted to the investigation of stability and stabilization tasks [–].

The present article is a direct extension of []. The remainder of this paper is organized as follows. In Section  the absolute stability problem of neutral type indirect nonlinear control system is formulated, some notation is deﬁned, and a result from [, ] is stated. Section  introduces the concepts of stability and stabilization with respect to a given functional for the case of a linear control system with delay. In Section  the scalar case of a neutral system with nonlinear indirect control is treated. The indirect control system of neutral type in the general matrix form is considered in Section . Finally, some conclusion are drawn in Section .

2 Problem formulation and preliminaries

In this paperR+_= [,∞),Rnis then-dimensional vector space over the real numbers; Rm×n_{will be used for the set of all}_m_×_n_{matrices, I}

n×nis then×nidentity matrix; m×n

is anm×nmatrix ﬁlled with zeros; a superscript T marks the transpose of a vector or a matrix; andek,nis the unit vector along thekth coordinate direction in ann-dimensional

space. Subscriptsnandn×n, which indicate the dimension of the space or the matrix, will be dropped whenever they are clear from the context. The Euclidean norm of a vector

a∈Rn_{will be written as}_|_a_|_{, so}

|a|= _n

i= a_i

 

and for a square matrixA∈Rn×n,|A|will be the operator norm induced by the Euclidean vector norm. Recall that

|A|=λmaxATA

 _,

where λmax is the largest eigenvalue of ATA. We will write Cn,τ for the Banach space

C([–τ, ],Rn) of continuous functions from [–τ, ] toRnwith norm

x∞= sup

s∈[–τ,]

_x₍_s₎

and useC

n,τ =C([–τ, ],Rn) for the Banach spaceC([–τ, ],Rn) of continuous functions

from [–τ, ] toRn_{with a continuous derivative with norm}

x∞,= sup

s∈[–τ,]

x(s),x˙(s) .

We will also need the time shift operator, which operates on time dependent functions and is given by

Ttx=s→x(s+t).

For a functionf with domainX, the functiongwith domainY ⊂Xthat coincides with

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delay, we will use the abbreviated notationxtfor the time shifted functionx, restricted to

the domain [–τ, ], so

xt=Ttx|[–τ,].

In this paper we will consider a Lur’e system of neutral type with indirect control,

d dt

x(t) –Dx(t–τ)=Ax(t) +Ax(t–τ) +bfσ(t), t≥t, ()

d dtσ(t) =c

T_x₍_t_{) –}_ρ_f_σ₍_t₎_, _t_≥_t_, _()

xt=φ ()

with φ ∈Cn,τ, A,A,D∈Rn×n, b,c∈Rn,ρ,τ ∈R, f ∈C(R,R) such thatρ> , τ > , |D|< , and

kσ≤σf(σ)≤kσ, ()

wherek,k∈Randk>k> . This is a special case of the more general autonomous

neutral functional-diﬀerential equation

d dt

x(t) –Dx(t–τ)=F(xt), ()

whereD∈Rn×n_and_F_∈_C₍_C

n,τ,Rn) with initial condition

xt=φ, ()

whereφ∈C_n_,_τ. If we need to refer to a speciﬁc solution of () with () then we will use the notationxt,φ.

Deﬁnition  A pair (x,σ)∈C([t–τ,∞),Rn)×C([t,∞),R) is asolution of (), (), ()

on [t,∞) ifxsatisﬁes () and the pair satisﬁes the system () and ().

Evidently, as discussed in [], p., there are obviously two families of metrics or mea-sures for stability in this case, one based onxalone and another based onxand its deriva-tive. A general theory of stability in two metrics or measures was ﬁrst given by [] and extended by []; see also [, ]. We use the deﬁnition of measure given in [].

Deﬁnition  A functionh∈C(R+

×X,R+), whereXis a Banach space, is called a measure

inXif

inf

(t,x)∈R×Xh(t,x) = 

and the set of all measures inXis denoted by(X).

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Deﬁnition  For givenh∈(C

n,τ) andh∈(Cn,τ) the solutionxt,φof () with () is (h,h)stableif

∀> ∃δ> ∀ψ∈Cn,τ :h(t,φ–ψ)≤δ⇒h(t,xt,φt–xt,ψt)≤.

Deﬁnition  For givenh∈(Cn,τ) andh∈(Cn,τ) the solutionxt,φof () with () is

(h,h)asymptotically stableif it is (h,h)stableand

∃δ> ∀> ∃T>t∀t≥T∀ψ∈Cn,τ :

h(t,φ–ψ)≤δ⇒h(t,xt,φt–xt,ψt)≤

or equivalently if it is (h,h)stableand

∃δ> ∀ψ∈Cn,τ:h(t,φ–ψ)≤δ⇒ lim

t→∞h(t,xt,φt–xt,ψt)→.

Deﬁnition  For givenh∈(C_n_,τ) andh∈(Cn,τ) the solutionxt,φof () with () is (h,h)exponentially stable(after for instance [, ]) if

∃ρ> ∃K> ∃λ> ∀t≥T∀ψ∈Cn,τ:

h(t,φ–ψ)≤ρ⇒h(t,xt,φt–xt,ψt)≤Kh(t,φ–ψ)e–λ(t–t).

Deﬁnition  For givenh∈(Cn,τ) andh∈(Cn,τ) the solutionxt,φof () with () is

(h,h)globally asymptotically stableif

∀> ∀ψ∈Cn,τ∃T>t∀t≥T:h(t,xt,φt–xt,ψt)≤.

Deﬁnition  We call the zero solutionx:t→n×,σ:t→ of (), ()stableif it is (h,h)

stable forh(t,φ) =φ∞andh(t,xt,σt) =

|xt()|+|σt()|.

Deﬁnition  We call the zero solution x:t→n×, σ : t→ of (), () asymptot-ically stable if it is (h,h) asymptotically stable for h(t,φ) =φ∞ andh(t,xt,σt) =

|xt()|+|σt()|.

Deﬁnition  We call the zero solution x : t →n×, σ :t →  of (), () globally asymptotically stableif it is (h,h) globally asymptotically stable forh(t,φ) =φ∞and

h(t,xt,σt) =

|xt()|+|σt()|.

Deﬁnition  We call the zero solutionx:t→n×,σ:t→ of (), ()globally asymp-totically stable in metricC_{if it is (}_h_,_h_{) globally asymptotically stable for}_h₍_t_,_φ_{) =}_φ_∞

and

ht,xt,σt

=maxxt()



+σt()



,x˙t()



+σ˙t()



.

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To investigate the system (), () we use a Lyapunov-Krasovskii functional of the form

V[x,σ,t] =xT(t)Hx(t) +

t s=t–τ

e–ζ(t–s)_xT₍_s₎_G

x(s) +x˙T(s)Gx˙(s) ds

+β σ(t)

w=

f(w)dw, ()

whereH,G,G∈Rn×nandβ,γ ∈R,β> ,ζ> .

We deﬁne the matrix

S[A,A,b,c,ρ,τ,H,G,G,β,ζ] = ⎡ ⎢ ⎢ ⎢ ⎣

S S S S ST_ S S S ST

 ST S S ST

 ST ST S

⎤ ⎥ ⎥ ⎥

⎦, ()

where

S= –AT_H–HA–G–AT_GA, S= –HA–AT_GA,

S= –HD–AT_GD, S= –Hb–AT_Gb– βc,

S=e–ζ τG–ATGA, S=AGD, S= –ATGb, S=e–ζ τG–DTGD, S= –DTGb, S=βρ–bTGb.

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In [, ] a general theorem was proved, that provided suﬃcient conditions for absolute stability and estimates of the exponential decay for the solutions of the system (), (), when the elements of the matricesA andAwere only known to lie in given intervals. WhenAandAare known exactly the following theorem follows immediately.

Theorem  Let|D|< ,ρ,τ> and suppose that there exist positive deﬁnite matrices G,

G,H,and constantsζ > ,β> such that the matrix S[A,A,b,c,ρ,τ,H,G,G,β,ζ]is positive deﬁnite.In that case the system(), ()is absolutely stable in metricwith respect to the metric deﬁned earlier forC_.

Corollary  Let|D|< ,ρ,τ> and suppose that there exist positive deﬁnite matrices G, G,H,and constants <λ< ,β> such that the matrixS˜[A,A,b,c,ρ,τ,H,G,G,β,λ]

given by Sijfor(i,j) /∈ {(, ), (, )}andS˜ =λG–ATGA,S˜ =λG–DTGD is positive deﬁnite.In that case the system(), ()is absolutely stable in metricin metricC _{for all} ﬁnite delaysτ.

Proof For eachτthis follows from Theorem  by takingζ=τ–_log_λ_.

Note  In this corollary there are no conditions on the delay other thanτ> .

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new type of stability. The deﬁnition is based on the inequality

d

dtV[x,t]≤–γV[x,t]. ()

Deﬁnition  A system isstable with respect to the functional V with exponentγ >  if inequality () holds for the total derivative of the functionalV[x,t] along any solution of

x:t→x(t) of the system.

For some systems it can be proﬁtable to examine the possibility of stabilizing the system by allowing a speciﬁc type of linear state feedback.

Deﬁnition  A system isstabilizable with respect to functional V and state feedback of a given typeif the adding state feedback of that type results in a system that is stable with respect to the functionalVwith exponentγ > .

To illustrate the use of these definitions in the next two sections we will apply these definitions first in the case of a linear system with delay and then in the case of a scalar nonlinear neutral system with indirect control.

3 A Lyapunov-Krasovkii functional approach to a linear problem with delay

Before considering the general problem of stabilization of nonlinear control systems, an example of a linear control system with delay is used to introduce the concept of stability and stabilization with respect to a given functional and to demonstrate the methodology. Let us consider the control system

˙

x(t) =Ax(t) +Ax(t–τ) +bu(t) ()

withA,A∈Rn×n_,_b_,_c_∈_Rn_{, and}_u₍_t_{) is a scalar function and}_τ_{>  is constant. To}

inves-tigate the system () we use a Lyapunov-Krasovskii functional of the form

Vx(t)=xT(t)Hx(t) + 

s=–τ

eγsxT(t+s)Gx(t+s)ds, ()

whereH,G∈Rn×n_and_γ _∈_R_,_ζ_{> . We will consider controls of the form}

u(t) =cTx(t) +dTx(t–τ), ()

wherec,d∈Rn_{. First, let us consider stability with respect to the functional ().}

Theorem  Consider()for b= and with given A,A.Let there be positive deﬁnite matrices G and H and a constantγ > such that the matrix

M[A,A,G,H,γ,τ]

=

–(AT

H+HA+G+γH) –HA

–AT_H e–γ τ_G

()

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Proof Letx(t) be a solution of (). We introduce the vector

y(t) =

x(t)

x(t–τ)

.

We can now write

x(t) = [I ]y(t), x(t–τ) = [ I]y(t), x˙(t) = [A A]y(t). ()

To show that the system () is stable with respect to functional () we need to show that () holds. For this we need to take the derivative ofV[x(t)]:

d dtV

x(t)=x˙T(t)Hx(t) +xT(t)Hx˙(t)

+ d

dt

t

ξ=t–τ

e–γ(t–ξ)xT(ξ)Gx(ξ)dξ

. ()

The terms containingHin () can be rewritten in terms ofy(t) by using ()

˙

xT(t)Hx(t) +xT(t)Hx˙(t) =yT(t)

(AT

H+HA) HA AT

H 

y(t). ()

To rewrite the terms in () containing the integral we will use

Vx(t)–xT(t)Hx(t) = t

s=t–τ

eγ(s–t)xT(s)Gx(s)ds

and

d dt

t s=t–τ

eγ(s–t)_g₍_s₎_ds_{= –}_γ

t s=t–τ

eγ(s–t)_g₍_s₎_ds₊_g₍_t_{) –}_e–γ τ_g₍_t_–_τ_).

If we insertg(t) =xT₍_s₎_Gx₍_s_{) then this results in}

d dt

Vx(t)–xT(t)Hx(t)

= (–γ)Vx(t)–xT(t)Hx(t) +xT(t)Gx(t) +e–γ τ–xT(t–τ)Gx(t–τ),

which with the aid of () can be put into matrix form

d dt

Vx(t)–xT(t)Hx(t)

= (–γ)Vx(t)–xT(t)Hx(t)+yT(t)

G 

 –e–γ τ_G

y(t).

If we combine these results, then we get

d dtV

x(t)= –yT₍_t₎_My₍_t_{) –}_γ_V_x₍_t₎

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Example  If in Theorem  we take

A=

–     –

; A=  ,

–   –

; τ = ,

then the conditions of the theorem are satisﬁed when we take

G=

 – – 

, H=

 – – 

, γ = .

Corollary  Let there be positive deﬁnite matrices H and G,vectors c and d,and a constant

γ > such that the matrix

MA+bcT,A+bdT,G,H

is also positive deﬁnite.In that case the system()is stabilizable with respect to functional

()with state feedback of type(),matrices G,H,and exponentγ.

Proof This follows immediately from Theorem .

Corollary  If the pair(A,b)is controllable and

R=b Ab Ab · · · An–b An–b

anddet(λI–A) =λn₊_p_λn–₊_{· · ·}₊_p

nand we deﬁne

˜ A=

(n–)×

I(n–)×(n–)

–p

=R–AR,

˜

A=R–AR, H˜ =RTHR, G˜=RTGR, ˜

b=R–b=en, c˜=RTc, d˜=RTd,

and the matrix

˜ M=

–(A˜ +_γI)TH˜ –H˜(A˜ +_γI) –G –HA˜

–AT

H e–γ τG

is positive deﬁnite,then the system()is stabilizable with respect to functional()with state feedback of type(),matrices G,H,and exponentγ.

Proof If we apply the change of basisy(t) =R–_x₍_t_{), then this corollary follows immediately}

from the previous corollary.

4 A scalar Lur’e system of neutral type with indirect control

Let us consider an indirect control system of neutral type described by a two scalar equa-tions

d dt

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d

dtσ(t) =cx(t) –ρf

σ(t), ()

wheret≥t≥,xis the state function,σ is the control deﬁned on [t,∞),a,a,b,c, – <d< ,ρ> ,τ>  are constants, andf is a continuous function onRthat satisﬁes the sector condition ().

For this case the Lyapunov-Krasovskii functional () can be written as

Vx(t),σ(t),t=h·x(t)

+ t

s=t–τ

e–ζ(t–s)gx(t)+gx˙(s) ds

+ σ(t)

w=

f(w)dw,

whereh> ,g> ,g> ,ζ >  are constants, (x,σ) is a solution of (), (), andt≥t≥

. We deﬁne

s= –ah–g–a_g, s= –ah–aag, s= –hd–adg,

s= –hb–ag, s=e–ζ τg–ag, s= –agd, s= –agb,

s=e–ζ τ–dg, s= –dgb, s=βρ–bg,

and the symmetric matrix

S[a,a,b,c,ρ,h,g,g,β,ζ] = ⎡ ⎢ ⎢ ⎢ ⎣

s s s s s s s s s s s s s s s s

⎤ ⎥ ⎥ ⎥ ⎦,

wheresij=sji. Our ﬁrst result is a theorem on the absolute stability for the system (),

().

Theorem  If there exist constants h> ,g> ,g> ,β> ,ζ > such that the matrix S[a,a,b,c,ρ,h,g,g,β,ζ]is positive deﬁnite,then the system(), ()is absolutely stable.

Proof The proof of this theorem follows directly from Theorem .

Example  If in Theorem  we take

a= –, a= 

, d= – 

, b= 

, c= 

, ρ= , τ= ,

then the conditions of the theorem are satisﬁed for

g=



, g= 

, h= 

, β= 

, ζ=  .

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are positive, that is,

s> , ()

ss– (s)> , ()

det ⎛ ⎜ ⎝

s s s s s s s s s

⎞ ⎟

⎠> , ()

det(S) > . ()

From inequalities () to () we can determine whether or not the matrixSis positive deﬁnite. If it is then the system (), () is absolutely stable. Another approach is based on the lemma on the properties of block matrices given below.

Lemma  Let A be a regular n×n matrix,B be an n×q matrix,and C be a regular q×q

matrix.Let a Hermitian matrix S be represented as

S=

A B

B∗ C

.

This matrix S is positive deﬁnite if and only if the matrices A and

C–B∗A–B

are positive deﬁnite.Here B∗denotes the Hermitian adjoint.

Proof See [], Theorem ..

Now we can use this to formulate another set of stability conditions.

Theorem  For S=S[a,a,b,c,ρ,h,g,g,β,ζ]let

W= [I× ×]S

I×

×

,

W= [× I×]S

×

I×

,

W= [I× ×]S

×

I×

,

and suppose there exist constants h> ,g> ,g> ,β> ,ζ > such that the inequalities

(), ()hold and the matrix

W–W_TW_–W ()

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Proof According to Lemma ,Sis positive deﬁnite if and only ifWandW–WT

W–W

are positive deﬁnite. This completes the proof.

The crucial assumption in Theorem  is the assumption of positive deﬁniteness of the matrixS[a,a,b,c,ρ,h,g,g,β,ζ]. If we cannot ﬁnd suitable constantsc> ,h> ,g> ,

g> ,β> ,ζ >  to ensure positive deﬁniteness, then we cannot apply Theorem . If that is the case, then we can consider modiﬁcation of the control function in (), () by adding a linear combination of the state attand att–τ

d dt

x(t) –dx(t–τ)=ax(t) +ax(t–τ) +bfσ(t)+u(t), ()

d

dtσ(t) =cx(t) –ρf

σ(t)+v(t), ()

where

u(t) =cx(t) +cx(t–τ),

v(t) =cx(t),

andc,c, andcare suitable constants. Inserting the deﬁnitions ofuandvin system (), () results in

d dt

x(t) –dx(t–τ)= (a+c)x(t) + (a+c)x(t–τ) +bf

σ(t), ()

d

dtσ(t) = (c+c)x(t) –ρf

σ(t). ()

In this case the matrix of the total derivative takes of the functional along the solution will be of the form

S[a+c,a+c,b,c+c,ρ,h,g,g,β,ζ].

To stabilize the system we need to ﬁndc,c, andcsuch that

S[a+c,a+c,b,c+c,ρ,h,g,g,β,ζ]

is positive deﬁnite. We can now either use the Sylvester criterion [] and look forc,c, andcsuch that the leading principal minors ofSare positive or use Lemma  by deﬁning

W= [I× ×]S

I×

×

,

W= [× I×]S

×

I×

,

W= [I× ×]S

×

I×

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where

S=S[a+c,a+c,b,c+c,ρ,h,g,g,β,ζ]

and look forc,c, andcsuch that the matricesSandS–ST

S–Sare positive deﬁnite.

5 Stabilization

Let us return to our original system (), (). According to Theorem  for absolute stability of the system (), () we need the matrix

S[A,A,b,c,ρ,τ,H,G,G,β,ζ]

to be positive definite. From the Sylvester criterion [] it follows that we can verify that the matrix is positive definite by calculating its leading principal minors, that is, by verifying the positivity of n+  determinants. Using the results of Lemma  we will give another set of absolute stability conditions. To do so we give names to specific blocks in matrix () as follows:

Theorem  The suﬃcient conditions of absolute stability of neutral-type indirect con-trol system (), () are the existence of the positive deﬁnite matrices W and W– (W)T₍_W₎–_W_.

Proof According to Lemma  the condition imposed on the matrices W andW– (W)T₍_W₎–_W_{implies that}_S_{is positive deﬁnite. Theorem  now implies that the}

sys-tem is stable.

Therefore, the absolute stability investigation problem is reduced to the task of checking of positive deﬁniteness for two matrices, one of which is n-dimensional and the other is

n+ -dimensional. Note that we can use Lemma  to reduce the proof of positive deﬁnite-ness of the n-dimensional case to positive deﬁniteness of twon-dimensional matrices.

Example  When the matrices have special properties, Theorem  can be quite useful. For example suppose we have

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NowAis negative deﬁnite and

AT_D= , DTb= , AT_b= ,

or in other wordsbis in the intersection of the null spaces ofDT_and_AT

 and the image of Dis in the null spaceAT

.

If we could takeH= –hA,G=gI, andG=gI, then our matrixS would be of the form

⎡ ⎢ ⎢ ⎢ ⎣

(h–g)ATA–gI (h–g)ATA (h–g)ATD (h–g)ATb–βc

(h–g)ATA ge–ζ τI–gATA  

(h–g)DTA  g(e–ζ τ_I_–_DT_D₎ _

(h–g)bT_A_–

βcT   βρ–gbTb

⎤ ⎥ ⎥ ⎥ ⎦.

It is interesting to examine under what conditions we could actually do this and still prove positive deﬁniteness of the matrix. To apply Theorem  to this matrix we need the follow-ing matrices to be positive deﬁnite:

W=

(h–g)ATA–gI (h–g)ATA

(h–g)ATA ge–ζ τI–gATA

and

W–WTW–W,

where

W=

g(e–ζ τ_I_–_DT_D₎ _

 βρ–gbT_b

and

W=

(h–g)ATD (h–g)ATb–βc

 

.

Note that we can apply Lemma  toW, so the proof of positive deﬁniteness ofWreduces

to the proofs that

(h–g)AT_A–gI

and

ge–ζ τI–gAT_A– (h–g)AT_AS_–AT_A

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A tempting further simpliﬁcation would beh=g, which would simplifyWandW

to get a positive deﬁniteS. Under these assumptions we would need the following matrices to be positive deﬁnite:

which is solvable if and only if

ρ>gβcThA_TA–gI–cbTb.

For our example we ﬁnd

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S=h

which, after insertion of the inverse matrix,

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applicable. In such a case we can try to construct a feedback controlu,v, such that the modiﬁed system

d dt

x(t) –Dx(t–τ)=Ax(t) +Ax(t–τ) +bfσ(t)+u(t), ()

d dtσ(t) =c

T_x₍_t_{) –}_ρ_f_σ₍_t₎₊_v₍_t_), _()

u(t) =Cx(t) +Cx(t–τ), ()

v(t) =Cx(t) ()

will be absolutely stable, whereCandCaren×nmatrices andCis a ×nmatrix.

Deﬁne

S=SA+C,A+C,b,c+C_T,ρ,τ,H,G,G,β,ζ.

We give a generalization of the two previous options of ﬁnding of the stabilization condi-tions to the case of the system (), ().

Theorem  Suppose that there are matrices C,C,and C,such that the matrix Sis pos-itive deﬁnite.In that case the system(), ()is stabilizable with respect to the state feedback shown in(), (),and the functional().

Proof The proof follows immediately from Theorem .

Using the results of Lemma , we can replace verification of positive definiteness of ma-trixSby verification of positive definiteness of two matrices of lower dimensionality.

Theorem  Deﬁne

˜

S= [In×n (n+)×(n+)]S

In×n

(n+)×(n+)

,

˜

S= [In×n (n+)×(n+)]S

n×n

I(n+)×(n+)

,

˜

S= [n×n I(n+)×(n+)]S

n×n

I(n+)×(n+)

.

Suppose that there are matrices C, C, and C, such that the matricesS˜ and S˜–

(S˜)T(S˜)–S˜are positive deﬁnite.In that case the system(), ()is stabilizable with re-spect to the state feedback shown in(), (),and the functional().

6 Conclusions

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Complex System Simulation, Faculty of Cybernetic, Taras Shevchenko National University of Kyiv,}

Volodymyrska str., 64, Kiev, 01033, Ukraine. 2_{Faculty of Civil Engineering and Geo Sciences, Delft University of Technology,}

Stevinweg 1, Delft, 2628 CN, Netherlands.

Acknowledgements

The authors would like to thank for their support: the ﬁrst and the fourth authors were supported by the budget program 2201250 of the Ministry of Education and Science of Ukraine ‘Study and training of students, scientist and teachers on abroad’ of 2012.

Received: 16 September 2014 Accepted: 5 February 2015

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