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Null controllability of neutral system with infinite delays
Author’s Accepted Manuscript Null controllability of neutral system with infinite delays I. Davies, O.C.L. Haas
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S0947-3580(15)00104-1 http://dx.doi.org/10.1016/j.ejcon.2015.09.001 EJCON137
To appear in: European Journal of Control Received date: 5 December 2014 Revised date: 8 June 2015 Accepted date: 1 September 2015 Cite this article as: I. Davies and O.C.L. Haas, Null controllability of neutral system with infinite delays, European Journal of Control, http://dx.doi.org/10.1016/j.ejcon.2015.09.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Null controllability of neutral system with infinite delays I. Davies*† and O. C. L. Haas* * Control Theory and Application Centre (CTAC), The Futures Institute, Faculty of Engineering and Computing, Coventry University, 10 Coventry Innovation village, Coventry University Technology Park, Cheetah Road, Coventry, CV1 2TL, United Kingdom (Authors e-mail: [email protected]; [email protected])
Abstract Sufficient conditions are developed for the null controllability of neutral control systems with infinite delays when the values of the control lie in an -dimensional unit cube. Conditions are placed on the perturbation function which guarantee that if the uncontrolled system is uniformly asymptotically stable and the control system satisfies a full rank ( )) condition, so that ( ) ( , for every complex , where ( ) is an polynomial matrix in constructed from the coefficient matrices of the control system and ( ( )) is the transpose of [ ( ) ( ( ) )] , then the control system is null controllable with constraint. Keywords: Neutral systems, infinite delay, null controllability, rank condition, stable Subject classification codes : 93Cxx, 93C23, 34k40, 93Bxx, 93B05 1. Introduction Controllability is one of the most important structural properties of dynamical systems used to design model based controllers and estimators. Neutral functional differential systems provide a useful modelling framework for a range of applications in science and engineering, see Corduneanu (2002), Khartovskii and Pavlovskaya (2013) and references therein. The controllability of neutral functional differential systems in particular, null controllability has been studied by many researchers including Onwuatu (1984), and Underwood and Chukwu (1988). Onwuatu (1984) established sufficient conditions for null controllability in function space for a class of linear and nonlinear neutral systems by showing that if the uncontrolled systems are uniformly asymptotic stable and the linear control base systems are controllable, then the null controllability of the systems is guaranteed in function space with constraints, provided that nonlinearities satisfy the necessary conditions imposed on them. In Underwood and Chukwu (1988) it was shown that, the null controllability of a class of nonlinear neutral system is implied by the null controllability of its linear approximations under very general conditions for which the analogous result is not in general true for retarded systems. The condition under which null controllability of the system implies local controllability was also demonstrated. These studies have been extended to linear and nonlinear neutral functional differential systems with infinite delays (Onwuatu, 1993; Balachandran and Leelamani, 2006; Umana, 2008, 2011; Dauer et al., 1998; Davies, 2006). The methods employed to address this problem include fixed point theorems (Dauer et al., 1998), non-singularity of the controllability grammiam (Umana, 2008), and rank criterion for properness (Davies, 2006). Controllability results can be established with the controls assumed to be either restrained or unrestrained but is required only to be square integrable on finite intervals (Chukwu, 1979). In the latter case, a non-singularity assumption for the controllability matrix of the system is a necessary and sufficient condition for null controllability. In the former case, such conditions are however no longer sufficient for null controllability and an additional condition of stability for the uncontrolled system is required (Chukwu, 1979). In Dauer et al. (1998), null controllability result is obtained by using the Schauder fixed point theorem based on the uniform asymptotic stability of the uncontrolled system and properness †
Corresponding author e-mail: [email protected]
1
assumption on the linear control system, the latter being equivalent to the non-singularity of controllability matrix. However, evaluating controllability analytically for linear time varying systems unlike time-invariant system is challenging even for very simple systems since it involves the evaluation of the controllability matrices. The controllability matrix may be calculated by computational methods provided that all the exact time-varying elements in the linear time varying systems are known. In this paper, the controls are assumed to be restrained and the null controllability result is obtained using the Schauder’s fixed point theorem. It extends the results from (Underwood and Chukwu, 1988; Jacobs and Langenhop, 1976; Rivera Rodas and Langenhop, 1978) to neutral functional differential system having infinite delays by using the Schauder fixed point theorem. Growth and continuity conditions are paced on the perturbation function which guarantees that if the linear control base system has full rank with the condition that ( ) ( ( )) for every complex , where ( ) is an polynomial matrix in ( )) is the constructed from the coefficient matrices of the control system and ( ( ) ( ( ) )], and the functional difference operator transpose of [ for the system uniformly stable, with the linear uncontrolled system uniformly asymptotically stable, then the perturbed neutral system with infinite delay is null controllable with constraint. The remainder of the paper is organised as follows. Section 2 contains the mathematical notations, preliminaries and problem definition. Section 3 presents the stability theorems for the system. Section 4 develops and proves the controllability theorems for the system; the main result of the paper is also developed and proved in this section. Finally, Section 5 contains numerical examples of the theoretical results prior to the discussions and conclusion. 2. Basic notations, preliminaries and definitions Suppose is a given number, ( ), is a real – dimensional Euclidean ( ) ( ) represents the space with norm | |. Let be any interval in , the convention ( ) ([ ] ) is the Lebesgue space of square-integrable functions from to , and ] Sobolev space of all absolutely continuous functions [ whose derivatives are ([ ] ) is the space of continuous function mapping the square integrable. ] into | ( )| . Define the interval [ with the norm ‖ ‖ , where ‖ ‖ symbol by ( ) ( ), . This paper will consider neutral functional differential system with infinite delays of the form ( )
(
)
∫ ( ) (
)
)
∫ ( ) (
)
( )
}
and its perturbation ( )
(
(
( ))
( )
through its linear base control system ( )
(
)
( )
and its free system ( )
(
)
∫ ( ) (
)
2
( )
where the functional difference operator ( ) ( ) ( ) ( ) , and ( with the following assumptions:
for the system is defined by ( ) ( ) ( ) ( ) ( ) ( ),
)
( ) ( ) and ( ) are continuous matrices ( ) is a continuous matrix ( ) is an ] matrix whose elements are square integrable on ( ( ) [ ) is a nonlinear continuous matrix function.
(i) (ii) (iii) (iv)
It is assumed that satisfy enough smoothness conditions to ensure that a solution of (2) ), is unique, and depends continuously upon ( ) and can be exists through each ( ) extended to the right as long as the trajectory remains in a bounded set [ . These conditions are given in Cruz and Hale, (1970). ( ) ) be a solution of (3) with ( ), Let ( and the set ( ) . Then ( ) ( ) ( ) is a continuous linear operator from . There is an matrix function ( ) which is defined on [ ), continuous in from the right of , bounded variation in ; ( ) , such that ( ) satisfies
( ) (
Now, define the
)
(
matrix function ( )
(
) )
as {
where is the identity matrix. Write ( ) ( ) ( ) ( ). ) satisfies the equation A solution of (3) through ( (
)
(
)
∫ (
(
)
)
(
)( ) , so that
( ) ( )
or (
)
(
)
∫ (
) ( ) ( )
( )
In a similar manner, any solution of system (2) will be given by (
)
( ∫ (
) ) (
∫ (
) ( ) ( ) ( ))
∫ (
) ∫ ( ) (
) ( )
Define the matrix functions by ( ) ( ) ( ) for , it follows then from (6) that
( )
3
(
)
(
)
∫ (
∫ (
) (
) ( )
∫ (
) ∫ ( ) (
( ))
) ( )
) The controls of interest, denoted by will be functions [ which are square integrable on finite intervals with values in , where | | . ] if ( ) Definition 1. The system (3) is proper on [ almost everywhere [ ] implies for , where is the transpose of . If (3) is proper on each ], then the system is said to be proper in interval [ ], if for each function Definition 2. System (3) is said to be controllable on [ ( ) ([ ] ), there is a control ([ ] ) such that the ( ) . ], if for each function Definition 3. System (3) is said to be completely controllable on [ ( ) ([ ] ) such that the solution , there is an admissible control ( ) of (3) satisfies ( ) ( ) , . It is completely [ ] controllable on with constraints, if the above holds with . ] if for each Definition 4. The system (2) is null-controllable on [ ( ) ([ ] ), there exists a ([ ] ) such that the solution of (2) satisfies ( ) ( ) , . The system (2) is null-controllable with constraints if the above holds with control Definition 5. The domain of null-controllability of (3) with constraints is the set of all ( ) ) of (3) with ( ) initial functions for which the solution ( , ( ) at some Definition 6. The reachable set of (3) is a subset of ( ([
If the controls are in
( where
([
)
]
{∫ ( ]
)
given by
) ( )
([
)}
), we define the constraint reachable set by {∫ (
) ( )
([
).
Definition 7. The controllability matrix of (3) will be given by ( where
]
)
∫ (
is the transpose of .
4
)
(
)
]
)}
3.
STABILITY RESULTS
Here, some definitions, lemmas and theorem which are fundamental to the development of the stability results for the system (4) are given. Consider system (3) defined by ( )
(
)
( )
Definition 8. The solution ( ) such that if ‖ ‖
of (9) is called stable at if ) exists for , then the solution ( (
Also, for each there exists a ( ) of (9) satisfies ‖ ( )‖
)
for
such that if ‖ ‖
and there exists a . , then the solution
.
The trivial solution of (9) is called stable if it is stable for each . It is called uniformly stable if it is stable and does not depend on . It is uniformly asymptotically stable if it is uniformly stable for every and for every there exists ( ) independent of and ‖ ‖ )‖ independent of such that implies ‖ ( , for all ( ). Definition 9. The solution exists constant Definition 10. Let ( {
of (9) is uniformly asymptotically stable if and only if there )‖ [ ( )]‖ ‖, for all such that ‖ ( .
)
, and consider the homogeneous difference equation
( )
(
( ) is uniformly stable if there are constants )‖ [ ( solution of (10) satisfies ‖ (
such that for )]‖ ‖, for all
) ( )
the
.
The next two lemmas are due to Cruz and Hale (1970). They are important for the analysis and development of the operator ( ) properties and the overall stability result in this section. Lemma 1. Let be an constant matrix. The operator [ ] uniformly stable if all the roots of the equation This holds if ‖ ‖ . Lemma 2. is uniformly stable if there are constants [ ), the solution ( ) of (10) satisfies ‖ ( )‖
( ) ( ) is have moduli less than .
such that for any ‖ ‖ ( ) .
( )
,
The next theorem is developed following Theorem 1 of Sinha (1985) and Corollary 2 of Hale (1974) for functional differential equations with infinite delay; see also Corollary 3.8 of Davies (2006) and references therein for neutral functional differential systems with infinite delays. Theorem 1. In system (4), assume there is a , and a constant such that | ( )| ( ) ( ] and if for ( ) ( ) , where ( )
(
(
))
( 5
)
∫
(
) ( )
then the solution of (4) is uniformly asymptotically stable if | ( )| )), ( ( , 4.
Controllability Results
This section develops and proves necessary and sufficient controllability conditions for system (3) by exploiting the method in (Jacobs and Langenhop, 1976; Rivera Rodas and Langenhop, 1978). Some controllability results which are relevant to this study are also given. ) is non-singular Lemma 3. The system (3) is completely controllable if and only if ( Proof. The proof can be observed from Proposition 3.1 in (Dauer and Gahl, 1977). □ Proposition 1. The system (3) is controllable on [
] if and only if
(
)
Proof. This is Lemma 3.2 in (Davies, 2006). □ Proposition 2. The system (3) is completely controllable with constraint on [ ( ). only if
] if and
Proof. This is Theorem 3.3 in (Davies, 2006). □ Proposition 3. The following are equivalent for system (3). (i) (ii) (iii)
( ) is non-singular system (3) is completely controllable on [ ] system (3) is proper on [
]
Proof. This is Proposition 3 in (Onwuatu, 1993). □ Lemma 4. The system (3) is completely controllable on [ ]. on [
] if and only if it is controllable
Proof. The proof follows immediately from Proposition 1 and 2. □ ] Let [ a positive integer, be absolutely continuous and define the differential ]. operator for neutral systems by ( )( ) ̇( ) ( )⁄ , almost everywhere on [ Higher powers of the operator are defined inductively by , with domain equal ] ]. Note that, by to all [ , such that is absolutely continuous on [ the ( ), [ ]. identity ( )( ) ( ) Define a positive integer and ( ] such that ( ) for ([ ] ) and adopt the convention shift operator by (
)( )
(
a nonnegative integer to be the collection of all ([ ] ), and the restriction |[ ] is in ([ ] ) . For ( ) define the
)
(
)
( ) and take Define to be the identity operator on by inductively using (11). Observe from the definition of the differential operator , and the shift operator that for ( ) is taken as a common domain for the operators and , if the function space
6
, then and commute in this setting and each commutes with multiplication by a scalar (element in ). For any matrix and matrix , one can define matrix by [ ] [ ] for integers . Consider the neutral system with delay of the form ( ⁄ )( ( )) ( ) ( ) ( ) ( ) where are constant matrices, is chosen to be constant real ) of (12) is the restriction to [ ] of the solution matrix. The solution ( ( ) ( ) of the equation ( ) ) by the equation Now define the matrix ( ( ) and let ( ) ( ) where “ ” denotes the transposed matrix of cofactors. Some basic relationship exists between these two operators which by Jacobs and Langenhop (1976) can be expressed as (
)
∑ ̂( )
( )
∑
(
where the matrix polynomials ( ) , ̂ ( ) are at most of degree argument. Using the polynomial ( ) in (13) define a unique matrix operator by ( ) [ ( ) ( ) ( ) ] Now, the operator ( ) can be written in the form of a polynomial to get ( )
∑
where, transpose of [
in their
( (
Theorem 2. Let sufficient for
are
)
[
constant real matrices, and let ( ( ( ) )] for all complex numbers
(
)
)
)) be the .
, then for (12) to be controllable on [ ] it is necessary and ] ( )) and ( ) ( for every complex .
Proof. This is Theorem 3.4 of Rivera Rodas and Langenhop, (1978). □ ,[
Corollary 1. Let (i) (ii)
( ) (
[
(
]
))
] and assume that system (12) satisfies the following ; , for every complex .
Then, system (12) is completely controllable on [
].
Proof. If condition (i) and (ii) holds, then by Theorem 2, the system (12) is controllable on [ ] . This by Lemma 4 implies that system (12) is completely controllable on [ ] . Conversely, if system (12) is completely controllable on [ ], then it is controllable by [ ] ( )) Lemma 4, and by Theorem 2; and ( ) ( for every complex , and the proof is complete. □ Theorem 3. In system (1), assume the following (i)
are
constant matrices, 7
is
constant real matrix
(ii) (iii) (iv)
(v)
[
] ; )) , for every complex , ( ) ,
for ( ) (
, ( ( )
( )
(
(
( )
(
))
(
)
(
∫
) ( )
) is uniformly stable.
Then, system (1) is null controllable with constraints on (
)
.
Proof. Because of (i), (ii) and (iii) system (1) is controllable on [ ] by Theorem 2. Hence, ( ) by Proposition 1. By condition (iv), and (v) the system (1) with ) ( ) ( ) and satisfies ( as . Hence, at some , hence , the domain of null controllability of (1). Suppose for the contrary that . Since is a solution of (1) with , then . This implies that, there ( ) exists a sequence such that as and , for any , therefore . It follows from the variation of constant formula (5) that: ( ) ( ) ∫ ( ) ( ) ( ). Then, since Let , ( ) ( ), for any , for any , and so and . However ( ) which is a contradiction. Therefore, as , and , and hence there exists a ball around 0 which is contained in . Again, by (iv) there exists some , ( ) ( ) . Therefore, using as initial point and as initial function, ( ) ) of (1) satisfies there exists and such that, the solution ( ( ) , and the proof is complete. □ Remark 1. The conditions imposed on Theorem 3 constrain in a box but these conditions can also allow the constraint set to be an arbitrary compact set as shown in Theorem 4. This is possible because the null controllability of linear neutral system, in general, depends on the length of the time interval over which the system operates (Jacobs and Langenhop, 1976). Therefore restrictions on interval could be made based on the requirements for the controllability of the linear controllable base system. Again, these different constraints by the [ ] definition of is possible because by Theorem 2, if the conditions , and ( ) ( ( )) ] for every complex holds on [ then (12) is controllable. ( ) ( ) This means ( ) . Now define a map taking [ ] by ( ) ( ) ( ) . Because is a continuous linear transformation of onto it is open (Hale, 1977). From the definition of , there is an open ball around zero, so that ( ) ( ) ( ). Therefore, ( ) ( ). This implies that ( ) is open, ( ) . Moreover, if it is assumed that (12) is since is open and therefore ] then by Lemma 3, ( ) is non-singular which in turn is completely controllable on [ ( ) ] , and equivalent to almost everywhere on [ . This implies ( ) by Proposition 2 and therefore properness of (12) by Definition 1, hence ) completely controllable on ( with constraint. 4.1.
Main result
The main result for the neutral control system with infinite delay will now be developed and proved in this section. 8
Theorem 4. Assume for system (2) that (i) (ii) (iii) (iv) (v)
the constraint set is an arbitrary compact set of ( ) ( ) is uniformly stable the system (4) is uniformly asymptotically stable; so that the solution of (4) )‖ ))‖ ‖, satisfies ‖ ( ( ( . the system (3) is completely controllable ( ) ( ( ) ( )) , The continuous function satisfies | ( ( ) ( ))| ( ) ) for all ( ( ) ( )) [ , where ∫ ( ( ) ( )) , and .
Then, the system (2) is null controllable. Proof. By (iv), exists for each to the integral equations ( )
(
)
(
)[
∫ (
(
) (
(
)
)
∫ (
( ) ( ))
∫ (
∫ (
) ( )
) (
) ∫ ( ) (
forms a pair
)
(
]
, ( )
for some suitably chosen ( )
. Assuming the pair of functions
( ), ∫ (
[
]
) ∫ ( ) (
)
( ) ( ))
(
( ) ( ), [ ]. ] and is a solution of (2) corresponding to Then is square integrable on [ [ ] initial state ( ) . Also, ( ) . It is necessary to show now that the arbitrary compact constraint subset of , that is | ( )| , for some constant By (ii) and (iii), and the continuity of in compact intervals, it follows that for some | ( | ( Hence, | ( )|
) )
∫
[
(
)|
(
)∫
(
(
)
)
with is in . ,
, ( ) (
))
)
|
∫
(
(
(
(
))
(
)) .
) ( ( ) ( ))
]
and therefore, | ( )| )) ( ) ( ( ( ( )) since and . Hence, from (17) can be chosen sufficiently large such [ ], showing that is admissible control. It remains to prove the that | ( )| , ( ) existence of a pair of the integral equations (15) and (16). Let represent the Banach ) [ ] [ ] space of all functions ( , where ( ) ([ ] ); ([ ] ) with the norm defined by
9
‖(
)‖
‖ ‖
‖ ‖ , where ‖ ‖ ( )
Define the operator ( )
(
)
(
)[
∫ ( [
for ( )
( )
)
(
( ))
) (
)
), where
( ∫ (
;‖ ‖
{∫
) ∫ ( ) (
| ( )|
}
[
]
| ( )|
(
.
(
)
(
)
].
∫ (
) ∫ ( ) (
].
[ and also ) . Again,
for
⁄
)
)
( ))
( ) for for and ( ) It is clear from (17) that | ( )| | ( )| ( . Hence, ‖ ‖ (
}
[
) ( )
⁄
| ( )|
)
( ) for
∫ (
∫ (
by (
) (
] and ( )
(
{∫
]
⁄
))
∫| ( )|
, so that
(
( ))
| ( )|. Since where , it follows that | ( )| ( ) | ( )| [ ]. and | ( )| ⁄ [ ], then ‖ ‖ ( ) [ ]. Hence, if . Let ( ) ) ‖ ‖ ‖ ‖ ( ) ( ). Then letting ( ) {( }, it follows that Now, since ( ) is closed, bounded and convex, by Riesz theorem (see Kantorovich and Akilov, 1982 ), it is relatively compact under the transformation . Hence, the Schauder’s fixed point theorem implies that has a fixed point. Hence, system (2) is null controllable. 5.
Example
Consider the neutral control system ( ⁄ )( ( )
(
))
( )
(
)
∫
(
) (
)
( )
(
)
(
)
(
)
and its perturbation ( ⁄ )( ( )
(
))
( )
( ) ( ( ) ( Its linear control base system is given by ( ⁄ )( ( ) ( )) ( ) and its free system
(
)
∫
) ( )) (
10
)
( )
(
) (
)
( ⁄ )( ( ) where
( ⁄
(
⁄ (
))
( )
(
)
)
( ) (
) ( ))
(
(
) ⁄ ) ⁄
( (
)
(
∫
( ( )
(
) (
(
)
⁄
(
)
),
( ))
))
( ) Now, use Lemma 1 to check that, the operator [ ] condition of Lemma 1 gives, (
is uniformly stable as follows: The
)
( ⁄ ) ( ⁄ ) ⁄ . Hence, the operator which implies , and stable if . Next, observe by Theorem 1 that the characteristic root of (23) is ( ( )) ( ( ) ( )) (
) ∫
(
is uniformly
)
(
)
and all the roots of (23) have negative real part. Hence by Theorem 1, system (23) is uniformly asymptotically stable. Finally, check that (22) is controllable as follows: ⁄ [ ] ( ) ( )
(
) (
( )
)
(
) [(
( ) ( ) (
(
))
(
Observe that for all complex
⁄
[ ( ) (
⁄
)
( (
( ) ] )⁄
)(
)
(
⁄ )] ⁄ (
(
⁄ ⁄ )⁄
(
)⁄
)
)
))
(
, ( ) (
(
))
( )
Therefore system (22) is controllable on ( ) Moreover, ) ( ))| | ( ) ( ( ) ( )) | ( ( ) ( ( )| ( ) ( Hence, all the conditions of Theorem 4 are satisfied and system (21) is null controllable.
)
Remark 2. This work can be extended to study conditions that would preserve the controllability/null controllability results when both system and input matrices undergo some parameter uncertainties. The problem of controllability of linear parameter uncertain systems 11
has received considerable research effort from control audience because of its significance in theory and its applications. These uncertainties in control systems analysis and design can be structured, if the uncertain parameter is an elemental part of the system and input matrices or unstructured, if the parameter uncertainties are contained in the systems matrices only. If is introduced into system (2) as uncertain parameters, and the constant , matrices ) respectively, depends linearly on for information, then (2) will and , ( be of the form ( ⁄ )( ( )) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ( ) (
)
(
( ))
(
)
∑ ∑ ∑ ). where , , ,( [ ] The question of null controllability for (26) on would depend on the stability of (26) when which can be estimated by using matrix norm or spectral radius (see Tai et al. (2009)), and the controllability of (12) which is relative to the controllable base for (26) amongst other assumptions on Theorem 4. That is, suppose (12) is controllable on the interval [ ], then the uncertain system (26) is controllable on [ ] for all sufficiently small, [ ] if it satisfies Corollary 1 and the conditions that and [ ] ( ). , This criterion can be easily ) ( ) matrix G, with an proved by introducing an ( matrix ; establishing their dependence and exploring their algebraic properties. 5.1
Simulation studies
The stability and controllability of the open loop system (21) can be illustrated using Simulink® and MATLAB® based simulation studies. The simulation model parameters are as given (21) with the default parameter setting and a square wave input where and are chosen to be and respectively with . Fig. 5.1 depicts the stability and controllability of the states when the simulation is performed with the linear control base system i.e. (22), and when the simulation is done with the perturbation function (see (21)). The amplitude of the internal state from the system response is observed to be slightly higher with the perturbation function whilst it exhibits a faster response when the simulation is done without the perturbation function. The settling times for the systems without the perturbation function are also observed to be faster; this is as expected and depends on the assumptions placed on the perturbation function (see (25)). The simulation showed that, the system (23) is stable and the overall control system (21) is controllable.
12
Figure 5.1. Simulation of control input and system states with perturbation function and linear control base system
6.
Discussion
Unlike the conditions in (Dauer et al., 1998; Umana, 2008; Davies, 2006), the controllability conditions introduced in this paper are explicit and computationally more effective. The computation of the controllability matrix is not required since it is obtained by an equivalent rank condition. It generalises to neutral systems the rank condition in Davies (2006). Indeed, applying the rank condition from (Davies 2006) to systems (12) with on [ ], ([ ] ) satisfying ( ) , would result in which limits the results to retarded systems. This paper introduces a different rank condition, see (Corollary 1, (i)) based on Rivera Rodas and Langenhop (1978). The rank condition alone is not considered to be necessary and sufficient; the algebraic requirement in (Corollary 1, (ii)) makes it sufficient as well as necessary for controllability. Therefore the controllability condition on this paper generalizes the results of null controllability to neutral systems with infinite delays and yields a less conservative result. The conditions (ii), which relates to the initial condition or structure of the information for the system considered, and (iii) of Theorem 4 ensures that the error signals are contained within the neighbourhood of the origin as time increases and not asymptotically tends to zero. Conditions (ii) and (iii) are required for null controllability when the controls are restrained and make the system less conservative, being able to handle internal and external disturbances that may prevent signals from converging asymptotically to zero. Perturbations cause conservatism as they do not varnish in some cases when the states approximates the origin. It makes uniform asymptotic stability impossible for such systems that have non-vanishing perturbations (Kofman, 2005). Therefore, condition (v) of Theorem 4 is imposed to preserve systems properties and directional consideration. Ignoring or altering condition (v) may affect the system’s properties differently and even lead to conservatism.
13
Conclusion Null controllability for neutral functional differential system with infinite delay have been developed and proved. The results were obtained with respect to the stability of the free system and the linear control base system having full rank with the condition that ( ) ( ( )) , with the assumption that the perturbation function satisfies the smoothness and growth condition placed on it. Acknowledgement The authors are thankful to the Rivers State Scholarship Board, Rivers State Secretariat Complex, Port Harcourt, Rivers State, Nigeria for supporting this research. The authors are also grateful to the reviewers for their useful comments References Balachandran, K., Leelamani, A., 2006. Null controllability of neutral evolution integrodifferential systems with infinite delay. Mathematical Problems in Engineering 2006, 1-18. Chukwu, E. N., 1979. Euclidean controllability of linear delay systems with limited controls. IEEE Transactions on Automatic Control AC-24, 798-800. Corduneanu, C., 2002. Absolute stability for neutral differential systems. European Journal of Control 8, 209-212 Cruz, M. A., Hale, J. K., 1970. Stability of functional differential equations of neutral type. Journal of Differential Equations 7, 334-355. Dauer, J., Balachandran, K., Anthoni, S., 1998. Null controllability of nonlinear infinite neutral systems with delays in control. Computers & Mathematics with Applications 36, 39-50. Dauer, J., Gahl, R., 1977. Controllability of nonlinear delay systems. Journal of Optimization Theory and Applications 21, 59-70. Davies , I., 2006. Euclidean null controllability of infinite neutral differential systems. Anziam Journal 48, 285-294. Kofman, E., 2005. Non-conservative ultimate bound estimation in LTI perturbed systems. Automatica 41, 1835-1838. Hale, J. K., 1977. Theory of Functional Differential Equations, New York, Springer Hale, J. K., 1974. Functional differential equations with infinite delays. Journal of Mathematical Analysis and Applications 48, 276-283. Jacobs, M. Q. Langenhop, C., 1976. Criteria for function space controllability of linear neutral systems. SIAM Journal on Control and Optimization 14, 1009-1048. Kantorovich, L. M., Akilov, G. P., 1982. Functional Analysis, Pergamon Press, Oxford. Khartovskii, V., Pavlovskaya, A., 2013. Complete controllability and controllability for linear autonomous systems of neutral type. Automation and Remote Control 74, 769-784. Onwuatu, J. U., 1984. On the null-controllability in function space of nonlinear systems of neutral functional differential equations with limited controls. Journal of Optimization Theory and Applications 42 (3), 397-420. Onwuatu, J. U., 1993. Null controllability of nonlinear infinite neutral system. Kybernetika 29, 325-338. Rivera Rodas, H., Langenhop, C., 1978. A sufficient condition for function space controllability of a linear neutral system. SIAM Journal on Control and Optimization 16, 429-435. 14
Sinha, A.S. C., 1985. Null controllability of nonlinear infinite delays systems with restrained controls. International Journal of Control 42, 735-741. Tai, Z., Wang, X., Wang, D., (2009). Robust Stability Analysis of Uncertain Neutral Functional Systems. Int.J.Innov.Comput.Inf.Control 5, 3013-3024. Umana, R., 2008. Null controllability of nonlinear infinite neutral systems with multiple delays in control. Journal of Computational Analysis & Applications 10, 509-522. Umana, R., 2011. Null controllability of nonlinear neutral Volterra integrodifferential systems with infinite delay. Differential Equations & Control Processes (ISSN 18172172) 2, 48-56. Underwood, R. G., Chukwu, E. N., 1988. Null Controllability of Nonlinear Neutral Differential Equations. Journal of Mathematical Analysis and Applications 129(2), 326345.
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S0947-3580(15)00104-1 http://dx.doi.org/10.1016/j.ejcon.2015.09.001 EJCON137
To appear in: European Journal of Control Received date: 5 December 2014 Revised date: 8 June 2015 Accepted date: 1 September 2015 Cite this article as: I. Davies and O.C.L. Haas, Null controllability of neutral system with infinite delays, European Journal of Control, http://dx.doi.org/10.1016/j.ejcon.2015.09.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Null controllability of neutral system with infinite delays I. Davies*† and O. C. L. Haas* * Control Theory and Application Centre (CTAC), The Futures Institute, Faculty of Engineering and Computing, Coventry University, 10 Coventry Innovation village, Coventry University Technology Park, Cheetah Road, Coventry, CV1 2TL, United Kingdom (Authors e-mail: [email protected]; [email protected])
Abstract Sufficient conditions are developed for the null controllability of neutral control systems with infinite delays when the values of the control lie in an -dimensional unit cube. Conditions are placed on the perturbation function which guarantee that if the uncontrolled system is uniformly asymptotically stable and the control system satisfies a full rank ( )) condition, so that ( ) ( , for every complex , where ( ) is an polynomial matrix in constructed from the coefficient matrices of the control system and ( ( )) is the transpose of [ ( ) ( ( ) )] , then the control system is null controllable with constraint. Keywords: Neutral systems, infinite delay, null controllability, rank condition, stable Subject classification codes : 93Cxx, 93C23, 34k40, 93Bxx, 93B05 1. Introduction Controllability is one of the most important structural properties of dynamical systems used to design model based controllers and estimators. Neutral functional differential systems provide a useful modelling framework for a range of applications in science and engineering, see Corduneanu (2002), Khartovskii and Pavlovskaya (2013) and references therein. The controllability of neutral functional differential systems in particular, null controllability has been studied by many researchers including Onwuatu (1984), and Underwood and Chukwu (1988). Onwuatu (1984) established sufficient conditions for null controllability in function space for a class of linear and nonlinear neutral systems by showing that if the uncontrolled systems are uniformly asymptotic stable and the linear control base systems are controllable, then the null controllability of the systems is guaranteed in function space with constraints, provided that nonlinearities satisfy the necessary conditions imposed on them. In Underwood and Chukwu (1988) it was shown that, the null controllability of a class of nonlinear neutral system is implied by the null controllability of its linear approximations under very general conditions for which the analogous result is not in general true for retarded systems. The condition under which null controllability of the system implies local controllability was also demonstrated. These studies have been extended to linear and nonlinear neutral functional differential systems with infinite delays (Onwuatu, 1993; Balachandran and Leelamani, 2006; Umana, 2008, 2011; Dauer et al., 1998; Davies, 2006). The methods employed to address this problem include fixed point theorems (Dauer et al., 1998), non-singularity of the controllability grammiam (Umana, 2008), and rank criterion for properness (Davies, 2006). Controllability results can be established with the controls assumed to be either restrained or unrestrained but is required only to be square integrable on finite intervals (Chukwu, 1979). In the latter case, a non-singularity assumption for the controllability matrix of the system is a necessary and sufficient condition for null controllability. In the former case, such conditions are however no longer sufficient for null controllability and an additional condition of stability for the uncontrolled system is required (Chukwu, 1979). In Dauer et al. (1998), null controllability result is obtained by using the Schauder fixed point theorem based on the uniform asymptotic stability of the uncontrolled system and properness †
Corresponding author e-mail: [email protected]
1
assumption on the linear control system, the latter being equivalent to the non-singularity of controllability matrix. However, evaluating controllability analytically for linear time varying systems unlike time-invariant system is challenging even for very simple systems since it involves the evaluation of the controllability matrices. The controllability matrix may be calculated by computational methods provided that all the exact time-varying elements in the linear time varying systems are known. In this paper, the controls are assumed to be restrained and the null controllability result is obtained using the Schauder’s fixed point theorem. It extends the results from (Underwood and Chukwu, 1988; Jacobs and Langenhop, 1976; Rivera Rodas and Langenhop, 1978) to neutral functional differential system having infinite delays by using the Schauder fixed point theorem. Growth and continuity conditions are paced on the perturbation function which guarantees that if the linear control base system has full rank with the condition that ( ) ( ( )) for every complex , where ( ) is an polynomial matrix in ( )) is the constructed from the coefficient matrices of the control system and ( ( ) ( ( ) )], and the functional difference operator transpose of [ for the system uniformly stable, with the linear uncontrolled system uniformly asymptotically stable, then the perturbed neutral system with infinite delay is null controllable with constraint. The remainder of the paper is organised as follows. Section 2 contains the mathematical notations, preliminaries and problem definition. Section 3 presents the stability theorems for the system. Section 4 develops and proves the controllability theorems for the system; the main result of the paper is also developed and proved in this section. Finally, Section 5 contains numerical examples of the theoretical results prior to the discussions and conclusion. 2. Basic notations, preliminaries and definitions Suppose is a given number, ( ), is a real – dimensional Euclidean ( ) ( ) represents the space with norm | |. Let be any interval in , the convention ( ) ([ ] ) is the Lebesgue space of square-integrable functions from to , and ] Sobolev space of all absolutely continuous functions [ whose derivatives are ([ ] ) is the space of continuous function mapping the square integrable. ] into | ( )| . Define the interval [ with the norm ‖ ‖ , where ‖ ‖ symbol by ( ) ( ), . This paper will consider neutral functional differential system with infinite delays of the form ( )
(
)
∫ ( ) (
)
)
∫ ( ) (
)
( )
}
and its perturbation ( )
(
(
( ))
( )
through its linear base control system ( )
(
)
( )
and its free system ( )
(
)
∫ ( ) (
)
2
( )
where the functional difference operator ( ) ( ) ( ) ( ) , and ( with the following assumptions:
for the system is defined by ( ) ( ) ( ) ( ) ( ) ( ),
)
( ) ( ) and ( ) are continuous matrices ( ) is a continuous matrix ( ) is an ] matrix whose elements are square integrable on ( ( ) [ ) is a nonlinear continuous matrix function.
(i) (ii) (iii) (iv)
It is assumed that satisfy enough smoothness conditions to ensure that a solution of (2) ), is unique, and depends continuously upon ( ) and can be exists through each ( ) extended to the right as long as the trajectory remains in a bounded set [ . These conditions are given in Cruz and Hale, (1970). ( ) ) be a solution of (3) with ( ), Let ( and the set ( ) . Then ( ) ( ) ( ) is a continuous linear operator from . There is an matrix function ( ) which is defined on [ ), continuous in from the right of , bounded variation in ; ( ) , such that ( ) satisfies
( ) (
Now, define the
)
(
matrix function ( )
(
) )
as {
where is the identity matrix. Write ( ) ( ) ( ) ( ). ) satisfies the equation A solution of (3) through ( (
)
(
)
∫ (
(
)
)
(
)( ) , so that
( ) ( )
or (
)
(
)
∫ (
) ( ) ( )
( )
In a similar manner, any solution of system (2) will be given by (
)
( ∫ (
) ) (
∫ (
) ( ) ( ) ( ))
∫ (
) ∫ ( ) (
) ( )
Define the matrix functions by ( ) ( ) ( ) for , it follows then from (6) that
( )
3
(
)
(
)
∫ (
∫ (
) (
) ( )
∫ (
) ∫ ( ) (
( ))
) ( )
) The controls of interest, denoted by will be functions [ which are square integrable on finite intervals with values in , where | | . ] if ( ) Definition 1. The system (3) is proper on [ almost everywhere [ ] implies for , where is the transpose of . If (3) is proper on each ], then the system is said to be proper in interval [ ], if for each function Definition 2. System (3) is said to be controllable on [ ( ) ([ ] ), there is a control ([ ] ) such that the ( ) . ], if for each function Definition 3. System (3) is said to be completely controllable on [ ( ) ([ ] ) such that the solution , there is an admissible control ( ) of (3) satisfies ( ) ( ) , . It is completely [ ] controllable on with constraints, if the above holds with . ] if for each Definition 4. The system (2) is null-controllable on [ ( ) ([ ] ), there exists a ([ ] ) such that the solution of (2) satisfies ( ) ( ) , . The system (2) is null-controllable with constraints if the above holds with control Definition 5. The domain of null-controllability of (3) with constraints is the set of all ( ) ) of (3) with ( ) initial functions for which the solution ( , ( ) at some Definition 6. The reachable set of (3) is a subset of ( ([
If the controls are in
( where
([
)
]
{∫ ( ]
)
given by
) ( )
([
)}
), we define the constraint reachable set by {∫ (
) ( )
([
).
Definition 7. The controllability matrix of (3) will be given by ( where
]
)
∫ (
is the transpose of .
4
)
(
)
]
)}
3.
STABILITY RESULTS
Here, some definitions, lemmas and theorem which are fundamental to the development of the stability results for the system (4) are given. Consider system (3) defined by ( )
(
)
( )
Definition 8. The solution ( ) such that if ‖ ‖
of (9) is called stable at if ) exists for , then the solution ( (
Also, for each there exists a ( ) of (9) satisfies ‖ ( )‖
)
for
such that if ‖ ‖
and there exists a . , then the solution
.
The trivial solution of (9) is called stable if it is stable for each . It is called uniformly stable if it is stable and does not depend on . It is uniformly asymptotically stable if it is uniformly stable for every and for every there exists ( ) independent of and ‖ ‖ )‖ independent of such that implies ‖ ( , for all ( ). Definition 9. The solution exists constant Definition 10. Let ( {
of (9) is uniformly asymptotically stable if and only if there )‖ [ ( )]‖ ‖, for all such that ‖ ( .
)
, and consider the homogeneous difference equation
( )
(
( ) is uniformly stable if there are constants )‖ [ ( solution of (10) satisfies ‖ (
such that for )]‖ ‖, for all
) ( )
the
.
The next two lemmas are due to Cruz and Hale (1970). They are important for the analysis and development of the operator ( ) properties and the overall stability result in this section. Lemma 1. Let be an constant matrix. The operator [ ] uniformly stable if all the roots of the equation This holds if ‖ ‖ . Lemma 2. is uniformly stable if there are constants [ ), the solution ( ) of (10) satisfies ‖ ( )‖
( ) ( ) is have moduli less than .
such that for any ‖ ‖ ( ) .
( )
,
The next theorem is developed following Theorem 1 of Sinha (1985) and Corollary 2 of Hale (1974) for functional differential equations with infinite delay; see also Corollary 3.8 of Davies (2006) and references therein for neutral functional differential systems with infinite delays. Theorem 1. In system (4), assume there is a , and a constant such that | ( )| ( ) ( ] and if for ( ) ( ) , where ( )
(
(
))
( 5
)
∫
(
) ( )
then the solution of (4) is uniformly asymptotically stable if | ( )| )), ( ( , 4.
Controllability Results
This section develops and proves necessary and sufficient controllability conditions for system (3) by exploiting the method in (Jacobs and Langenhop, 1976; Rivera Rodas and Langenhop, 1978). Some controllability results which are relevant to this study are also given. ) is non-singular Lemma 3. The system (3) is completely controllable if and only if ( Proof. The proof can be observed from Proposition 3.1 in (Dauer and Gahl, 1977). □ Proposition 1. The system (3) is controllable on [
] if and only if
(
)
Proof. This is Lemma 3.2 in (Davies, 2006). □ Proposition 2. The system (3) is completely controllable with constraint on [ ( ). only if
] if and
Proof. This is Theorem 3.3 in (Davies, 2006). □ Proposition 3. The following are equivalent for system (3). (i) (ii) (iii)
( ) is non-singular system (3) is completely controllable on [ ] system (3) is proper on [
]
Proof. This is Proposition 3 in (Onwuatu, 1993). □ Lemma 4. The system (3) is completely controllable on [ ]. on [
] if and only if it is controllable
Proof. The proof follows immediately from Proposition 1 and 2. □ ] Let [ a positive integer, be absolutely continuous and define the differential ]. operator for neutral systems by ( )( ) ̇( ) ( )⁄ , almost everywhere on [ Higher powers of the operator are defined inductively by , with domain equal ] ]. Note that, by to all [ , such that is absolutely continuous on [ the ( ), [ ]. identity ( )( ) ( ) Define a positive integer and ( ] such that ( ) for ([ ] ) and adopt the convention shift operator by (
)( )
(
a nonnegative integer to be the collection of all ([ ] ), and the restriction |[ ] is in ([ ] ) . For ( ) define the
)
(
)
( ) and take Define to be the identity operator on by inductively using (11). Observe from the definition of the differential operator , and the shift operator that for ( ) is taken as a common domain for the operators and , if the function space
6
, then and commute in this setting and each commutes with multiplication by a scalar (element in ). For any matrix and matrix , one can define matrix by [ ] [ ] for integers . Consider the neutral system with delay of the form ( ⁄ )( ( )) ( ) ( ) ( ) ( ) where are constant matrices, is chosen to be constant real ) of (12) is the restriction to [ ] of the solution matrix. The solution ( ( ) ( ) of the equation ( ) ) by the equation Now define the matrix ( ( ) and let ( ) ( ) where “ ” denotes the transposed matrix of cofactors. Some basic relationship exists between these two operators which by Jacobs and Langenhop (1976) can be expressed as (
)
∑ ̂( )
( )
∑
(
where the matrix polynomials ( ) , ̂ ( ) are at most of degree argument. Using the polynomial ( ) in (13) define a unique matrix operator by ( ) [ ( ) ( ) ( ) ] Now, the operator ( ) can be written in the form of a polynomial to get ( )
∑
where, transpose of [
in their
( (
Theorem 2. Let sufficient for
are
)
[
constant real matrices, and let ( ( ( ) )] for all complex numbers
(
)
)
)) be the .
, then for (12) to be controllable on [ ] it is necessary and ] ( )) and ( ) ( for every complex .
Proof. This is Theorem 3.4 of Rivera Rodas and Langenhop, (1978). □ ,[
Corollary 1. Let (i) (ii)
( ) (
[
(
]
))
] and assume that system (12) satisfies the following ; , for every complex .
Then, system (12) is completely controllable on [
].
Proof. If condition (i) and (ii) holds, then by Theorem 2, the system (12) is controllable on [ ] . This by Lemma 4 implies that system (12) is completely controllable on [ ] . Conversely, if system (12) is completely controllable on [ ], then it is controllable by [ ] ( )) Lemma 4, and by Theorem 2; and ( ) ( for every complex , and the proof is complete. □ Theorem 3. In system (1), assume the following (i)
are
constant matrices, 7
is
constant real matrix
(ii) (iii) (iv)
(v)
[
] ; )) , for every complex , ( ) ,
for ( ) (
, ( ( )
( )
(
(
( )
(
))
(
)
(
∫
) ( )
) is uniformly stable.
Then, system (1) is null controllable with constraints on (
)
.
Proof. Because of (i), (ii) and (iii) system (1) is controllable on [ ] by Theorem 2. Hence, ( ) by Proposition 1. By condition (iv), and (v) the system (1) with ) ( ) ( ) and satisfies ( as . Hence, at some , hence , the domain of null controllability of (1). Suppose for the contrary that . Since is a solution of (1) with , then . This implies that, there ( ) exists a sequence such that as and , for any , therefore . It follows from the variation of constant formula (5) that: ( ) ( ) ∫ ( ) ( ) ( ). Then, since Let , ( ) ( ), for any , for any , and so and . However ( ) which is a contradiction. Therefore, as , and , and hence there exists a ball around 0 which is contained in . Again, by (iv) there exists some , ( ) ( ) . Therefore, using as initial point and as initial function, ( ) ) of (1) satisfies there exists and such that, the solution ( ( ) , and the proof is complete. □ Remark 1. The conditions imposed on Theorem 3 constrain in a box but these conditions can also allow the constraint set to be an arbitrary compact set as shown in Theorem 4. This is possible because the null controllability of linear neutral system, in general, depends on the length of the time interval over which the system operates (Jacobs and Langenhop, 1976). Therefore restrictions on interval could be made based on the requirements for the controllability of the linear controllable base system. Again, these different constraints by the [ ] definition of is possible because by Theorem 2, if the conditions , and ( ) ( ( )) ] for every complex holds on [ then (12) is controllable. ( ) ( ) This means ( ) . Now define a map taking [ ] by ( ) ( ) ( ) . Because is a continuous linear transformation of onto it is open (Hale, 1977). From the definition of , there is an open ball around zero, so that ( ) ( ) ( ). Therefore, ( ) ( ). This implies that ( ) is open, ( ) . Moreover, if it is assumed that (12) is since is open and therefore ] then by Lemma 3, ( ) is non-singular which in turn is completely controllable on [ ( ) ] , and equivalent to almost everywhere on [ . This implies ( ) by Proposition 2 and therefore properness of (12) by Definition 1, hence ) completely controllable on ( with constraint. 4.1.
Main result
The main result for the neutral control system with infinite delay will now be developed and proved in this section. 8
Theorem 4. Assume for system (2) that (i) (ii) (iii) (iv) (v)
the constraint set is an arbitrary compact set of ( ) ( ) is uniformly stable the system (4) is uniformly asymptotically stable; so that the solution of (4) )‖ ))‖ ‖, satisfies ‖ ( ( ( . the system (3) is completely controllable ( ) ( ( ) ( )) , The continuous function satisfies | ( ( ) ( ))| ( ) ) for all ( ( ) ( )) [ , where ∫ ( ( ) ( )) , and .
Then, the system (2) is null controllable. Proof. By (iv), exists for each to the integral equations ( )
(
)
(
)[
∫ (
(
) (
(
)
)
∫ (
( ) ( ))
∫ (
∫ (
) ( )
) (
) ∫ ( ) (
forms a pair
)
(
]
, ( )
for some suitably chosen ( )
. Assuming the pair of functions
( ), ∫ (
[
]
) ∫ ( ) (
)
( ) ( ))
(
( ) ( ), [ ]. ] and is a solution of (2) corresponding to Then is square integrable on [ [ ] initial state ( ) . Also, ( ) . It is necessary to show now that the arbitrary compact constraint subset of , that is | ( )| , for some constant By (ii) and (iii), and the continuity of in compact intervals, it follows that for some | ( | ( Hence, | ( )|
) )
∫
[
(
)|
(
)∫
(
(
)
)
with is in . ,
, ( ) (
))
)
|
∫
(
(
(
(
))
(
)) .
) ( ( ) ( ))
]
and therefore, | ( )| )) ( ) ( ( ( ( )) since and . Hence, from (17) can be chosen sufficiently large such [ ], showing that is admissible control. It remains to prove the that | ( )| , ( ) existence of a pair of the integral equations (15) and (16). Let represent the Banach ) [ ] [ ] space of all functions ( , where ( ) ([ ] ); ([ ] ) with the norm defined by
9
‖(
)‖
‖ ‖
‖ ‖ , where ‖ ‖ ( )
Define the operator ( )
(
)
(
)[
∫ ( [
for ( )
( )
)
(
( ))
) (
)
), where
( ∫ (
;‖ ‖
{∫
) ∫ ( ) (
| ( )|
}
[
]
| ( )|
(
.
(
)
(
)
].
∫ (
) ∫ ( ) (
].
[ and also ) . Again,
for
⁄
)
)
( ))
( ) for for and ( ) It is clear from (17) that | ( )| | ( )| ( . Hence, ‖ ‖ (
}
[
) ( )
⁄
| ( )|
)
( ) for
∫ (
∫ (
by (
) (
] and ( )
(
{∫
]
⁄
))
∫| ( )|
, so that
(
( ))
| ( )|. Since where , it follows that | ( )| ( ) | ( )| [ ]. and | ( )| ⁄ [ ], then ‖ ‖ ( ) [ ]. Hence, if . Let ( ) ) ‖ ‖ ‖ ‖ ( ) ( ). Then letting ( ) {( }, it follows that Now, since ( ) is closed, bounded and convex, by Riesz theorem (see Kantorovich and Akilov, 1982 ), it is relatively compact under the transformation . Hence, the Schauder’s fixed point theorem implies that has a fixed point. Hence, system (2) is null controllable. 5.
Example
Consider the neutral control system ( ⁄ )( ( )
(
))
( )
(
)
∫
(
) (
)
( )
(
)
(
)
(
)
and its perturbation ( ⁄ )( ( )
(
))
( )
( ) ( ( ) ( Its linear control base system is given by ( ⁄ )( ( ) ( )) ( ) and its free system
(
)
∫
) ( )) (
10
)
( )
(
) (
)
( ⁄ )( ( ) where
( ⁄
(
⁄ (
))
( )
(
)
)
( ) (
) ( ))
(
(
) ⁄ ) ⁄
( (
)
(
∫
( ( )
(
) (
(
)
⁄
(
)
),
( ))
))
( ) Now, use Lemma 1 to check that, the operator [ ] condition of Lemma 1 gives, (
is uniformly stable as follows: The
)
( ⁄ ) ( ⁄ ) ⁄ . Hence, the operator which implies , and stable if . Next, observe by Theorem 1 that the characteristic root of (23) is ( ( )) ( ( ) ( )) (
) ∫
(
is uniformly
)
(
)
and all the roots of (23) have negative real part. Hence by Theorem 1, system (23) is uniformly asymptotically stable. Finally, check that (22) is controllable as follows: ⁄ [ ] ( ) ( )
(
) (
( )
)
(
) [(
( ) ( ) (
(
))
(
Observe that for all complex
⁄
[ ( ) (
⁄
)
( (
( ) ] )⁄
)(
)
(
⁄ )] ⁄ (
(
⁄ ⁄ )⁄
(
)⁄
)
)
))
(
, ( ) (
(
))
( )
Therefore system (22) is controllable on ( ) Moreover, ) ( ))| | ( ) ( ( ) ( )) | ( ( ) ( ( )| ( ) ( Hence, all the conditions of Theorem 4 are satisfied and system (21) is null controllable.
)
Remark 2. This work can be extended to study conditions that would preserve the controllability/null controllability results when both system and input matrices undergo some parameter uncertainties. The problem of controllability of linear parameter uncertain systems 11
has received considerable research effort from control audience because of its significance in theory and its applications. These uncertainties in control systems analysis and design can be structured, if the uncertain parameter is an elemental part of the system and input matrices or unstructured, if the parameter uncertainties are contained in the systems matrices only. If is introduced into system (2) as uncertain parameters, and the constant , matrices ) respectively, depends linearly on for information, then (2) will and , ( be of the form ( ⁄ )( ( )) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ( ) (
)
(
( ))
(
)
∑ ∑ ∑ ). where , , ,( [ ] The question of null controllability for (26) on would depend on the stability of (26) when which can be estimated by using matrix norm or spectral radius (see Tai et al. (2009)), and the controllability of (12) which is relative to the controllable base for (26) amongst other assumptions on Theorem 4. That is, suppose (12) is controllable on the interval [ ], then the uncertain system (26) is controllable on [ ] for all sufficiently small, [ ] if it satisfies Corollary 1 and the conditions that and [ ] ( ). , This criterion can be easily ) ( ) matrix G, with an proved by introducing an ( matrix ; establishing their dependence and exploring their algebraic properties. 5.1
Simulation studies
The stability and controllability of the open loop system (21) can be illustrated using Simulink® and MATLAB® based simulation studies. The simulation model parameters are as given (21) with the default parameter setting and a square wave input where and are chosen to be and respectively with . Fig. 5.1 depicts the stability and controllability of the states when the simulation is performed with the linear control base system i.e. (22), and when the simulation is done with the perturbation function (see (21)). The amplitude of the internal state from the system response is observed to be slightly higher with the perturbation function whilst it exhibits a faster response when the simulation is done without the perturbation function. The settling times for the systems without the perturbation function are also observed to be faster; this is as expected and depends on the assumptions placed on the perturbation function (see (25)). The simulation showed that, the system (23) is stable and the overall control system (21) is controllable.
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Figure 5.1. Simulation of control input and system states with perturbation function and linear control base system
6.
Discussion
Unlike the conditions in (Dauer et al., 1998; Umana, 2008; Davies, 2006), the controllability conditions introduced in this paper are explicit and computationally more effective. The computation of the controllability matrix is not required since it is obtained by an equivalent rank condition. It generalises to neutral systems the rank condition in Davies (2006). Indeed, applying the rank condition from (Davies 2006) to systems (12) with on [ ], ([ ] ) satisfying ( ) , would result in which limits the results to retarded systems. This paper introduces a different rank condition, see (Corollary 1, (i)) based on Rivera Rodas and Langenhop (1978). The rank condition alone is not considered to be necessary and sufficient; the algebraic requirement in (Corollary 1, (ii)) makes it sufficient as well as necessary for controllability. Therefore the controllability condition on this paper generalizes the results of null controllability to neutral systems with infinite delays and yields a less conservative result. The conditions (ii), which relates to the initial condition or structure of the information for the system considered, and (iii) of Theorem 4 ensures that the error signals are contained within the neighbourhood of the origin as time increases and not asymptotically tends to zero. Conditions (ii) and (iii) are required for null controllability when the controls are restrained and make the system less conservative, being able to handle internal and external disturbances that may prevent signals from converging asymptotically to zero. Perturbations cause conservatism as they do not varnish in some cases when the states approximates the origin. It makes uniform asymptotic stability impossible for such systems that have non-vanishing perturbations (Kofman, 2005). Therefore, condition (v) of Theorem 4 is imposed to preserve systems properties and directional consideration. Ignoring or altering condition (v) may affect the system’s properties differently and even lead to conservatism.
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Conclusion Null controllability for neutral functional differential system with infinite delay have been developed and proved. The results were obtained with respect to the stability of the free system and the linear control base system having full rank with the condition that ( ) ( ( )) , with the assumption that the perturbation function satisfies the smoothness and growth condition placed on it. Acknowledgement The authors are thankful to the Rivers State Scholarship Board, Rivers State Secretariat Complex, Port Harcourt, Rivers State, Nigeria for supporting this research. The authors are also grateful to the reviewers for their useful comments References Balachandran, K., Leelamani, A., 2006. Null controllability of neutral evolution integrodifferential systems with infinite delay. Mathematical Problems in Engineering 2006, 1-18. Chukwu, E. N., 1979. Euclidean controllability of linear delay systems with limited controls. IEEE Transactions on Automatic Control AC-24, 798-800. Corduneanu, C., 2002. Absolute stability for neutral differential systems. European Journal of Control 8, 209-212 Cruz, M. A., Hale, J. K., 1970. Stability of functional differential equations of neutral type. Journal of Differential Equations 7, 334-355. Dauer, J., Balachandran, K., Anthoni, S., 1998. Null controllability of nonlinear infinite neutral systems with delays in control. Computers & Mathematics with Applications 36, 39-50. Dauer, J., Gahl, R., 1977. Controllability of nonlinear delay systems. Journal of Optimization Theory and Applications 21, 59-70. Davies , I., 2006. Euclidean null controllability of infinite neutral differential systems. Anziam Journal 48, 285-294. Kofman, E., 2005. Non-conservative ultimate bound estimation in LTI perturbed systems. Automatica 41, 1835-1838. Hale, J. K., 1977. Theory of Functional Differential Equations, New York, Springer Hale, J. K., 1974. Functional differential equations with infinite delays. Journal of Mathematical Analysis and Applications 48, 276-283. Jacobs, M. Q. Langenhop, C., 1976. Criteria for function space controllability of linear neutral systems. SIAM Journal on Control and Optimization 14, 1009-1048. Kantorovich, L. M., Akilov, G. P., 1982. Functional Analysis, Pergamon Press, Oxford. Khartovskii, V., Pavlovskaya, A., 2013. Complete controllability and controllability for linear autonomous systems of neutral type. Automation and Remote Control 74, 769-784. Onwuatu, J. U., 1984. On the null-controllability in function space of nonlinear systems of neutral functional differential equations with limited controls. Journal of Optimization Theory and Applications 42 (3), 397-420. Onwuatu, J. U., 1993. Null controllability of nonlinear infinite neutral system. Kybernetika 29, 325-338. Rivera Rodas, H., Langenhop, C., 1978. A sufficient condition for function space controllability of a linear neutral system. SIAM Journal on Control and Optimization 16, 429-435. 14
Sinha, A.S. C., 1985. Null controllability of nonlinear infinite delays systems with restrained controls. International Journal of Control 42, 735-741. Tai, Z., Wang, X., Wang, D., (2009). Robust Stability Analysis of Uncertain Neutral Functional Systems. Int.J.Innov.Comput.Inf.Control 5, 3013-3024. Umana, R., 2008. Null controllability of nonlinear infinite neutral systems with multiple delays in control. Journal of Computational Analysis & Applications 10, 509-522. Umana, R., 2011. Null controllability of nonlinear neutral Volterra integrodifferential systems with infinite delay. Differential Equations & Control Processes (ISSN 18172172) 2, 48-56. Underwood, R. G., Chukwu, E. N., 1988. Null Controllability of Nonlinear Neutral Differential Equations. Journal of Mathematical Analysis and Applications 129(2), 326345.
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