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Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 846590, 24 pages
http://dx.doi.org/10.1155/2010/846590
Research Article

On Chebyshev's Systems and Non-Uniform Sampling Related to Caputo Fractional Dynamic Time-Invariant Systems

Institute for Research and Development of Processes, Faculty of Science and Technology, University of Basque Country, Campus of Leioa, Aptdo. 544, 48080 Bilbao, Spain

Received 4 October 2010; Accepted 18 December 2010

Academic Editor: Jinde Cao

Copyright © 2010 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with the investigation of the controllability and observability of Caputo fractional differential linear systems of any real order . Expressions for the expansions of the evolution operators in powers of the matrix of dynamics are first obtained. Sets of linearly independent continuous functions or matrix functions, which are also Chebyshev's systems, appear in such expansions in a natural way. Based on the properties of such functions, the controllability and observability of the Caputo fractional differential system of real order are discussed as related to their counterpart properties in the corresponding standard system defined for . Extensions are given to the fulfilment of those properties under non-uniform sampling. It is proved that the choice of the appropriate sampling instants is not restrictive as a result of the properties of the associate Chebyshev's systems.

1. Introduction

Caputo fractional calculus is a very useful tool to calculate alternative solutions to the classical ones in many applications as, for instance, in dynamic systems (see, e.g., [13]). Since the fractional order can be a nonpositive integer, even real or complex, the technique can be used to better fix the trajectory solution of mathematical models to obtained experimental data due to modelling or measuring errors. On the other hand, the so-called Chebyshev system of linearly independent functions in the Banach space of continuous functions , endowed with the supremum norm, has the important property, due to Haar, that each nontrivial polynomial of this system has at most () distinct zeros in , [4]. There are many sets of linearly independent functions which are Chebyshev’s systems as, for instance, (a) the sets , respectively, on any real interval of nonzero finite measure, respectively, on , and (b) the set with being continuous and with no zeros on is a Chebyshev system on . The linearly independent functions defining the expansions of the -semigroup , of differential generator , in powers of are a Chebyshev system on any interval where is the maximum eigenfrequency; that is, the maximum absolute imaginary part of any complex eigenvalues, if any, and otherwise (see [57], firstly discussed in [6]). In [5], it is proven that the property holds even if is a complex matrix which is not related by a similarity transformation to some real one, that is, some potential complex eigenvalues may not have their complex conjugate counterparts as eigenvalues. In [6, 7], the Haar property of Chebyshev systems is used to formulate the following properties:(a)controllability to the origin of standard linear time-invariant dynamic systems (roughly speaking, the ability to steer any initial condition to the equilibrium in any finite time by injecting some admissible control) under, in general, non-uniform (also referred to as aperiodic or nonperiodic) sampling, (b)observability of standard linear time-invariant dynamic (i.e., the ability to calculate initial conditions from past values of a measured output trajectory) under, in general, non-uniform sampling, (c)identifiability of standard linear dynamic time-invariant systems (i.e., the ability to compute the values of the parameters from output measurements and eventually time-derivatives up till some order of the output trajectory) and model matching of linear and time-invariant systems also under, in general, non-uniform sampling.

It has been proven in the above papers that the Haar property ensures that the corresponding properties are transferred from the continuous-time dynamic system to its discrete-time counterpart for infinitely many choices of sets of sufficient large cardinal (exceeding a lower-bound being dependent on the degree of the minimal polynomial of and the dimensions of the output and input spaces) of distinct sampling instants. The particular choices of the sampling instants may be done by some practical considerations as, for instance, to achieve a well-conditioned coefficient matrix of the resulting algebraic problem related to controllability, observability, local identifiability and so forth. The study of the properties of controllability, reachability (related to controllability to any final state), observability and constructability (closely related to observability) under non-uniform sampling has been extended more recently to nonfractional positive linear continuous time systems, namely, those having nonnegative state and output trajectory for any given non-negative initial conditions and controls [8]. Many other theoretical studies and applications have been performed. In [9], the reconstruction problem from non-uniform data is focused on. In [10], the general non-uniform sampling in a stochastic framework is investigated. The problem of sequential estimation under non-uniform sampling is studied in [11]. In [1214], some estimation properties and application under non-uniform sampling are described. The sampling efficiency in event-based sampling laws of signals is discussed in [15]. Finally, some filtering properties under non-uniformly sampled signals are investigated in [16]. All those commented studies were referred to the standard (i.e., nonfractional) case. Therefore, the extension given in this manuscript which includes the fractional case of positive real order is novel in the context of non-uniform sampling. The paper is organized as follows. Section 2 is devoted to some preliminary results concerning the expansions of the evolution operators of the solution in powers of the matrix of dynamics of a linear time-invariant dynamic system. Sets of Chebyshev’s systems of functions appear in a natural way in such expansions. Such sets have the property of keeping nonsingular, under very generic sampling, the coefficient matrices of the algebraic systems of equations associate to controllability, respectively, observability problems in the case that the continuous-time counterpart is controllable or, respectively, observable. The results are extended in Section 3 for Caputo fractional dynamic systems on any real order . Those results are used in Sections 4-5 to discuss the properties of observability and controllability and their counterparts under non-uniform sampling. Finally, an example is given in Section 6.

Some notations used are ; . A corresponding notation are used for the corresponding subsets of the integer set . , for all , and denotes that the matrix is positive definite.

2. Preliminaries on the Expansions of exp(At) and Associate Chebyshev’s Systems

This section is devoted to preliminary basic results about useful sets of linearly independent real functions and real matrix functions of time, which are also Chebyshev systems [47], and used for the expansions of the evolution operators defining the solution of the differential system.

Lemma 2.1. Assume that is the degree of the minimal polynomial of . Then is a linearly independent set of matrices.

Proof. Proceed by contradiction by assuming that the set is linearly dependent so that there is a nonidentically zero set such that . Assume that for so that Choose which always exists since is a nonidentically zero set and . Thus, and some is then the degree of the minimal polynomial of which contradicts that such a degree is .

Lemma 2.2. The following formula holds with unique complex coefficients , for all , for all .

Proof. Note from Cayley-Hamilton theorem that , thus, so that the above identity holds irrespective of if and only if Proceeding recursively with the above formulas, one gets for all with The uniqueness of the coefficients follows from Lemma 2.1 as follows. Assume that there are two sets of coefficients such that . Then, which implies for all for all from Lemma 2.1 since is a linearly independent set.

Alternative Proof
Equivalently, proceed by complete induction by assuming that with Then, it follows that (2.4) holds with

A particular case of interest of (2.8) is that allows to interpret the two following common formulas for the Cayley-Hamilton theorem: where and are, respectively, the degrees of the minimal and characteristic polynomials of . On the other hand, it is possible to extend the expansion of powers of up till the degree of the characteristic polynomial of through the use of modified coefficients as follows: so that , for all and it turns out that the extension also works for , for all with , for all , and for . There has been proven the following.

Lemma 2.3. There exist complex coefficients and , for all , for all , such that the formulas are true. Those sets of coefficients coincide if .

It turns out that he sets of expanding coefficients are real if the matrix is real or if it is complex being similar to a real one, that is, all complex values, if any, appear by complex conjugate pairs or identical multiplicities. Now, the above results are related to the fundamental matrices of time-invariant differential systems as follows. Consider -semigroup of infinitesimal generator is an evolution operator in point-wise defined by the defining the trajectory solutions of the differential system of order n ; . An equivalent description is that is the fundamental matrix function (or the state-transition matrix function) of the above differential system. The expansions of in finite powers of , being not less than its minimal polynomial, where studied in detail for in [47] by using Chebyshev sets of complex functions.

Lemma 2.4. The formulas , for all for linearly independent sets of functions which are unique and analytic in for each and which satisfy a -order linear time-invariant differential system on .

Proof. It follows after using Lemma 2.3 with that with , ; for all , where Using the recursive expressions (2.9) for the in the above formula and using also the recursions (2.12) for the , for all , one gets These functions satisfy sets of linear time-invariant ordinary differential equations as follows: that implies The above constraint holds for any matrix so that functions in the sets , for all are linearly independent since they are the solutions of the linear time-invariant differential system for all ,for all , with , for all , whose respective unique solutions are where with , and Then, there exist unique sets of linearly independent functions, which are unique and analytic in for each from (2.19)-(2.20), such that the formulas hold, for all , for all .

Remark 2.5. Note that it has been proven that there exist sets and ; of complex coefficients, with , for all , such that the following polynomial expansions are true , where and are the degrees of the minimal and the characteristic polynomial of , respectively. In general, for . Those sets are unique for each . The above formulas imply that there exist unique sets of linearly independent functions, which are unique and analytic in for each from (2.19)-(2.20), such that the formulas hold, for all .

The following result is useful to relate the coefficients of the minimal and characteristic polynomials of to the two more relevant versions of the Cayley-Hamilton theorem.

Lemma 2.6. Assume that is the characteristic polynomial of a nonzero matrix of order and Jordan form . Then, .
Assume that , with , is the degree of the minimal polynomial of , . Then, , where is a -square matrix and T and are -square nonsingular and matrices, respectively, which are unique except multiplication by a nonzero scalar and is the characteristic and minimal polynomial of .

Proof. The first part follows from the Cayley-Hamilton theorem in the form and the fact that the matrix satisfies its own characteristic polynomial. If is the degree of the minimal polynomial of , then there exist a nonsingular real -matrix and a real matrix , formed by identity and zero-block matrices of appropriate orders which pickup from the higher-order Jordan blocks of each eigenvalue of , whose orders are given by the respective index of each eigenvalue, such that the identities hold. The last identity holds from the Cayley-Hamilton theorem since the characteristic polynomial of and the minimal polynomial of and coincide.

3. The Expansions of Evolution Operators of Functional Fractional Caputo Differential Systems and Their Associate Chebyshev Systems

The above results are extended to the following fractional Caputo differential systems of order : with , for some and is the control matrix. If , then (3.1) is referred to in the sequel as the standard system. The initial conditions are , for all . The admissible function vector is any given bounded piecewise continuous control function. The following result is concerned with the unique solution on of the above differential fractional system (3.1). The proof follows directly from a parallel existing result from the background literature on fractional differential systems by grouping all the additive forcing terms of (3.1) in a unique one (see, e.g., [1], equations (1.8.17), (3.1.34)– (3.1.49), with ).The linear and time-invariant differential functional fractional differential system (3.1) of any order has the following unique solution on for each given set of initial conditions and each given control being a bounded piecewise continuous control function, [1]: with if and if . Direct calculations yield that for any set of initial conditions the solution (3.2) verifies (3.1). Uniqueness is direct from Picard-Lindeloff theorem. The matrix functions , for all and from to are calculated via the Mittag-Leffler functions which, after using the identities , for all from Lemma 2.3 for the matrix , lead to for all , where, provided that , is the -function of definition domain restricted to and the elements of the sets of functions , for all and , for all are defined for as follows:

Remark 3.1. Note that the homogeneous solution of the fractional differential system is given by Then, (see, e.g., [1, 3]). On the other hand, if only one of the above point initial conditions is nonzero for some arbitrary , then (3.7) is decomposable for each additive term resulting in: Thus, is a fundamental matrix for all which satisfies the fractional differential system of order for all . Note that the fractional system becomes the standard one for and so that (defined in Lemma 2.4) and .

The following result of Section 2 extends the linear independence of the functions expanding to those appearing in the expansions of , for all and

Lemma 3.2. All the sets of functions and , for all are analytic and linearly independent on for any given , fulfilling and for all , for all . Furthermore, the sets satisfy a -order linear time-invariant fractional differential system on , for all .

Proof. The sets of functions are analytic since from their defining formulas, it follows that they are infinitely differentiable on . Their linear independence is proven by contradiction. Since the rows of the fundamental matrices of solutions ; for all of the Caputo fractional differential system of order are linearly independent on , it follows that for any ; for all . But, if is a linearly dependent set on , since is a full row rank matrix, then it exists such that for all which contradicts that in order that be a fundamental matrix, that is, their rows are linearly independent functions. The functions in the set are also linearly independent from the above considerations for the particular case , of . Note that the functions in the various sets are zero for so that linear independence is restricted to. On the other hand, the functions in the set , satisfy the differential system for all , subject to initial conditions , , for all , for all , for all . The functions in the sets satisfy initial conditions for all , , for all , for all , for all .

Remark 3.3. Note that, in the nonfractional standard case, the solution is also obtained from the fractional solution of order (3.2) as a particular case for , , which results to yield , for all , that is, the fundamental matrix appears simultaneously for the homogeneous solution and for the zero-initial-state forced one and in such a case is the exponential matrix function which also defines a -semigroup of infinitesimal generator . It can be pointed out that in the fractional case , the fundamental matrix is not an exponential matrix function as it follows from (3.3). The linear independence of functions in Lemma 3.2 applies for the general fractional case as for the nonfractional case. This is an essential point for the non-uniform sampling case discussed later on in this paper.

4. Observability and Controllability of Linear Fractional Differential Time-Invariant Systems

Consider the Caputo fractional differential system (3.1) with a measurable output defined by for some with . The following observability property is characterized.

Definition 4.1. The Caputo fractional differential system of order is said to be observable in the observation time interval if , for all can be uniquely calculated from the measurable output ; for some real interval of nonzero measure.

Theorem 4.2. The Caputo fractional differential system (3.1), (4.1) of order is observable in for any finite only if the standard dynamic system, that is, that resulting when , that is, is observable.

Proof. First, consider the homogeneous Caputo fractional differential system (3.1), that is, . One gets from (3.2)-(3.3) that where , and for all is a real matrix function; for all where . Since (4.2) is observable, then Furthermore, since the sets are linearly independent on from Lemma 3.2, the map from to , for all from the initial conditions to the output-trajectory of the homogeneous system defined by (4.3) is injective. Thus, can be uniquely calculated from . The sufficiency part has been proven. The necessity is obvious since if the rank in (4.4) is less than , so that (4.2) is not observable, then the above mentioned map from to is not injective. The above considerations on (4.3)–(4.5) may be extended directly to the case that the admissible control is nonzero with its proof remaining valid, by replacing via (3.2) and (4.1)

Remark 4.3. Note that Definition 4.1 and Theorem 4.2 characterize the observability property on any time interval of nonzero measure so that the property is independent of the time interval used for observation purposes. On the other hand, the property is independent of because it implies the observability of the standard system (4.2) and it is implied by such an observability.

Remark 4.3 is formally enounced in the subsequent result.

Corollary 4.4. The observability property of the Caputo fractional differential system (3.1), (4.1) is independent of any real fractional order , with for any given , and independent of the observation interval provided it is of nonzero measure.

The observability property can be tested as follows.

Corollary 4.5. The Caputo fractional differential system (3.1), (4.1) of any real order , with for any given , is observable if and only if any of the three equivalent conditions holds(1)(2) where is the spectrum of ,(3), for any finite time interval of nonzero measure

Proof. The equivalence of Conditions (1)-(2) follows from the equivalence of the observability test for the standard system (i.e., ) with the Popov-Belevitch-Hautus spectral observability test, [8, 17], with the observability of any Caputo fractional system of order with by using Corollary 4.4. The equivalence of Conditions (1)-(2) with Condition 3 follows from the fact that the observability of the fractional system of order is uniform with respect to time (Theorem 4.2 or Corollary 4.4) and the fact that (3.6) into (4.1) with (4.9) yields for the unique (nontrivial) measurable output trajectory of the homogeneous fractional system of order for each   for any time interval finite of nonzero measure. Thus, is unique from (4.10), and then the system is observable, if and only if . Thus, Condition is equivalent to Conditions (1)-(2).

Remark 4.6. Note from Condition of Corollary 4.5 that if the Caputo fractional system of order , with , is observable, then for any finite time interval of nonzero measure. This is easily seen by taking initial conditions such that for some while , for all .

Remark 4.7. Note that Theorem 4.2 is alternatively proven from the spectral observability test as follows. Since the rank in (4.8) can only be lost for , then (4.8) is equivalent to Using Laplace transform of (3.1), [2], the spectral observability of the Caputo fractional system of order is lost if and only if for some . Thus, note that Then, the Caputo fractional system of order , with , is (is not) spectrally observable (equivalently, observable) if and only if the standard system is ( is not) spectrally observable.

The controllability property is now discussed.

Definition 4.8. The Caputo fractional differential system of order is said to be controllable in the time interval of nonzero finite measure if it exists an admissible control which steers the state-trajectory solution to any prescribed value for any given initial conditions.

Theorem 4.9. The Caputo fractional differential system (3.1), (4.1) of any real order , with for any given, is controllable if and only if any of the three equivalent conditions holds:(1)(2) where is the spectrum of ,(3), for any finite time interval of nonzero measure.

Proof. Assume the forced solution of (3.2) under zero initial conditions, namely, ; for all so that for all , for all after using (3.4) where are -vector functions being dependent on the control; for all ; for all , and are, respectively, a real matrix and a real -vector function. Since the functions in are linearly independent and analytic on , for all for any given (with –Lemma 3.2), then , for all is, by construction, a linearly independent set of analytic -vector functions on a finite nonzero measure time interval for any given , provided that the control is nonzero on some subset of nonzero measure of . Thus, note from (4.17) that for arbitrary for some linearly independent set on defined for some admissible nonzero control the Caputo fractional differential system of order is controllable, independent of , if and only if what follows from Rouché-Froebenius theorem from Linear Algebra. Then, the controllability is guaranteed by that of the standard system with . The equivalence of (4.15) with the spectral controllability condition (4.16) follows from Theorem 4.2 and the well-known duality result and the state trajectory solution (3.2). The pair is observable in the sense that (4.7), equivalently (4.8), holds if its dual pair is controllable. The equivalence of Conditions (1)-(2) with Condition is proven as follows from (4.16). Assume that the control is generated as ; for some . Then, the controllability constraint is solvable for any prescribed if and only if with the control solution being Then, the Caputo fractional differential system of any order is controllable independent of on if . Sufficiency has been proven. Necessity follows by contradiction. Assume that the system is controllable and fails for a given of nonzero finite measure. Then, the columns of the matrix function are not linearly independent vector functions on for any admissible control . Then, from (4.17)-(4.18a), (4.18b) either which contradicts the controllability Condition or the set is not linearly independent on for any admissible control what is impossible if . Then, Condition is equivalent to the equivalent Conditions (1)–(3). All the proof may be easily readdressed for nonzero initial conditions by replacing

Note that, although the controllability property is independent of , the control (4.21) depends on . The matrices and are referred to as the controllability matrix of the pair () and the observability matrix of the pair (), respectively, and have to be of rank in order that the standard system by controllable (see Theorem 4.9, (4.14)), respectively, observable (see Corollary 4.5, (4.7)).

Remark 4.10. Note from Corollary 4.5 that the rank condition (4.7) requires the necessary condition which is then a necessary condition for observability. In the same way, the rank condition (4.15) in Theorem 4.9 requires which is then a necessary condition for controllability.

5. Observability and Controllability under Non-Uniform Sampling

The results of the above section on controllability and observability are now extended for, in general, non-uniform sampling. It has proven in Section 4 that if the standard continuous-time system (i.e., ) is controllable/observable then, any fractional system of real order keeps the corresponding property from that of the standard system. It is now proven that the properties are still kept under non-uniform sampling for almost all choices of the sampling instants provided that their numbers are nonless than the degree of the minimal polynomial of the matrix . Such generic choices are possible from the following important property: the linearly independent matrix/vector functions which expand the evolution and control operators defining the state and output trajectory solutions of the Caputo fractional system of order in polynomial functions of the matrix in (3.1) are Chebyshev’s systems, [47].

Theorem 5.1. Assume that the standard system is observable (controllable). Then, the sampled Caputo fractional system of order is observable (controllable) for almost any choices of distinct sampling instants subjects to distinct in-between sampling intervals.

Proof. Note from (4.3) and (4.17) that the measurable output of the homogeneous system of the Caputo fractional differential system of order and the transpose of its forced solution under zero initial conditions at any time instant are, respectively: for all , for all , with , where , with , and . Now, define for some set of sampling instants ; and for some set of sampling instants . One gets from (5.3) via (5.1) and (5.4), and from (5.5) via (5.2) and (5.6) the following linear algebraic systems: Note that is full-rank equal to if and (i.e., if the standard system with is observable). A unique solution exists to (5.7) if and only if the set of sampling instants is chosen so that the square real matrix is nonsingular provided that since then so that (5.7) is a compatible linear algebraic system (then, the map from to   defined via (5.7) is injective) leading to a unique solution being a least-square estimation of given by where from Rouché-Froebenius theorem for compatibility from Linear Algebra. Then, the observability property is preserved from the standard system for such a set of samples. Also, if if , so that the standard system is controllable, then in (5.8), if and only if the set of sampling instants is chosen so that the square real matrix is nonsingular. Then, the controllability property is preserved from the standard system for such a set of samples. Since the linearly independent real matrix vector functions and linearly independent real vector functions , for all , for all of domain are also Chebyshev’s systems, [47] both and are nonsingular matrices for almost all choices of the sampling instants.

Remark 5.2. It suffices to choose the sampling periods (i.e. the time intervals in-between any two consecutive sampling instants) in Theorem 5.1 as being mutually distinct and belonging to real intervals of the form , where