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A Note on the Sequential Linear Fractional Dynamical Systems from the Control System Viewpoint and L2 -Theory Abolhassan Razminia*, Vahid Johari Majd** * Intelligent Control Systems Laboratory, School of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran (e-mail: [email protected] ). ** Intelligent Control Systems Laboratory, School of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran (Tel: +21 8288-3353; e-mail: [email protected] ) Abstract: In this paper, we consider a sequential linear fractional differential equation with constant coefficients. After deriving a closed analytical solution based on the sequential fractional derivatives, we propose a theorem that gives some weak conditions under which any arbitrary function in L2 can be approximated by an output of a linear time-invariant fractional differential equation. More precisely, we prove that all L2 functions can be represented as the L2 limit of functions that are the outputs of fractional linear control systems. The analysis is developed in the Hilbert space. Keywords: Linear fractional differential equations, LTI control system, L2 theory. spaces. They are sometimes called Lebesgue spaces. One of the main topics in Lp space is approximation theory [11], which is employed here for deriving a relationship between an arbitrary function from L2 space and the output of a linear time invariant fractional control system. The dynamic of the control system is stated in the sequential format. Sequential fractional operator is a special case among other fractional operators which is discussed in the next section. 1. INTRODUCTION Fractional calculus is a generalization of ordinary differentiation and integration to an arbitrary real-valued order. This subject is as old as the ordinary differential calculus, and goes back to times when Leibniz and Newton invented differential calculus. The problem raised by Leibniz in a letter dated September 30, 1695 for a fractional derivative has become an ongoing topic for more than hundreds of years [1-2]. Due to the computing demands of the fractional differentiation and integration and the need for approximating some complicated functions which are used in physical process, we present a novel method in this paper by which any arbitrary function from L2 space can be approximated by the output of a linear time invariant time fractional control system (LTIFCS). Using a rigorous proof, we provide the construction procedure of LTIFCS whose output approximated the given function in the sense of L2 -norm. This subject has attracted the attention of researchers from different fields in the recent years. While it was developed by mathematicians few hundred years ago, efforts on its usage in practical applications have been made only recently. It is known that many real systems are fractional in nature, thus, it is more effective to model them by means of fractional order than integer order systems. Applications such as modeling of damping behavior of viscoelastic materials [3], cell diffusion processes [4], transmission of signals through strong magnetic fields [5], and finance systems [6] are some examples. Moreover, fractional order dynamic systems have been used in both design and implementation of control systems. Studies have shown that a fractional order controller can provide better performance than an integer order one and lead to more robust control performance [7]. This paper is organized as follows: after this introduction in section 1, a historical review is presented in section 2. Some basic concepts and mathematical preliminaries are summarized in section 3. Section 4 studies the main result which is stated in a theorem form. Finally conclusions in section 5 close the paper. 2. HISTORICAL REVIEW It is has been reported that specific fractional differential equations with order less than three, may exhibit complex dynamical evolution even chaotic dynamics [8-9]; however, this is not the case for ordinary differential equations due to the famous Poincaré–Bendixson theorem [10]. L2 space as a special case of Lp space is considered here. The Lp spaces are function spaces defined using natural generalizations of p-norms for finite-dimensional vector How well can functions be approximated on a finite interval by the output of SISO controllable and observable fractional linear time-invariant system? This question was first considered in the framework of constructing the inverse of a linear system in the integer order system basin [12]. Here we want to extend this problem for the first time, in a more comprehensive field, i.e. fractional order systems. Although in [13], Unser’s method has been extended for fractional 1 systems, the method presented in this paper is completely different and has a more general form. for ordinary derivatives. Some of the current definitions for fractional derivatives are described in [21]. The Riemann– Liouville definition is the simplest definition to use. Based on Over the past few decades, the theory of linear control theory has centered on stability properties that only really make sense on an infinite interval. However, there are many problems in which the only thing of interest is a finite interval. Indeed, this work began with questions of constructing flyable trajectories for linear systems [14]. Since the appearance of those initial papers, there has been a growing interest on the construction of curve approximation of using linear control theory, which has developed into the theory of interpolating and smoothing splines. q th this definition, the order fractional integral of f L [0, t ] and the terminal value a is given by [22]: 1 (1) t 1 (t s ) q 1 f ( s ) ds ( q ) J a q f (t ) f q (t ) ; q R a where the Euler-Gamma function is defined as follows: (2) ( q ) e q u q 1du ; qR 0 The theory of interpolating splines began with the paper of Schoenberg [15] in 1946 and has begun to be widely employed since 1960s with the availability of digital computers. The theory of smoothing splines began with the work of [16] and received a tremendous impetus with the publication of her CBMS lectures. A more theoretic problem can be found in [14]. As another definition for the fractional derivative we propose the Caputo’s definition [22]: t f (m) ( ) 1 d ( m ) (t ) 1 m D f (t ) J a m D m f (t ) a * dm f (t ) dt m Almost all of the literature has been devoted to polynomial splines and an impressive body of tools has been developed for this application. A body of literature has developed around the concept of control theoretic splines. The initial work was with respect to flight control and is represented by the works of Crouch and Jackson [11]. m 1 m (3) m In the same way we can prove the following property for the Laplace transform of the Caputo derivative [23]: L{D f (t )} s L{ f (t )} * m 1 s k 1 f (k ) (0) (4) k 0 As one can see, in the Caputo derivative we need the integer order derivatives of the initial conditions while in the Riemann-Liouville we need the fractional derivative of the initial conditions. This is a merit of Caputo definition. In the area of smoothing splines Martin and colleagues have developed an theory of smoothing splines based on linear control theory [17]. Some modifications in a more general prospective under weaker conditions and more general spaces, e.g. Banach spaces, of the problem studied in [17], were treated in [17] and [18] recently. These works were motivated by the trajectory planning problem in order to avoid high accelerations that were observed when it was required to be exactly at point at a given time. The constraint of interpolation was relaxed to approximation and a simple optimal control problem yielded generalized smoothing splines. This technique was shown to be adaptable to a number of situations not easily handled with traditional polynomial smoothing spline techniques. The question that has been in the background of all of the control theoretic splines has been its convergence. This problem was directly attacked in [19]. In particular the paper [20] gives a fairly comprehensive theory of convergence of Miller and Ross in [24] introduced the so called sequential fractional derivative in the following way: MR D (5) MR D D , 0 1 MR D k MR D MR D (k 1) , k 2, 3,... where D is the Caputo fractional derivative. Next we define a sequential fractional differential equation in an operator format as follows: n a (n 1) ... a (6) MR D n 1 MR D 1 MR D a0 y(t ) f (t ) without loss of generality we restrict 0 1 . As in the usual case, it is easy to obtain the connection between (6) and the corresponding system of linear fractional differential equations. If we apply the following change of variables to (6): (7) x (t ) y(t ), ( D x )(t ) x (t ), ( j 1, 2, ..., n 1) 2 splines in a L sense. However, it was necessary to impose some rather restrictive conditions on the linear system, which although were natural from an approximation view point, were not natural from the viewpoint of control theory. Beside these restrictions and developments, we want to extend these tasks to a more general form, i.e. fractional order control systems. The new theory might have an important impact on the interpolation and spline theory. Moreover, this methodology can be used fairly in the signal processing and system identification. 1 j j 1 j where we have used the Caputo fractional derivative. Using the above notation we have: D x1 x2 , D x2 x3 , (8) D xn 1 xn , D xn 2. PRELIMINARIES n 1 a x k k f (t ) k 0 Or in a matrix form with a companion matrix A , D x(t ) Ax (t ) f (t ) Fractional calculus as an extension to ordinary calculus possesses definitions that stem from the definitions existing 2 (9) This is a standard linear fractional differential equation (LFDE). It is easy to see that the vector M is the initial conditions of x . Also one can show that the particular solution is as follows [28]: Definition 1 [25]. Let 1 p and S , , be a measure space. Consider the set of all measurable functions from S to C (or R) whose absolute value raised to the p-th power has finite integral, or equivalently, that: 1 (10) p p f p : f d t xp 0 k 0 S k 0 Ak (t ) (1 k ) 1 f ( )d [(k 1) ] (16) t Ak (t ) ( k 1) 1 f ( )d [(k 1) ] 0 In this paper we consider a special case with p 2 which is Therefore the general solution is: the most important case in the Lp spaces. Consider the dynamical system: x(t ) xh (t ) x p (t ) k 0 D x(t ) Ax (t ) bu (t ) y Cx (11) k 0 Definition 1 [26]. The system (11) is observable on [t0 , t f ] t k A (t ) ( k 1) 1 f ( )d [(k 1) ] y (t ) and u (t ) on [t0 , t f ] . 0 D x(t ) Ax (t ) bu (t ) y Cx Definition 2 [26]. System (11) is controllable iff for any ( x0 , x1) there exists a control u (t ) defined on [t0 , t f ] which y Cx C. k 0 3. MAIN RESULTS As in the integer order case, consider xh and x p are the k 0 homogeneous and particular solutions of (9), respectively. 0 (13) uL2 [ 0,T ] k 0 2 (21) 0 (u ) is dense in L2[0, T ] . Suppose that (u ) is not dense in L2[0, T ] . Thus there M exists a nonzero function v L2 [0, T ] such that (u ), v 0 for Ak (t ) (1 k ) 1 (t ) (1 k ) 1 D M Ak D M [(k 1) ] [(k 1) ] k 0 k 0 (m(t ) y(t )) dt 0 Proof. It is enough to establish that the range space of the all u L2 [0, T ] . So To show that this can be the homogenous solution of (9), based on the differintegrability of xh it is easy to differentiate (14)[27]: (20) T inf [(k 1) ] k 0 Ak (t ) (1 k ) 1 bu ( )d [(k 1) ] Therefore for every m(t ) L2 [0, T ] there exists a u L2[0, T ] such that: So based on this proposition it is sufficient to find the homogeneous and particular solutions separately. Using a similar method in finding the homogeneous solution for integer order differential equations, we can find the homogeneous solution for LFDE as (14) Ak (t ) (1 k ) 1 Ak A( xh x p ) f Ax f . 0 t D x D ( xh x p ) D xh D x p Axh ( Ax p f ) k 0 (19) CA (t ) ( k 1) 1 bu ( )d [(k 1) ] (u )(t ) C D xh Axh , 0 Theorem. Consider the system (18) which is controllable and observable. Defining a linear operator as follows: xh (t ) t k Proof. Inserting xh and x p in (9) we have: t Ak (t ) ( k 1) 1 bu ( )d [(k 1) ] Now we present the main result of the paper: Proposition 1. For the LFDE (9) we have the general solution as follows: (12) x(t ) xh (t ) x p (t ) (18) According to the previously developed theory we can write: drives the initial state x(t0 ) x0 to the final state x(t f ) x1 . (17) Now we return to a control system problem. Consider a fractional control system with zero initial conditions and companion matrix which is not a restrictive assumption: iff x (t ) for t [to , t f ] can be deduced from knowledge of D x p Ax p f , Ak (t ) (! k ) 1 M [(k 1) ] (t ) k 1 ((1 k ) ) M A. [(k 1) ](k ) j 0 (15) A j (t ) (1 j ) 1 M [( j 1) ] 3 T (u ), u T v(t ) C (t ) 1 0 k 0 k 0 k 0 0 T Now introducing the vector valued function ( ) as follows, we can manipulate the relations easily: (29) C c1 ... cn , ( ) 1( ) 2 ( ) ... n ( )' A k (t ) k bu( )d dt [(k 1) ] T CA k b [(k 1) ] CA k b [(k 1) ] v(t ) (t ) (k 1) 1 u ( )d dt 0 TT Using these notations we have (t ) (k 1) 1 u ( )d v(t )dt u ( )d (22) since ( A, b) and ( A, C ) are controllable and observable, respectively. So we can write: n (31) u , * (v ) c D ( ) 0 where i * (v)( ) T [(k 1) ] (t ) k CA b k 0 T ( k 1) 1 k CA b ( k 1) 1 v(t )dt Introducing the following state variables: (23) D1 i 1, i 1, 2, ..., n 1 v(t )dt 0 1 0 D 2 c0 n 1 cn So it is concluded that * (v)( ) 0 . Now we use the fact that (u ) is onto if * (v) is one-to-one [29]. It suffices to show that v 0 almost everywhere. Let k 0 Ak b (t )( k 1) 1 v(t )dt [(k 1) ] (24) ( ) (l 1) l (l 1 ) 0 0 D( ) D k 0 0 1 0 c1 cn (Q) k ( k 1) 1 (0) ((k 1) ) (26) 0 k 0 ( A) k ( ) ( k 1) 1 bv( )d 0 ((k 1) ) ( A) k D 0 ( )( k 1) 1bv( )d (( k 1 ) ) k 0 0 Or briefly D0 (27) k 0 ( A) k D ( )( k 1) 1bv( )d ((k 1) ) 0 lim D ( )( k 1) 1bv( ) 0 of the original system ( A, b) it is obtained that for an initial value 0 we have v( ) 0 a.e. k 0 k 0 ( A) k ((k 1) ) 0 ( ) (k 1) 1 bv( )d (37) 0 So from * (v)( ) 0 and using the controllability condition ( A) k (k 1) 1 0 ((k 1) ) (36) 0 Differintegrating of (36) once we have: A( ) bv( ) (35) and hence that D( ) A( ) bv( ), (T ) 0 (33) (34) ( A) k (k 1) 1 ( A) k 0 ( ) (k 1) 1 bv( )d 0 ((k 1) ) ((k 1) ) k 0 k 0 0 Ak b (t ) (k 1) 1 v(t )dt [(k 1) ] T Ak b ( k 1 ) 1 D (t ) v(t )dt bv( ) [( k 1 ) ] k 0 T A jb A (t ) ( j 1) 1 v(t )dt bv( ) [(k 1) ] j 0 0 1 0 1 1 0 2 Q 2 1 c1 n n cn We deduce that T 0 Since (T ) 0 we conclude that (0) 0 and consequently ( ) 0 . This establishment is based on the assumption that cn 0 , but this approach can be easily extended to more general case. Anyway we have: (25) 2. D K ( , )d D K ( , )d lim D 1K ( , ) 0 k 0 1 Using (17), we have: Using the following important facts [30] 1. D l (32) we have: k 0 T i i 0 [(k 1) ] (t ) ( ) cii ( ) 0 i 0 0 (30) n 0 So this shows that * is one-to-one and hence that is onto (28) a dense subspace of L2[0, T ] .□ 0 0 This idea indicates an important point in control theory. Indeed given a function in L2 space, we can propose a fractional order transfer function which can approximate that C( ) 0 4 function in term of L2 norm. One of the main advantages of this idea is that we can analyze a very complex function in L2 space by transferring it to a fractional order transfer function space which may exhibits a more general transfer function than the usual ones. Transferring it to the second space we can develop a n analysis using well-known existing methods such as frequency methods, root-locus methods, and state space techniques, which are well established and easy to use. belongs to L2 can be approximated by an output of a linear time-invariant fractional differential control system. More precisely we gave a proof that every L2 function can be represented as the L2 limit of functions that are the outputs of fractional linear control systems. Our analysis was developed in the Hilbert space time-invariant fractional differential control system. More precisely we gave a proof that every L2 function can be represented as the L2 limit of functions that are the outputs of fractional linear control systems. Our analysis was developed in the Hilbert space. 4. 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