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.
1. INTRODUCTION spaces. They are sometimes called Lebesgue spaces. One of the main topics in Lp space is approximation theory [11], Fractional calculus is a generalization of ordinary which is employed here for deriving a relationship between differentiation and integration to an arbitrary real-valued an arbitrary function from L2 space and the output of a linear order. This subject is as old as the ordinary differential time invariant fractional control system. The dynamic of the calculus, and goes back to times when Leibniz and Newton control system is stated in the sequential format. Sequential invented differential calculus. The problem raised by Leibniz fractional operator is a special case among other fractional in a letter dated September 30, 1695 for a fractional operators which is discussed in the next section. 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 This subject has attracted the attention of researchers some complicated functions which are used in physical from different fields in the recent years. While it was process, we present a novel method in this paper by which developed by mathematicians few hundred years ago, efforts 2 on its usage in practical applications have been made only any arbitrary function from L space can be approximated by recently. It is known that many real systems are fractional in the output of a linear time invariant time fractional control nature, thus, it is more effective to model them by means of system (LTIFCS). Using a rigorous proof, we provide the fractional order than integer order systems. Applications such construction procedure of LTIFCS whose output as modeling of damping behavior of viscoelastic materials approximated the given function in the sense of L2 -norm. [3], cell diffusion processes [4], transmission of signals This paper is organized as follows: after this introduction through strong magnetic fields [5], and finance systems [6] are in section 1, a historical review is presented in section 2. some examples. Moreover, fractional order dynamic systems Some basic concepts and mathematical preliminaries are have been used in both design and implementation of control summarized in section 3. Section 4 studies the main result systems. Studies have shown that a fractional order controller which is stated in a theorem form. Finally conclusions in can provide better performance than an integer order one and section 5 close the paper. lead to more robust control performance [7]. 2. HISTORICAL REVIEW It is has been reported that specific fractional differential equations with order less than three, may exhibit complex How well can functions be approximated on a finite dynamical evolution even chaotic dynamics [8-9]; however, interval by the output of SISO controllable and observable this is not the case for ordinary differential equations due to fractional linear time-invariant system? This question was the famous Poincaré–Bendixson theorem [10]. first considered in the framework of constructing the inverse 2 p L space as a special case of L space is considered here. of a linear system in the integer order system basin [12]. Here The Lp spaces are function spaces defined using natural we want to extend this problem for the first time, in a more generalizations of p-norms for finite-dimensional vector comprehensive field, i.e. fractional order systems. Although
1 in [13], Unser’s method has been extended for fractional methodology can be used fairly in the signal processing and systems, the method presented in this paper is completely system identification. different and has a more general form. 2. PRELIMINARIES Over the past few decades, the theory of linear control theory has centered on stability properties that only really Fractional calculus as an extension to ordinary calculus make sense on an infinite interval. However, there are many possesses definitions that stem from the definitions existing problems in which the only thing of interest is a finite for ordinary derivatives. Some of the current definitions for interval. Indeed, this work began with questions of fractional derivatives are described in [21]. The Riemann– constructing flyable trajectories for linear systems [14]. Since Liouville definition is the simplest definition to use. Based on the appearance of those initial papers, there has been a th 1 growing interest on the construction of curve approximation this definition, the q order fractional integral of f L [0,t] of using linear control theory, which has developed into the and the terminal value a is given by [22]: theory of interpolating and smoothing splines. t (1) 1 J q f (t) f (t) (t s)q1 f (s)ds ; q R The theory of interpolating splines began with the paper a q (q) of Schoenberg [15] in 1946 and has begun to be widely a employed since 1960s with the availability of digital where the Euler-Gamma function is defined as follows: (2) computers. The theory of smoothing splines began with the q q1 work of [16] and received a tremendous impetus with the (q) e u du ; q R publication of her CBMS lectures. A more theoretic problem 0 can be found in [Error: Reference source not found]. As another definition for the fractional derivative we Almost all of the literature has been devoted to propose the Caputo’s definition [Error: Reference source not polynomial splines and an impressive body of tools has been found]: developed for this application. A body of literature has t 1 f (m) ( ) developed around the concept of control theoretic splines. d m 1 m (m ) 1m D f (t) J m Dm f (t) (t ) The initial work was with respect to flight control and is * a a (3) d m represented by the works of Crouch and Jackson [Error: f (t) m m Reference source not found]. dt In the same way we can prove the following property for the In the area of smoothing splines Martin and colleagues 23 have developed an theory of smoothing splines based on Laplace transform of the Caputo derivative [ ]: linear control theory [17]. Some modifications in a more m1 L{D f (t)} s L{ f (t)} s k 1 f (k)(0) general prospective under weaker conditions and more * (4) general spaces, e.g. Banach spaces, of the problem studied in k 0 [Error: Reference source not found], were treated in [Error: As one can see, in the Caputo derivative we need the Reference source not found] and [18] recently. These works integer order derivatives of the initial conditions while in the were motivated by the trajectory planning problem in order to Riemann-Liouville we need the fractional derivative of the avoid high accelerations that were observed when it was initial conditions. This is a merit of Caputo definition. required to be exactly at point at a given time. The constraint 24 of interpolation was relaxed to approximation and a simple Miller and Ross in [ ] introduced the so called sequential optimal control problem yielded generalized smoothing fractional derivative in the following way: MR D splines. This technique was shown to be adaptable to a MR D D , 0 1 (5) k (k1) number of situations not easily handled with traditional MR D MRD MR D , k 2,3,... polynomial smoothing spline techniques. where D is the Caputo fractional derivative. Next we define The question that has been in the background of all of the a sequential fractional differential equation in an operator control theoretic splines has been its convergence. This format as follows: 19 problem was directly attacked in [ ]. In particular the paper n (n1) (6) MR D an1MR D ... a1MR D a0 y(t) f (t) [20] gives a fairly comprehensive theory of convergence of 0 1 splines in a L2 sense. However, it was necessary to impose without loss of generality we restrict . some rather restrictive conditions on the linear system, which As in the usual case, it is easy to obtain the connection although were natural from an approximation view point, between (6) and the corresponding system of linear fractional were not natural from the viewpoint of control theory. Beside differential equations. If we apply the following change of these restrictions and developments, we want to extend these variables to (6): tasks to a more general form, i.e. fractional order control x (t) y(t), (D x )(t) x (t), ( j 1,2,...,n 1) (7) systems. The new theory might have an important impact on 1 j j j 1 the interpolation and spline theory. Moreover, this where we have used the Caputo fractional derivative. Using the above notation we have:
2 integer order differential equations, we can find the D x1 x2, homogeneous solution for LFDE as D x2 x3, (8) k (1k) 1 (14) ⋮ A (t) xh (t) M [(k 1)] D xn1 xn , k 0 n1 To show that this can be the homogenous solution of (9), D xn ak xk f (t) based on the differintegrability of xh it is easy to differentiate k 0 (14)[27]: Or in a matrix form with a companion matrix A , (9) Ak (t)(1k) 1 (t)(1k) 1 D x(t) Ax(t) f (t) D M Ak D M (15) [(k 1)] [(k 1)] This is a standard linear fractional differential equation k 0 k 0 (LFDE). k 1 j (1 j) 1 k (t) ((1 k)) A (t) 25 A M A. M Definition 1 [ ]. Let 1 p and S,, be a measure [(k 1)](k) [( j 1)] k 0 j0 space. Consider the set of all measurable functions from S to It is easy to see that the vector M is the initial conditions of C (or R) whose absolute value raised to the p-th power has x finite integral, or equivalently, that: . Also one can show that the particular solution is as 28 1 (10) follows [ ]: p p t f : f d k (1k) 1 p A (t ) S x p f ( )d [(k 1)] 0 k 0 p 2 (16) In this paper we consider a special case with which is t Ak the most important case in the Lp spaces. Consider the (t )(k 1) 1 f ( )d [(k 1)] dynamical system: k 0 0 D x(t) Ax(t) bu(t) Therefore the general solution is: (11) y Cx Ak (t)(!k) 1 x(t) xh (t) x p (t) M 26 [(k 1)] Definition 1 [ ]. The system (11) is observable on [t0,t f ] iff k 0 t x(t) for t [to,t f ] can be deduced from knowledge of y(t) and Ak (17) (t )(k 1) 1 f ( )d u(t) on [t0,t f ] . [(k 1)] k 0 0 Definition 2 [Error: Reference source not found]. System Now we return to a control system problem. Consider a (11) is controllable iff for any (x0, x1) there exists a control fractional control system with zero initial conditions and u(t) defined on [t0,t f ] which drives the initial state x(t0) x0 companion matrix which is not a restrictive assumption: to the final state x(t f ) x1 . D x(t) Ax(t) bu(t) (18) y Cx 3. MAIN RESULTS According to the previously developed theory we can write: k t A (k 1) 1 As in the integer order case, consider xh and xp are the y Cx C. (t ) bu( )d [(k 1) ] homogeneous and particular solutions of (9), respectively. k 0 0 (19) t Proposition 1. For the LFDE (9) we have the general CAk (t )(k 1) 1bu( )d solution as follows: [(k 1)] k 0 0 x(t) xh (t) xp (t) (12) Now we present the main result of the paper: Proof. Inserting xh and xp in (9) we have: Theorem. Consider the system (18) which is controllable and D x Ax , h h observable. Defining a linear operator as follows: D xp Axp f , (13) t k (1k) 1 (20) A (t ) D x D (xh xp ) D xh D xp Axh (Axp f ) (u)(t) C bu( )d [(k 1)] k 0 A(xh xp ) f Ax f . 0 Therefore for every m(t) L2[0,T] there exists a u L2[0,T] So based on this proposition it is sufficient to find the homogeneous and particular solutions separately. Using a such that: similar method in finding the homogeneous solution for
3 T Or briefly inf (m(t) y(t))2 dt 0 (27) (21) D ( ) A( ) bv( ), (T ) 0 uL2 [0,T ] 0 So from * and using the controllability condition Proof. It is enough to establish that the range space of the (v)( ) 0 of the original system (A,b) it is obtained that for an initial (u) is dense in L2[0,T ] . value 0 we have 2 Suppose that (u) is not dense in L [0,T ] . Thus there k (k 1) 1 k (A) (A) (k 1) 1 0 ( ) bv( )d (28) exists a nonzero function v L2[0,T ] such that (u),v 0 for ((k 1)) ((k 1)) k 0 k 0 0 2 all u L [0,T] . So 0 C( ) 0 T T Ak (t )k (u),u v(t) C(t ) 1 bu( )d dt [(k 1)] Now introducing the vector valued function ( ) as follows, 0 0 k 0 we can manipulate the relations easily: T T ' CAkb C c ... c , ( ) ( ) ( ) ... ( ) (29) v(t) (t )(k 1) 1u( )d dt 1 n 1 2 n [(k 1)] k 0 0 0 Using these notations we have T T n CAkb (22) (30) (t )(k 1) 1u( )d v(t)dt u( )d c ( ) 0 [(k 1)] i i k 0 0 i0 * u, (v) since (A,b) and (A,C) are controllable and observable, respectively. So we can write: where n (31) T CAkb ciD i ( ) 0 *(v)( ) (t )(k 1) 1v(t)dt [(k 1)] i0 k 0 (23) Introducing the following state variables: T CAkb (32) (t )(k 1) 1v(t)dt D 1 i1, i 1,2,...,n 1 [(k 1)] k 0 we have: * 0 1 0 ⋯ 0 So it is concluded that (v)( ) 0 . Now we use the fact that 1 0 0 1 0 ⋮ 1 1 * 29 (u) is onto if (v) is one-to-one [ ]. It suffices to show D 2 ⋮ ⋱ ⋱ 1 0 2 Q 2 that v 0 almost everywhere. Let ⋮ ⋮ ⋮ (33) 0 0 ⋯ ⋯ 1 T k c1 c1 c1 A b (k 1) 1 n ⋯ ⋯ n n ( ) (t ) v(t)dt (24) c c c [(k 1)] n n n k 0 Using (17), we have: Using the following important facts [30] (Q)k (k 1) 1 l (l 1) l ( ) (0) (34) 1.D ((k 1)) (l 1) k 0 (25) (T ) 0 (0) 0 1 Since we conclude that and consequently 2.D K( , )d D K( , )d lim D K( , ) 0 ( ) 0 . This establishment is based on the assumption that 0 0 cn 0 , but this approach can be easily extended to more We deduce that general case. Anyway we have: T k k (k 1) 1 k A b (k 1) 1 (A) (A) (k 1) 1 D ( ) D (t ) v(t)dt 0 ( ) bv( )d 0 (35) [(k 1)] ((k 1)) ((k 1) ) k 0 k 0 k 0 0 T k A b and hence that D (t )(k 1) 1v(t)dt bv( ) (26) [(k 1)] k k 0 (A) (k 1) 1 0 ( ) bv( )d 0 (36) T ((k 1)) A jb k 0 0 A (t )( j1) 1v(t)dt bv( ) [(k 1)] Differintegrating of (36) once we have: j0 A( ) bv( )
4 k state space techniques, which are well established and easy to (A) (k 1) 1 use. D 0 ( ) bv( )d ((k 1)) k 0 0 4. CONCLUSION (A)k (37) D D ( )(k 1) 1bv( )d 0 ((k 1)) In this paper, considering a fractional differential equation k 0 0 system with constant matrices and using some facts from (k 1) 1 0 lim D ( ) bv( ) 0 functional analysis, we proved that any arbitrary function 0 belongs to L2 can be approximated by an output of a linear v( ) 0 a.e. time-invariant fractional differential control system. More 2 So this shows that * is one-to-one and hence that is onto precisely we gave a proof that every L function can be 2 a dense subspace of L2[0,T ] .□ represented as the L limit of functions that are the outputs of fractional linear control systems. Our analysis was developed This idea indicates an important point in control theory. in the Hilbert space time-invariant fractional differential 2 Indeed given a function in L space, we can propose a control system. More precisely we gave a proof that every L2 fractional order transfer function which can approximate that function can be represented as the L2 limit of functions that function in term of L2 norm. One of the main advantages of are the outputs of fractional linear control systems. Our this idea is that we can analyze a very complex function in analysis was developed in the Hilbert space. 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
REFERENCES
5 1[] D.Cafagna, “Fractional calculus: a mathematical tool from the past for the present engineer”, IEEE Trans. Indust. Electronic, pp. 35-40, 2007. 2[] X.Gao, “Chaos in fractional order autonomous system and its control”, IEEE, pp. 2577-2581, 2006. 3[] Yu. A. Rossikhin, M.V. Shitikova, “Analysis of damped vibrations of linear viscoelastic plates with damping modeled with fractional derivatives”, Signal Processing, Volume 86, Issue 10, October 2006, Pages 2703-2711. 4[] T.A.M. Langlands, “Solution of a modified fractional diffusion equation”, Physica A: Statistical Mechanics and its Applications, Volume 367, pp. 136-144, 15 July 2006. 5[] J.A. Tenreiro Machado, Isabel S. Jesus, Alexandra Galhano, J. Boaventura Cunha, “Fractional order electromagnetics”, Signal Processing, Volume 86, Issue 10, October 2006, Pages 2637-2644. 6[] N. Laskin, “Fractional market dynamics”, Physica A 287, pp. 482-492, 2000. 7[] I. Podlubny, “Fractional order systems and -controllers”, IEEE Trans. Autom. Control 44 (1), pp. 208–214, 1999. 8[] P. Baranyi and Y. Yam, “Case study of the TP-model transformation in the control of a complex dynamic model with structural nonlinearity,” IEEE Trans. Ind. Electron., vol. 53, no. 3, pp. 895–904, Jun. 2006. 9[] D.Baleanu, “Fractional Hamiltonian analysis of irregular systems”, Signal Processing, Volume 86, Issue 10, October 2006, Pages 2632-2636. 10[] M.S.Tavazoei, et.al, “Some applications of fractional calculus in suppression of chaotic oscillations”, IEEE trans. Indus.electronic, pp.4094- 5001, 2008. 11[] S. Sun, M. Egerstedt, C. Martin, Control theoretic smoothing splines, IEEE Trans. Automat. Control 45 (12) pp.2271–2279, 2000. 12[] R.W. Brockett, and M.D. Mesarovic, “The reproducibility of multivariable systems”, J. Math. Anal. Appl. 11 pp.548– 563, 1965. 13[] R.Panda, M.Dash, “Fractional generalized splines and signal processing”, Signal Processing, Volume 86, Issue 9, September 2006, Pages 2340- 2350. 14[] Z. Zhang, J. Tomlinson, P. Enqvist, C. Martin, “Linear control theory, splines and approximation”, J. Lund (Eds.), Computation and Control III, Birkhäuser, Boston, 1995, pp. 269–288. 15[] I. Schoenberg, “Contributions to the problem of approximation of equidistant data by analytic functions”, Quart. Appl. Math. 4, pp.45–49, 1946. 16[] G.Wahba, ”Spline Models for Observational Data”, CBMS–NSF Regional Conf. Ser. in Appl. Math., SIAM, Philadelphia, 1990. 17[] U.Jonsson, and C.Martin, “Approximation with the output of linear control systems”, J. Math. Anal. Appl. 329, pp. 798-821, 2007. 18[] J.H. Shapiro, “On the outputs of linear control systems”, J. Math. Anal. Appl. 340, pp.116–125, 2008. 19[] M. Egerstedt, C. Martin, “Statistical estimates for generalized splines”, ESAIM Control Optim. Calc. Var. 9, pp. 553–562 (electronic), 2003. 20[] Y. Zhou, M. Egerstedt, C. Martin, “Optimal approximation of functions”, Commun. Inf. Syst. 1 (1), pp. 101–112, 2001. 21[] A.A.Kilbas, H.M.Srivastava, and J.J.Trujillo, “Theory and applications of fractional differential equations”, Elsevier, 2006. 22[] A.Loverro, “ Fractional Calculus: History, Definitions and Applications for the Engineer”, university of Notre Dame, 2004. 23[] I.Podlubny, “ Fractional derivatives: History, Theory, Application”, Utah State University, 2005. 24[] K.S. Miller, B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations”, Wiley and Sons, New York, 1993. 25[] Adams, Robert A.; Fournier, John F. Sobolev Spaces (Second ed.), Academic Press, 2003. 26[] D.Matignon, and B.D.Novel, “some results on controllblity and observability of finite dimensional fractional differential equations”, 27[] K.B.Oldham, and J.Spanier, “The fractional Calculus”, Mathematics in science and engineering, Academic press, Inc. 1974. 28[] B. Bonilla, M. Rivero, J.J. Trujillo, “On systems of linear fractional differential equations with constant coefficients”, Applied Mathematics and Computation 187, pp. 68–78, 2007. 29[] G.B.Folland, “Real analysis, modern techniques and their applications”, John Wiley and sons, 1999. 30[] I.Podlubny, “ Fractional differential equations”, mathematics in science and engineering V198, Acadamic press 1999.