RESEARCH Open Access Point to point control of fractional differential linear control systems Andrzej Dzieliński * and Wiktor Malesza * Correspondence: adziel@ee.pw. edu.pl Institute of Control and Industrial Electronics Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland Abstract In the article, an alternative elementary method for steering a controllable fractional linear control system with open-loop control is presented. It takes a system from an initial point to a final point in a state space, in a given finite time interval. Keywords: fractional control systems, fractional calculus, point to point control 1 Introduction Fractional integration and differentiation are generalizations of the notions of integer- order integration and differentiation. It turns out that in many real-life cases, models described by fractional differential equations much more better reflect the behavior of a phenomena than models expressed by means of the classical calculus (see, e.g., [1,2]). This idea was used successfully in various fields of science and engineering for model- ing numerous processes [3]. Mathematical fundamentals of fractional calculus are given in the monographs [4-9]. Some fractional-order controllers were developed in, e.g., [10,11]. It is also worth mentioning that there are interesting results in optimal control of fractional order systems, e.g., [12-14]. In this article, it will be shown how to steer a controllable single-input fractional lin- ear control system from a given initial state to a given final point of state space, in a given time interval. There is also shown how to derive hypothetical open-loop control functions, and some of them are presented. This method of control is an alternative to, e.g., introduced in [15], in which a derived open-loop control is based on controll- ability Gramian matrix, defined in [16] that seems to be much more complex to calcu- late than in our approach. The article is divided into two main parts: in Sect. 2 we study control systems described by the Riemann-Liouville derivatives and in Sect. 3–systems expressed by means of the Caputo derivatives. In each of these sections, we consider three cases of linear control systems: in the form of an integrator of fractional order a,intheform of sequential na-integrator, and finally, in a general (controllable) vector state space form. In Sect. 3.3, an illustrative example is given. Conclusions are given in Sect. 4. 2 Fractional control systems with Riemann-Liouville derivative Let (I α t s + g)(t ) and (D α t s + h)(t ) denote the Riemann-Liouville fractional left-sided integral and fractional derivative, respectively, of order a Î ℂ, on a finite interval of the real line [4,9]: Dzieliński and Malesza Advances in Difference Equations 2011, 2011:13 http://www.advancesindifferenceequations.com/content/2011/1/13 © 2011 Dzielińński and Malesza; licensee Springer. Thi s is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecomm ons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproductio n in any medium, provided the original work is prop erly cited. (I α t s + g)(t):= 1 (α) t  t s g(τ ) (t − τ ) 1−α dτ for (α) > 0, t > t s , (D α t s + h)(t):= 1 (n − α) d n dt n t  t s h(τ ) (t − τ ) α−n+1 dτ for (α) ≥ 0, t > t s , where n =[ℜ(a)] + 1, and [ℜ(a)] denotes the integer part of ℜ(a). Let us consider a fract ional-order (a Î ℝ and a > 0) differential equation of the form: (D α t s + x)(t )=f (t , x(t)), t > t s , (2:1) with the initial conditions (D α−k t s + x)(t s +) = w k , k =1, , n , (2:2) where n =[a]+1fora ∉ N,andn = a for a Î N.By (D α−k t s + x)(t s + ) ,wemeanthe following limit (D α−k t s + x)(t s +) = lim t→t s + (D α−k t s + x)(t ), k =1, , n , i.e., the limit taken in ]t s , t s + ε [for ε >0. The existence and uniqueness of solutions of (2.1) and (2.2) were considered by numerous authors, e.g., [4,8]. 2.1 Linear control system in the form of a-integrator Consider a control system of the form (D α t s + z)(t )=v(t), (2:3) where 0 <a <1,z(t) is a scalar solution of (2.3), and v(t) is a scalar control function. The aim of the control is to bring system (2.3), i.e., the state trajectory z(t), from the start point z( t s + ) = z s , (2:4) i.e., from the point z(t)=z(t s +) for t ® t s +, to the final point z( t f ) = z f , (2:5) in a finite time interval t f - t s . In other words, we are looki ng for such an open-loop control function v = v(t), which will achieve it in a finite time interval t f - t s . The start and final points will be also called the terminal points. In order to solve Equation 2.3, we need to use an initial condition of the form (D α−1 t s + z)(t s +) = (I 1−α t s + z)(t s +) = w 1 (2:6) that will correspond to condition (2.4), i.e.,wehavetofindanappropriatevaluew 1 corresponding to (2.4). To this end, initial condition (2.6) can be rewritten (see [4]) as lim t→t s + (t − t s ) 1−α z(t)= w 1  ( α ) , Dzieliński and Malesza Advances in Difference Equations 2011, 2011:13 http://www.advancesindifferenceequations.com/content/2011/1/13 Page 2 of 17 from which w 1 = (α) lim t→t s + z(t) lim t→t s + (t − t s ) 1−α = z s (α) lim t→t s + (t − t s ) 1−α . (2:7) Propositio n 1. Acontrolv(t) that steers system (2.3) from the start point (2.4) to the final point (2.5) is of the form v(t)=(D α t s + ϕ)(t) , (2:8) where j(t) is an arbitrary C 1 -function satisfying ϕ ( t s ) = z s and ϕ ( t f ) = z f . (2:9) Proof. Take (2.8) as a control applied to (2.3), i.e., (D α t s + z)(t )=(D α t s + ϕ)(t) . (2:10) Integrating both sides of (2.10) by means of I α t s + , i.e., (I α t s + D α t s + z)(t )=(I α t s + D α t s + ϕ)(t) , we get (using the rule of integration given, e.g., in [4]) z(t) − (I 1 −α t s + z)(t s +)  ( α ) (t − t s ) α−1 = ϕ(t) − (I 1 −α t s + ϕ)(t s +)  ( α ) (t − t s ) α−1 . (2:11) Since j(t s )=z s , and the system starts from z(t s )=z s , we get (I 1−α t s + z)(t s +) = (I 1−α t s + ϕ)(t s +) , which finally yields z(t)=j(t). In particular, z(t f )=j(t f )=z f . □ Example 2. We want to steer system (2.3) from the start point (2.4) to the final point (2.5) by means of the control given by (2.8), where ϕ ( t ) = a 1 ( t − t s ) + a 0 , a 0 , a 1 ∈ R . (2:12) The values of coefficients a 0 and a 1 have to be chosen such that conditions (2.9) hold, i.e., from ϕ(t s )=a 0 = z s , ϕ ( t f ) = a 1 ( t f − t s ) + a 0 = z f , we calculate, for t f >t s , a 0 = z s , a 1 = z f − z s t f − t s . (2:13) Thus, polynomial (2.12) has the form ϕ(t)= z f − z s t f − t s (t − t s )+z s , and then, Equation 2.3, with control v(t)=(D α t s + ϕ)(t ) , is the following (D α t s + z)(t )=a 1 (2)  ( 2 − α ) (t − t s ) 1−α + a 0 1  ( 1 − α ) (t − t s ) −α . (2:14) Dzieliński and Malesza Advances in Difference Equations 2011, 2011:13 http://www.advancesindifferenceequations.com/content/2011/1/13 Page 3 of 17 In order to show that the above-calculated control v(t) is right, we integrate (2.14) by means of I α t s + , giving z(t)− (I 1 −α t s + z)(t s +)  ( α ) (t − t s ) α−1 = a 1 (2)  ( 2 − α ) (I α t s + (t − t s ) 1−α )(t)+a 0 1  ( 1 − α ) (I α t s + (t − t s ) −α )(t) . Since the value of initial condition (I 1 −α t s + z)(t s + ) corresponding to the start point z s is given by (2.6) and (2.7), and substituting already calculated coefficients a 0 and a 1 given by (2.13), we get z(t) − z s (t − t s ) α−1 lim t→t s + (t − t s ) 1−α = z f − z s t f − t s (t − t s )+z s . (2:15) Since lim t→t s + (t − t s ) 1 −α =0 for a < 1, evaluating (2.15) at t = t s yields z(t s )=z s , and for t = t f gives z(t f )=z f . 2.2 Linear control system in the form of na-integrator Consider a control system of order na,for0<a <1,n Î N + such that na <1,given by (D n α t s + z)(t )=v(t ) (2:16) with the initial conditions (I 1−α t s + D kα t s + z)(t s +) = w k , w k ∈ R, k =0, , n − 1 , (2:17) where z(t) is a scalar solution of (2.16), (2.17), and v(t) is a scalar control function. By D k α t s + z we mean D α t s + z =D α t s + z, D kα t s + z =D α t s + D (k−1)α t s + z, k =2,3, , n . (2:18) We introduce the notion of D α t s + z (see Property 2.4 in [4]), because, in general, D α t s + D α t s + ···D α t s + z    n - t im es =D nα t s + z . Initial conditions (2.17) are equivalent (see [4]) to lim t→t s + (t − t s ) 1−α (D kα t s + z)(t )= w k  ( α ) , w k ∈ R, k =0, , n − 1 . (2:19) The aim of the control is to bring system (2.16) from the start point Z(t s ):=(z(t s ), (D α t s + z)(t s ), ,(D (n−1)α t s + z)(t s )) T =(z s0 , z s1 , , z sn−1 ) T =: Z s (2:20) at time t s , to the final point Z(t f ):=(z(t f ), (D α t s + z)(t f ), ,(D (n−1)α t s + z)(t f )) T =(z f0 , z f1 , , z fn−1 ) T =: Z f (2:21) at time t f , in the finite time interval t f - t s . For initial conditions (2.17) to correspond to the start point Z s , we cal culate (from (2.19)) Dzieliński and Malesza Advances in Difference Equations 2011, 2011:13 http://www.advancesindifferenceequations.com/content/2011/1/13 Page 4 of 17 w k = (α) lim t→t s + (t − t s ) 1−α lim t→t s + (D kα t s + z)(t ) = (α) lim t→t s + (t − t s ) 1−α (D kα t s + z)(t s ) = (α) lim t→t s + (t − t s ) 1−α z sk , k =0, , n − 1 . Proposition 3. A control v(t) that s teers system (2.16) from the start point (2.20) to the final point (2.21) is of the form v(t)=(D nα t s + ϕ)(t) , where j(t) is an arbitrary C n -function satisfying D kα t s + ϕ(t s )=z sk , D kα t s + ϕ(t f )=z fk ,0≤ k ≤ n −1 , (2:22) i.e., (ϕ(t s ), ,(D (n−1)α t s + ϕ)(t s )) T = Z s and (ϕ(t f ), ,(D (n−1)α t s + ϕ)(t f )) T = Z f . For such defined conditions (2.22), the initial conditions are (I 1−α t s + D k α t s + ϕ)(t s +) = ( α) lim t→t s + (t − t s ) 1−α (D k α t s + ϕ)(t), k =0, , n − 1 . (2:23) Proof. Apply the control v(t)=D nα t s + ϕ(t ) to (2.16), and we obtain (D nα t s + z)(t )=(D nα t s + ϕ)(t) . (2:24) Next, integrating (2.24) by means of I α t s + (I α t s + D α t s + D ( n−1 ) α t s + z)(t )=(I α t s + D α t s + D ( n−1 ) α t s + ϕ)(t) , we get ( D (n−1)α t s + z)(t)− (I 1−α t s + D ( n−1 ) α t s + z)(t s +)  ( α ) (t − t s ) α−1 =( D (n−1)α t s + ϕ)(t)− (I 1−α t s + D ( n−1 ) α t s + ϕ)(t s +)  ( α ) (t − t s ) α−1 . (2:25) Since the system starts from (2.20), and (2.22) holds, i.e., D (n−1)α t s + ϕ(t s )=z sn− 1 , we get (I 1−α t s + D (n−1)α t s + z)(t s +) = (I 1−α t s + D (n−1)α t s + ϕ)(t s +) , which yields (D ( n−1 ) α t s + z)(t )=(D ( n−1 ) α t s + ϕ)(t) . (2:26) In particular, for t = t f we obtain (D ( n−1 ) α t s + z)(t f )=(D ( n−1 ) α t s + ϕ)(t f )=z fn−1 . Analogously, consecutive integrations of (2.26) by m eans of I α t s + , together for all n integrations, yields (D k α t s + z)(t s )=(D k α t s + ϕ)(t s )=z sk , k =0, , n − 1 Dzieliński and Malesza Advances in Difference Equations 2011, 2011:13 http://www.advancesindifferenceequations.com/content/2011/1/13 Page 5 of 17 and (D kα t s + z)(t f )=(D kα t s + ϕ)(t f )=z fk , k =0, , n − 1 . □ One of the possible choices of function j(t)is ϕ(t)= 2n−1  i = 0 a i (I iα t s + 1)(t) , (2:27) where (I iα t s + 1)(t)= 1  ( iα +1 ) (t − t s ) iα ,0≤ i ≤ 2n −1, ((I 0 t s + 1)(t)=1 ) (2:28) satisfying (2.22). For a function of type (t - t s ) ia , the following holds (D α t s + ···D α t s +    n - t im es (t − t s ) iα )(t)=(D nα t s + (t − t s ) iα )(t)foriα +1> 0 , which is always satisfied, since we have i =0, ,2n - 1 and a >0(0<a <1).Itfol- lows that for the function (I iα t s + 1)(t ) (given by (2.28)), we have (D α t s + ···D α t s +    n - t im es I iα t s + 1)(t)=(D nα t s + I iα t s + 1)(t)=(I (i−n)α t s + 1)(t) . Thus, for the function j(t) given by (2.27), we have (D nα t s + ϕ)(t)=(D nα t s + ϕ)(t ) , and then v(t)=(D nα t s + ϕ)(t)= 2 n− 1  i = 0 a i (I (i−n)α t s + 1)(t) . Example 4. Consider control system (2.16) of order 2a (n = 2), i.e., (D 2α t s + z)(t )=v(t) , which we want to bring from the start point Z(t s ):=(z(t s ), (D α t s + z)(t s )) T =(z s0 , z s1 ) T =: Z s to the final point Z(t f ):=(z(t f ), (D α t s + z)(t f )) T =(z f0 , z f1 ) T =: Z f , in the finite time interval t f - t s . We take function j(t) in the form ϕ(t)= 3  i = 0 a i (I iα t s + 1)(t) , for which (D α t s + ϕ)(t)= 3  i = 0 a i 1 ((i −1)α +1) (t − t s ) (i−1)α . Dzieliński and Malesza Advances in Difference Equations 2011, 2011:13 http://www.advancesindifferenceequations.com/content/2011/1/13 Page 6 of 17 According to (2.22), the following must be satisfied ϕ(t s )=a 0 = z s0 , (D α t s + ϕ)(t s )=a 1 = z s1 , ϕ(t f )= 3  i=0 a i (I iα t s + 1)(t f )=z f0 , (D α t s + ϕ)(t f )= 3  i = 0 a i (I (i−1)α t s + 1)(t f )=z f1 , or, in the matrix form ⎛ ⎜ ⎜ ⎝ 1000 0100 1(I α t s + 1)(t f )(I 2α t s + 1)(t f )(I 3α t s + 1)(t f ) (I −α t s + 1)(t f )1(I α t s + 1)(t f )(I 2α t s + 1)(t f ) ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ a 0 a 1 a 2 a 3 ⎞ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎝ z s0 z s1 z f0 z f1 ⎞ ⎟ ⎟ ⎠ , (2:29) from which we can calculate coefficients a i ,0≤ i ≤ 3, assuming that t f >t s . Therefore, a control function steering the system from the start point Z s to the final point Z f ,is v(t)=(D 2α t s + ϕ)(t)= 3  i = 0 a i 1 ((i − 2)α +1) (t − t s ) (i−2)α , where a i ,0≤ i ≤ 3, are already calculated from (2.29). 2.3 Linear control system in the general state space form Consider a linear fractional control system of the form  :(D α t s + x)(t )=Ax + bu,0<α<1 , (2:30) where x(t)=(x 1 (t), , x n (t)) T Î ℝ n is a state space vector, A Î ℝ n×n , u(t) Î ℝ, b Î ℝ n×1 and (D α t s + x)(t ) = ((D α t s + x 1 )(t), ,(D α t s + x n )(t)) T . The initial conditions are (I 1−α t s + x i )(t s +) = w i , w i ∈ R,1≤ i ≤ n , or, in the equivalent form lim t→t s + (t − t s ) 1−α x i (t )= w i  ( α ) ,1≤ i ≤ n . The aim of the control is to bring the control system Λ from the start point x ( t s ) := ( x 1 ( t s ) , , x n ( t s )) T = ( x s1 , , x sn ) T =: x s (2:31) to the final point x ( t f ) := ( x 1 ( t f ) , , x n ( t f )) T = ( x f1 , , x fn ) T =: x f , (2:32) in the finite time interval t f - t s .Tothisend,sinceΛ is assumed to be controllable [15,16], i.e., rank R ( A, b ) =rank ( b, Ab, , A n−1 b ) = n, Dzieliński and Malesza Advances in Difference Equations 2011, 2011:13 http://www.advancesindifferenceequations.com/content/2011/1/13 Page 7 of 17 we can change the state coordinates x to new coordinates ˜ x , in the following linear way ˜ x = Tx,whereT ∈ R n×n ,detT  = 0 such that Λ expressed in the new coordinates ˜ x = ( ˜ x 1 , , ˜ x n ) T will be in the Frobe- nius form, i.e., ˜  Fr : ˙ ˜ x = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 010 0 001 0 . . . . . . . . . . . . . . . 000 1 − ˜ a 0 − ˜ a 1 − ˜ a 2 − ˜ a n−1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ˜ x + ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 . . . 0 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ u = ˜ A ˜ x + ˜ bu, ˜ x ∈ R n . In order to find a linear transformation T we take a row vector t 1 Î ℝ 1×n such that t 1 A j b =  00≤ j ≤ n − 2 1 j = n − 1, (2:33) which yields T = ⎛ ⎜ ⎜ ⎜ ⎝ t 1 t 1 A . . . t 1 A n−1 ⎞ ⎟ ⎟ ⎟ ⎠ . Indeed, if we take ˜ x = T x , where the first coordinate function is given by ˜ x 1 = t 1 x , and such that t 1 satisfies (2.33), then, using the linearity of Riemann-Liouville derivative, we have (D α t s + ˜ x i )(t)=t 1 A i−1 (D α t s + x)(t )=t 1 A i x = ˜ x i+1 ,1≤ i ≤ n − 1 , (D α t s + ˜ x n )(t)=t 1 A n−1 (D α t s + x)(t )=t 1 A n x + t 1 A n−1 bu = t 1 A n x + u getting ˜ x = ( t 1 , t 1 A, , t 1 A n−1 ) T x . Condition (2.33) can also be rewritten in the matrix form t 1 ( b, Ab, , A n−1 b ) = ( 0, 0, ,1 ) , which gives rise to t 1 = (0, 0, ,1)R −1 (A, b)=R − 1 ( n ) (A, b) , where R −1 ( n ) (A, b ) is the nth row of the matrix R -1 (A, b). Next, applying to the system ˜  F r a feedback of the form u( t ) = ˜ k ˜ x + v ( t ), (2:34) where ˜ k = −t 1 A n T −1 = ( ˜ a 0 , ˜ a 1 , ˜ a 2 , , ˜ a n−1 ) ∈ R 1× n and v(t) Î ℝ, we get (D α t s + ˜ x i )(t)= ˜ x i+1 ,1≤ i ≤ n − 1 , (D α t s + ˜ x n )(t)=v(t). Dzieliński and Malesza Advances in Difference Equations 2011, 2011:13 http://www.advancesindifferenceequations.com/content/2011/1/13 Page 8 of 17 Denoting z = ˜ x 1 , and using notation (2.18), we get (D α t s + ˜ x i )(t)=(D iα t s + z)(t )= ˜ x i+1 ,1≤ i ≤ n − 1 , (D α t s + ˜ x n )(t)=(D nα t s + z)(t )=v(t), then (D nα t s + z)(t )=v(t) . (2:35) Since the transformation ˜ x = T x is already known, for the given start point (2.31) and final point (2.32) we can calculate corresponding terminal points expressed in the new coordinates ˜ x , i.e., ˜ x ( t s ) := ( ˜ x 1 ( t s ) , , ˜ x n ( t s )) T = Tx s = ( ˜ x s1 , , ˜ x sn ) T =: ˜ x s and ˜ x ( t f ) := ( ˜ x 1 ( t f ) , , ˜ x n ( t f )) T = Tx f = ( ˜ x f1 , , ˜ x fn ) T =: ˜ x f . Then, for system (2.35) the terminal points are the following Z(t s ):=(z(t s ), (D α t s + z)(t s ), ,(D (n−1)α t s + z)(t s )) T =( ˜ x s1 , , ˜ x sn ) T =: ˜ x s = Z s (2:36) and Z(t f ):=(z(t f ), (D α t s + z)(t f ), ,(D ( n−1 ) α t s + z)(t f )) T =( ˜ x f1 , , ˜ x fn ) T =: ˜ x f = Z f . (2:37) In such a way, we have transformed the problem of finding a control u(t) for the sys- tem (2.30) steering from the start point (2.31) to the final point (2.32), into an equiva- lent problem of finding a control v(t) for system (2.35) steering from t he start point (2.36) to the final point (2.37), which has already been explained in Sect. 2.2. To this end, we take a C n -function j(t) satisfying (2.22) for given (2.36) and (2.37). For such a function j(t), the control is v(t)=(D nα t s + ϕ)(t) . Finally, using (2.34), the desired control u(t) taking system Λ from x s to x f is the fol- lowing u (t )= ˜ k ˜ x(t)+v(t)= ˜ kTx(t)+v(t)=−R −1 ( n ) (A, b)A n x(t)+(D nα t s + ϕ)(t) . 3 Fractional control systems with Caputo derivative We will use the following definition of Caputo derivative. Let a Î ℂ and ℜ(a) ≥ 0. If a ∉ N 0 , n =[ℜ(a)] + 1, and then ( C D α t s + f )(t):= 1 (n −α) t  t s f (n) (τ ) (t − τ ) α−n+1 dτ =: (I n−α t s + D n f )(t) . If a = n Î N 0 , then ( C D n t s + f )(t)=f (n) (t ) . Dzieliński and Malesza Advances in Difference Equations 2011, 2011:13 http://www.advancesindifferenceequations.com/content/2011/1/13 Page 9 of 17 Consider a differential equation, for a Î ℝ and a >0, ( C D α t s + x)(t )=f (t, x(t)), t s ≤ t ≤ t f , (3:1) with the initial conditions x (k) ( t s ) = w k , w k ∈ R, k =0, , n − 1 . (3:2) It has been already shown, e.g., in [4] that for (3.1) and (3.2) a solution exists. 3.1 Linear control system in the form of a-integrator Consider a linear fractional differential equation ( C D α t s + z)(t )=v(t), α ∈ R, α> 0 (3:3) with the initial conditions z (k) ( t s ) = w k , w k ∈ R, k =0, , n −1 , (3:4) where z(t) is a scalar solution and v(t) is a scalar control function. The aim of the control is to steer system (3.3) from the start point Z ( t s ) := ( z ( t s ) , ˙z ( t s ) , , z (n−1) ( t s )) T = ( z s0 , , z sn−1 ) T =: Z s (3:5) to the final point Z ( t f ) := ( z ( t f ) , ˙z ( t f ) , , z (n−1) ( t f )) T = ( z f0 , , z fn−1 ) T =: Z f (3:6) in a finite time interval t f - t s . In contrast to the e quation defined by means of Rie- mann-Liouville derivative, initial conditions (3.4) coincide with start point (3.5), i.e., w i = z si ,0≤ i ≤ n − 1 . Propositio n 5. Acontrolv(t) that steers system (3.3) from the start point (3.5) to the final point (3.6) is of the form v(t)=( C D α t s + ϕ)(t) , (3:7) where j(t) is an arbitrary C n -function satisfying ϕ (k) ( t s ) = z sk , ϕ (k) ( t f ) = z f k ,0≤ k ≤ n − 1 , (3:8) i.e.,  ( t s ) := ( ϕ ( t s ) , , ϕ (n−1) ( t s )) T = Z s and  ( t f ) := ( ϕ ( t f ) , , ϕ (n−1) ( t f )) T = Z f . Proof. As a control applied to (3.3) take (3.7), and then ( C D α t s + z)(t )=( C D α t s + ϕ)(t) . (3:9) Integrating (3.9) (according to the rule given by Lemma 2.22 in [4]) by means of I α t s + , i.e., (I α t s + C D α t s + z)(t )=(I α t s + C D α t s + ϕ)(t), Dzieliński and Malesza Advances in Difference Equations 2011, 2011:13 http://www.advancesindifferenceequations.com/content/2011/1/13 Page 10 of 17 [...]... model Procedings of the European Control Conference, Budapest, Hungary 2009 2 Dzieliński A, Sarwas G, Sierociuk D: Time domain validation of ultracapacitor fractional order model Procedings of the 49th IEEE Conference on Decision and Control, Atlanta, CA, USA 2010 3 Vinagre M, Monje C, Calderon A: Fractional order systems and fractional order control actions Lecture 3 of the IEEE CDC02: Fractional Calculus... 3.3 Linear control system in the general state space form Consider a controllable linear fractional control system of the form : (C Dαs + x)(t) = Ax + bu, t 0 < α < 1, where x(t) = (x1(t), , xn(t))T Î ℝn is the state space vector, A Î ℝn×n, u(t) Î ℝ, b Î ℝ n×1 and (C Dαs + x)(t) = ((C Dαs + x1 )(t), , (C Dαs + xn )(t))T The initial conditions are t t t x0 (ts ) = x0 , i i 1 ≤ i ≤ n The aim of control. .. CDC02: Fractional Calculus Applications in Automatic Control and Robotics 2002 4 Kilbas A, Srivastava H, Trujillo J: Theory and Appliccations of Fractional Differential Equations Elsevier, Amsterdam; 2006 5 Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations Wiley, New York; 1993 6 Oldham KB, Spanier J: The Fractional Calculus Academic Press, New York;... D: A hamiltonian formulation and a direct numerical scheme for fractional optimal control problems J Vibr Control 2007, 13(9-10):1269-1281 13 Agrawal OP, Defterli O, Baleanu D: Fractional optimal control problems with several state and control variables J Vibr Control 2010, 16(13):1967-1976 14 Baleanu D, Defterli O, Agrawal OP: A central difference numerical scheme for fractional optimal control problems... fractional differential systems Comput Eng Syst Appl 1996, 2:952-956 doi:10.1186/1687-1847-2011-13 Cite this article as: Dzieliński and Malesza: Point to point control of fractional differential linear control systems Advances in Difference Equations 2011 2011:13 Page 17 of 17 ... and (3.6) can be transformed to the following form (Dαs + y)(t) = v(t), t y(ts +) = 0, y(tf ) = zf − zs , (3:17) where y (t) = z (t) − zs (3:18) Indeed, control v(t) steering system (3.17) from the given point y(ts+) to the given point y(tf ), steers system (3.3) from the given point z(ts) to the given final point z(tf ), which follows from the inverse transformation of (3.18), i.e., z(t) = y(t) +... means that control (3.14) correctly steers the system from zs0 to zf0 Remark 7 For 0 . RESEARCH Open Access Point to point control of fractional differential linear control systems Andrzej Dzieliński * and Wiktor Malesza * Correspondence: adziel@ee.pw. edu.pl Institute of Control and Industrial Electronics. an initial point to a final point in a state space, in a given finite time interval. Keywords: fractional control systems, fractional calculus, point to point control 1 Introduction Fractional. consider three cases of linear control systems: in the form of an integrator of fractional order a,intheform of sequential na-integrator, and finally, in a general (controllable) vector state space form.