Minimum Energy Control of Fractional Positive Electrical Circuits with Bounded Inputs

Circuits, Systems, and Signal Processing, Oct 2015

Minimum energy control problem for the fractional positive electrical circuits with bounded inputs is formulated and solved. Sufficient conditions for the existence of solution to the problem are established. A procedure for solving of the problem is proposed and illustrated by example of fractional positive electrical circuit.

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Minimum Energy Control of Fractional Positive Electrical Circuits with Bounded Inputs

Minimum Energy Control of Fractional Positive Electrical Circuits with Bounded Inputs Tadeusz Kaczorek 0 0 Faculty of Electrical Engineering, Bialystok University of Technology , Wiejska 45D, 15-351 Bialystok , Poland Minimum energy control problem for the fractional positive electrical circuits with bounded inputs is formulated and solved. Sufficient conditions for the existence of solution to the problem are established. A procedure for solving of the problem is proposed and illustrated by example of fractional positive electrical circuit. Fractional; Positive; Electrical circuits; Minimum energy control; Bounded inputs; Procedure 1 Introduction A dynamical system is called positive if its trajectory starting from any nonnegative initial state remains forever in the positive orthant for all nonnegative inputs. An overview of state of the art in positive theory is given in the monographs [3, 17]. Variety of models having positive behavior can be found in engineering, economics, social sciences, biology and medicine, etc. Mathematical fundamentals of the fractional calculus are given in the monographs [27–29]. The positive fractional linear systems have been investigated in [5, 6, 8, 18, 19, 21]. Stability of fractional linear continuous-time systems has been investigated in the papers [1, 20, 21]. The notion of practical stability of positive fractional linear systems has been introduced in [20]. Some recent interesting results in fractional systems theory and its applications can be found in [2, 29, 30, 33, 34]. B Tadeusz Kaczorek The minimum energy control problem for standard linear systems has been formulated and solved by J. Klamka in [22–25] and for 2D linear systems with variable coefficients in [22]. The controllability and minimum energy control problem of fractional discrete-time linear systems has been investigated by Klamka in [26]. The minimum energy control of fractional positive continuous-time linear systems has been addressed in [4, 11, 13] and for descriptor positive discrete-time linear systems in [4, 6, 10, 12]. The minimum energy control problem for positive and positive fractional electrical circuits has been investigated in [13–15]. In this paper, the minimum energy control problem for fractional positive electrical circuits with bounded inputs will be formulated and solved. The paper is organized as follows. In Sect. 2, the basic definitions and theorems of the fractional positive electrical circuits are recalled and the necessary and sufficient conditions for the reachability of the electrical circuits are given. The main result of the paper is given in Sect. 3 where minimum energy control problem is formulated, sufficient conditions for its solution are established, and a procedure is proposed. Illustrating example of fractional positive electrical circuit is given in Sect. 4. An extension to the method is presented in Sect. 5. Concluding remarks are given in Sect. 6. The following notation will be used: —the set of real numbers, n×m —the set of n n+× =m ren+a×l 1m, aMtrnic—est,he n+se×tmo—fnth×e snetMoeftnzle×r mmamtriacterisce(rsewalitmhantroincnesegwaittihveneonntnreiegsatainvde off-diagonal entries), In —the n × n identity matrix. 2 Preliminaries f (n) (t − τ )α+1−n dτ , n − 1 < α ≤ n ∈ N = {1, 2, . . .} e−t t x−1dt is the gamma function. Dα x (t ) = Ax (t ) + Bu(t ), where x (t ) ∈ n , u(t ) ∈ n×n , B ∈ Theorem 2.1 [21] The solution of Eq. (2.2) is given by Φ(t − τ )Bu(τ )dτ, x (0) = x0 Φ0(t ) = Eα( At α) = k=0 Theorem 2.2 [21] The continuous-time fractional system (2.2) is (internally) positive if and only if A ∈ Mn, B ∈ iDnetfiimnietiotfn i2f.2theTreheexstiastteanx fin∈put un+(to)f ∈the fm+ra,ctiton∈al[s0y,sttefm] w(2h.i2c)hisstceaelrlsedthreeasctahtaebolef system (2.2) from zero initial state x0 = 0 to the state x f . A real square matrix is called monomial if each of its row and each of its column contain only one positive entry and the remaining entries are zero. Proof Sufficiency It is well known [17,21] that if A ∈ Mn is diagonal, then Φ(t ) ∈ n+×n is also diagonal and if B ∈ n+×m is monomial, then BBT ∈ n+×n is also monomial. In this case, the matrix R f = Φ(τ )BBTΦT(τ )dτ ∈ is also monomial and R−1 f ∈ u(t ) = BTΦT(t f − t )R −f1x f 0 t f steers the state of the system (2.2) from x0 = 0 to x f since using (2.3) for x0 = 0 and (2.5) we obtain x (t f ) = Φ(t f − τ )BBTΦT(t f − τ )dτ R −f1x f Φ(τ )BBTΦT(τ )dτ R −f1x f = x f . The proof of necessity is given in [11]. 3 Problem Formulation and Its Solution Consider the fractional positive electrical circuit (2.2) with A ∈ Mn and B ∈ n+×m monomial. If the system is reachable in time t ∈ [0, t f ], then usually there exist many dAimffeornegntthinepseutisnup(utt)s,∈we n+artehlaotoskteinergsftohreisntpatuet ouf(tth)e∈systn+e,m ftro∈m[x00, t =f]0satotisxfyfi∈ng thn+e. condition u(t ) ≤ U ∈ n+, t ∈ [0, t f ] that minimizes the performance index I (u) = where Q ∈ n+×n is a symmetric positive defined matrix and Q−1 ∈ n+×n. The performance index (3.2) is a measure of the energy used for steering the state of the systems from x0 = 0 to x f . The minimum energy control problem for the fractional positive electrical circuit (2.2) can be stated as follows. Given the matrices A ∈ Mn, B ∈ n+×m , α, U ∈ n+ and Q ∈ n+×n of the performance matrix (3.2), x f ∈ n+ and t > 0, find an input u(t ) ∈ n+ for t ∈ [n0,atnfd] satisfying (3.1) that steers the state vector of the system from x0 = 0 to x f ∈ + minimizes the performance index (3.2). To solve the problem, we define the matrix W (t f ) = Φ(t f − τ )B Q−1 BTΦT(t f − τ )dτ where Φ(t ) is defined by (2.5). From (3.3) and Theorem 2.3, it follows that the matrix (3.3) is monomial if and only if the fractional positive electrical circuit (2.2) is reachable in time [0, t f ]. In this case, we may define the input sTtahteeooref mthe3.f1racLteiotnu¯a(tl)p∈ositin+vefoerletc∈tri[c0a,ltcfi]rcbueitan(2i.n2p)ufrtosmatixs0fyi=ng0(3to.1x) fth∈at stn+ee.rTshtehne the input (3.4) satisfying (3.1) also steers the state of the electrical circuit from x0 = 0 to x f ∈ n+ and minimizes the performance index (3.2), i.e., I (uˆ) ≤ I (u¯). The minimal value of the performance index (3.2) is equal to I (uˆ) = x Tf W −1(t f )x f . x (t f )= x f = Q−1 ∈ Substitution of (3.4) into (3.8) yields since (3.3) holds. Using (3.9), it is easy to verify that [u¯(τ ) − uˆ(τ )]T Q[u¯(τ ) − uˆ(τ )]dτ . I (uˆ) = uˆT(τ )Qu(uˆ)dτ = x Tf W −1 (t f − τ )B Q−1 BT T(t f − τ )dτ W −1x f 4 Procedure and Example To find t ∈ [0, t f ] for which uˆ(t ) ∈ compute the derivative n reaches its minimal value using (3.4), we + (t ) and using the equality = Q−1 BTΨ (t )W −1(t f )x f , t ∈ [0, t f ] Ψ (t )W −1(t f )x f = 0 Procedure 4.1 Step 1 Knowing A ∈ Mn and using (2.5), compute (t ). Step 2 Using (3.3), compute the matrix W f for given A, B, Q, α and some t f . Step 3 Using (3.4) and (4.3), find t f for which uˆ(t ) satisfying (3.1) reaches its maximal value and the desired uˆ(t ) for given U ∈ n+ and x f ∈ n+. Step 4 Using (3.6), compute the maximal value of the performance index. Fig. 1 Electrical circuit Using the Kirchhoff’s laws, we can write the equations e2 = R3(i2 − i1) + R2i2 + L2 dt α A = , B = The fractional electrical circuit is positive since the matrix A is a Metzler matrix and the matrix B has nonnegative entries. In [17], it was shown that the electrical circuit is reachable if R3 = 0. In this case, the matrix A has the form A = R2 = 1, condition uˆ(t ) = for t ∈ [0, t f ] that steers the state of the electrical circuit from zero state to final state x f = [ 1 1 ]T (T denotes the transpose) and minimizes the performance index (3.1) with Using the Procedure 4.1, we obtain the following: Step 1 Taking into account that Q = A = and using (2.5), we obtain k=0 Γ [(k + 1)α] = k=0 t (k+1)0.7−1 W (t f ) = Φ(t f − τ ) B Q−1 BTΦT(t f − τ )dτ = k=0 t (k+1)0.7−1 = 2 0 t f 0 k=0 l=0 1 ∞ Step 3 Using (2.7) and (2.8), we obtain uˆ (t ) = Q−1 BTΦT(t f − t )W −1(t f )x f k=0 W −1(t f ) Step 4 From (3.6), we have the minimal value of the performance index I (uˆ ) = x Tf W −1(t f )x f where W (t f ) is given by (4.11). 5 Extension to Fractional Positive Electrical Circuits with Different Orders Consider an electrical circuit composed of resistors, n capacitors and m voltage (current) sources. Using the Kirchhoff’s laws, we may describe the transient states in the electrical circuit by the fractional differential equation where x (t ) ∈ n, u(t ) ∈ m , A ∈ n×n, B ∈ n×m . The components of the state vector x (t ) and input vector u(t ) are the voltages on the capacitors and source voltages, respectively. Similarly, using the Kirchhoff’s laws, we may describe the transient states in the electrical circuit composed of resistances, inductances and voltage (current) sources by the fractional differential equation = Ax (t ) + Bu(t ), 0 < α < 1, = Ax (t ) + Bu(t ), 0 < β < 1, has the form where x (t ) ∈ n, u(t ) ∈ m , A ∈ n×n, B ∈ n×m . In this case, the components of the state vector x (t ) are the currents in the coils. Now let us consider electrical circuit composed of resistors, capacitors, coils and voltage (current) sources. As the state variables (the components of the state vector x (t )), we choose the voltages on the capacitors and the currents in the coils. Using Eqs. (5.1), (5.2) and Kirchhoff’s laws, we may write for the fractional linear circuits in the transient states the state equation where the components of xC ∈ n1 are voltages on the capacitors, the components of xL ∈ n2 are currents in the coils and the components of u ∈ m are the source voltages and Ai j ∈ Bi ∈ ni ×m , i, j = 1, 2. xC (0) = x10 and xL (0) = x20 x (t ) = Tkl = 0(t ) = 1(t ) = 2(t ) = for k = 1, l = 0 ∞ ∞ k=0 l=0 Proof is given in [18,21]. The extension of Theorem 5.1 to systems consisting of n subsystems with different fractional orders is given in [18,21]. Example 5.1 Consider the fractional electrical circuit shown in Fig. 3 [19,31,32] with given source voltages e1, e2, ultracapacitor C1 = 1 of the fractional order α = 0.7, ultracapacitor C2 = 2 of the fractional order β = 0.6, conductances G1 = 4, G1 = 4, G2 = 3, G2 = 6, G12 = 0 and N = n1 + n2 = 2. Using the Kirchhoff’s laws, we can write the equations From (5.8), we obtain Substitution of (5.9) into we obtain A = ⎣ ⎡ B = ⎣ −1 −1 = A From (5.12), it follows that A is a diagonal Metzler matrix and the matrix B is monomial matrix with positive diagonal entries. Therefore, the fractional electrical circuit is positive for all values of the conductances and capacitances. Find the optimal input (source voltage) eˆ(t ) ∈ 2+, t ∈ [0, t f ] satisfying the condition eˆ(t ) = for t ∈ [0, t f ] for the performance index (3.2) with Q = diag[2, 2] which steers the system from initial state (voltage drop on capacitances) u0 = [ 0 0 ]T to the finite state u f = [ 2 3 ]T and minimizes the performance index (3.2) with (5.12). Using (5.6) and (5.12), we obtain 2(t )B01 = = M (t ), 22(t ) = k=0 l=0 k=0 l=0 t (k+1)0.7+l0.6−1 t k0.7+(l+1)0.6−1 From (5.6a), (5.12) and (3.3), we have W (t f ) = M (t f − τ )Q−1 M (t f − τ )dτ T Now using (3.4) and (5.16), we obtain eˆ(t) = Q−1 MT(t f − t)W −1(t f )u f t = t f . W −1(t f ) 1 1 T −1dτ [Φ11(τ )[ 11(τ )] ] and the minimal value of the performance index (3.2) is equal to for t ∈ [0, t f ] I (eˆ) = uTf W −1u f = [ 2 3 ] ⎢ ⎣ = 0.5 [Φ111(τ )[Φ111(τ )]T]−1dτ + 3 6 Concluding Remarks Acknowledgments This work was supported under work S/WE/1/11. 1. M. Busłowicz , Stability of linear continuous time fractional order systems with delays of the retarded type . Bull. Pol. Acad. Sci. Tech . 56 ( 4 ), 319 - 324 ( 2008 ) 2. A. Dzielin´ski , D. Sierociuk , G. Sarwas , Ultracapacitor parameters identification based on fractional order model , in Proceedings of ECC'09 , Budapest ( 2009 ) 3. L. Farina , S. 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Tadeusz Kaczorek. Minimum Energy Control of Fractional Positive Electrical Circuits with Bounded Inputs, Circuits, Systems, and Signal Processing, 2016, 1815-1829, DOI: 10.1007/s00034-015-0181-7