Minimum Energy Control of Fractional Positive Electrical Circuits with Bounded Inputs

Transcription

1 Circuits Syst Signal Process (26) 35: DOI.7/s Minimum Energy Control o Fractional Positive Electrical Circuits with Bounded Inputs Tadeusz Kaczorek Received: 24 April 25 / Revised: 5 October 25 / Accepted: 6 October 25 / Published online: 6 October 25 The Author(s) 25. This article is published with open access at Springerlink.com Abstract Minimum energy control problem or the ractional positive electrical circuits with bounded inputs is ormulated and solved. Suicient conditions or the existence o solution to the problem are established. A procedure or solving o the problem is proposed and illustrated by example o ractional positive electrical circuit. Keywords Fractional Positive Electrical circuits Minimum energy control Bounded inputs Procedure Introduction A dynamical system is called positive i its trajectory starting rom any nonnegative initial state remains orever in the positive orthant or all nonnegative inputs. An overview o state o the art in positive theory is given in the monographs [3,7]. Variety o models having positive behavior can be ound in engineering, economics, social sciences, biology and medicine, etc. Mathematical undamentals o the ractional calculus are given in the monographs [27 29]. The positive ractional linear systems have been investigated in [5,6,8,8, 9,2]. Stability o ractional linear continuous-time systems has been investigated in the papers [,2,2]. The notion o practical stability o positive ractional linear systems has been introduced in [2]. Some recent interesting results in ractional systems theory and its applications can be ound in [2,29,3,33,34]. B Tadeusz Kaczorek kaczorek@isep.pw.edu.pl Faculty o Electrical Engineering, Bialystok University o Technology, Wiejska 45D, 5-35 Bialystok, Poland

2 86 Circuits Syst Signal Process (26) 35: The minimum energy control problem or standard linear systems has been ormulated and solved by J. Klamka in [22 25] and or 2D linear systems with variable coeicients in [22]. The controllability and minimum energy control problem o ractional discrete-time linear systems has been investigated by Klamka in [26]. The minimum energy control o ractional positive continuous-time linear systems has been addressed in [4,,3] and or descriptor positive discrete-time linear systems in [4,6,,2]. The minimum energy control problem or positive and positive ractional electrical circuits has been investigated in [3 5]. In this paper, the minimum energy control problem or ractional positive electrical circuits with bounded inputs will be ormulated and solved. The paper is organized as ollows. In Sect. 2, the basic deinitions and theorems o the ractional positive electrical circuits are recalled and the necessary and suicient conditions or the reachability o the electrical circuits are given. The main result o the paper is given in Sect. 3 where minimum energy control problem is ormulated, suicient conditions or its solution are established, and a procedure is proposed. Illustrating example o ractional 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 ollowing notation will be used: R the set o real numbers, R n m the set o n m real matrices, R+ n m the set o n m matrices with nonnegative entries and R n + Rn +, M n the set o n n Metzler matrices (real matrices with nonnegative o-diagonal entries), I n the n n identity matrix. 2 Preliminaries The ollowing Caputo deinition o the ractional derivative will be used [2] t D α (t) dα dt α (t) Γ(n α) (n) dτ, n <α n N {, 2,...} (t τ) α+ n (2.) where α Ris the order o ractional derivative and (n) (τ) dn (τ) dτ n and Ɣ(x) e t t x dt is the gamma unction. Consider a ractional electrical circuit composed o resistors, coils, condensators and voltage (current) sources. Using the Kirchho s laws, we may describe the transient states in the electrical circuit by state equations [7,9,6,7,2,28] D α x(t) Ax(t) + Bu(t), <α (2.2) where x(t) R n, u(t) R m are the state and input vectors and A R n n, B R n m.

3 Circuits Syst Signal Process (26) 35: Theorem 2. [2] The solution o Eq. (2.2) is given by where t x(t) Φ (t)x + Φ(t τ)bu(τ)dτ, x() x (2.3) Φ (t) E α (At α ) Φ(t) k k A k t (k+)α Γ [(k + )α] A k t kα Γ(kα + ) (2.4) (2.5) and E α (At α ) is the Mittag Leler matrix unction [2]. Deinition 2. [2] The ractional system (2.2) is called the (internally) positive ractional system i and only i x(t) R n +, t or any initial conditions x R n + and all inputs u(t) R m +, t. Theorem 2.2 [2] The continuous-time ractional system (2.2) is (internally) positive i and only i A M n, B R+ n m. (2.6) Deinition 2.2 The state x R n + o the ractional system (2.2) is called reachable in time t i there exist an input u(t) R m +, t [, t ] which steers the state o system (2.2) rom zero initial state x to the state x. A real square matrix is called monomial i each o its row and each o its column contain only one positive entry and the remaining entries are zero. Theorem 2.3 The positive ractional system (2.2) is reachable in time t [, t ] i and only i the matrix A M n is diagonal and the matrix B R+ n n is monomial. Proo Suiciency It is well known [7,2] that i A M n is diagonal, then Φ(t) R+ n n is also diagonal and i B R+ n m is monomial, then BB T R+ n n is also monomial. In this case, the matrix is also monomial and R R Φ(τ)BB T Φ T (τ)dτ R n n + (2.7) R+ n n. The input u(t) B T Φ T (t t)r x (2.8)

4 88 Circuits Syst Signal Process (26) 35: steers the state o the system (2.2) rom x tox since using (2.3)orx and (2.5) we obtain x(t ) Φ(t τ)bu(τ)dτ Φ(t τ)bb T Φ T (t τ)dτ R x Φ(τ)BB T Φ T (τ)dτ R x x. (2.9) The proo o necessity is given in []. 3 Problem Formulation and Its Solution Consider the ractional positive electrical circuit (2.2) with A M n and B R+ n m monomial. I the system is reachable in time t [, t ], then usually there exist many dierent inputs u(t) R n + that steers the state o the system rom x tox R n +. Among these inputs, we are looking or input u(t) R n +, t [, t ] satisying the condition u(t) U R n +, t [, t ] (3.) that minimizes the perormance index I (u) u T (τ)qu(τ)dτ (3.2) where Q R+ n n is a symmetric positive deined matrix and Q R+ n n. The perormance index (3.2) is a measure o the energy used or steering the state o the systems rom x tox. The minimum energy control problem or the ractional positive electrical circuit (2.2) can be stated as ollows. Given the matrices A M n, B R+ n m, α, U Rn + and Q Rn n + o the perormance matrix (3.2), x R n + and t >, ind an input u(t) Rn + or t [, t ] satisying (3.) that steers the state vector o the system rom x tox R n + and minimizes the perormance index (3.2). To solve the problem, we deine the matrix W (t ) Φ(t τ)bq B T Φ T (t τ)dτ (3.3) where Φ(t) is deined by (2.5). From (3.3) and Theorem 2.3, it ollows that the matrix (3.3) is monomial i and only i the ractional positive electrical circuit (2.2) is reachable in time [, t ]. In this case, we may deine the input

5 Circuits Syst Signal Process (26) 35: û(t) Q B T Φ T (t t)w (t )x or t [, t ]. (3.4) Note that the input (3.4) satisies the condition u(t) R n + or t [, t ] i Q R n n + and W (t ) R n n +. (3.5) Theorem 3. Let ū(t) R n + or t [, t ] be an input satisying (3.) that steers the state o the ractional positive electrical circuit (2.2) rom x to x R n +. Then the input (3.4) satisying (3.) also steers the state o the electrical circuit rom x to x R n + and minimizes the perormance index (3.2), i.e., I (û) I (ū). The minimal value o the perormance index (3.2) is equal to I (û) x T W (t )x. (3.6) Proo I the conditions (3.5) are met, then the input (3.4) is well deined and û(t) R n + or t [, t ]. We shall show that the input steers the state o the electrical circuit rom x tox R n +. Substitution o (3.4)into(2.3)ort t and x yields x(t ) Φ(t τ)bû(τ)dτ Φ(t τ)bq B T Φ T (t τ)dτw x x (3.7a) since (3.3) holds. By assuming the inputs ū(t) and û(t), t [, t ] steers the state o the system rom x tox R n +, i.e., and x Φ(t τ)bū(τ)dτ Φ(t τ)bû(τ)dτ (t τ)b[ū(τ) û(τ)]dτ. By transposition o (3.7c) and postmultiplication by W (t )x, we obtain (3.7b) (3.7c) Substitution o (3.4)into(3.8) yields [ū(τ) û(τ)] T B T T (t τ)dτ W (t )x. (3.8) [ū(τ) û(τ)] T B T T (t τ)dτ W x [ū(τ) û(τ)] T Qû(τ)dτ. (3.9)

6 82 Circuits Syst Signal Process (26) 35: Using (3.9), it is easy to veriy that ū(τ) T Qū(τ)dτ û(τ) T Qû(τ)dτ + [ū(τ) û(τ)] T Q[ū(τ) û(τ)]dτ. (3.) From (3.), it ollows that I (û) <I (ū) since the second term in the right-hand side o the inequality is nonnegative. To ind the minimal value o the perormance index (3.2), we substitute (3.4) into(3.2) and we obtain I (û) û T (τ)qu(û)dτ x T W x T W x (t τ)bq B T T (t τ)dτ W x (3.) since (3.3) holds. 4 Procedure and Example To ind t [, t ] or which û(t) R n + reaches its minimal value using (3.4), we compute the derivative dû(t) dt Q B T Ψ(t)W (t )x, t [, t ] (4.) where Ψ(t) d dt [ΦT (t t)]. (4.2) Knowing (t) and using the equality Ψ(t)W (t )x (4.3) we can ind t [, t ] or which û(t) reaches its maximal value. Note that i the electrical circuit is asymptotically stable lim Φ(t), then û(t) t reaches its maximal value or t t and i it is unstable, then or t. From the above considerations, we have the ollowing procedure or computation o the optimal inputs satisying the condition (3.) that steers the state o the system rom x tox R n + and minimizes the perormance index (3.2). Procedure 4. Step Knowing A M n and using (2.5), compute (t). Step 2 Using (3.3), compute the matrix W or given A, B, Q, α and some t. Step 3 Using (3.4) and (4.3), ind t or which û(t) satisying (3.) reaches its maximal value and the desired û(t) or given U R n + and x R n +. Step 4 Using (3.6), compute the maximal value o the perormance index.

7 Circuits Syst Signal Process (26) 35: Fig. Electrical circuit Example 4. Following [7], consider the ractional electrical circuit shown in Fig. with given resistances R, R 2, R 3, inductances L, L 2 and source voltages e, e 2. Using the Kirchho s laws, we can write the equations d α i e R 3 (i i 2 ) + R i + L dt α, d α i 2 e 2 R 3 (i 2 i ) + R 2 i 2 + L 2 dt α (4.4a) (4.4b) where <α<, which can be written in the orm d α [ ] [ ] [ ] i i e dt α A + B i 2 i 2 e 2 (4.5a) where [ R +R 3 A L R 3 L R 3 L 2 R 2+R 3 L 2 ] [ ] L, B. (4.5b) L 2 The ractional electrical circuit is positive since the matrix A is a Metzler matrix and the matrix B has nonnegative entries. In [7], it was shown that the electrical circuit is reachable i R 3. In this case, the matrix A has the orm [ ] R A L R (4.6) 2 L 2 Consider the ractional positive reachable electrical circuit shown in Fig. or R R 2, R 3, L L 2 and α.7. Compute the input û(t) satisying the condition û(t) [ e ] e 2 [ ] 2 2 or t [, t ] (4.7)

8 822 Circuits Syst Signal Process (26) 35: that steers the state o the electrical circuit rom zero state to inal state x [] T (T denotes the transpose) and minimizes the perormance index (3.) with Q [ ] 2. (4.8) 2 Using the Procedure 4., we obtain the ollowing: Step Taking into account that A [ ] (4.9) and using (2.5), we obtain Φ(t) k A k t (k+)α Γ [(k + )α] [ ] ( ) k t (k+).7 ( ) k Γ [(k + ).7]. (4.) k Step 2 Using (3.3), (4.8), (4.9) and taking into account that B I 2, we get W (t ) Φ(t τ)bq B T Φ T (t τ)dτ Φ 2 (τ)dτ 2 k l k l [ [ k [ ( ) k+l ] ( ) k+l Φ(τ)BQ B T Φ T (τ)dτ [ ] ] 2 ( ) k t (k+).7 ( ) k dτ Γ [(k + ).7] τ (k+l).7.6 Γ [(k + ).7]Γ [(l + ).7] dτ [ ] ( ) k+l t (k+l).7+.4 ( ) k+l Γ [(k + ).7]Γ [(l + ).7] (k + l) ]. (4.) Step 3 Using (2.7) and (2.8), we obtain û(t) Q B T Φ T (t t)w (t )x ( [ ] ( ( ) k t t ) ) (k+).7 [ ] 2 ( ) k W (t ) Γ [(k + ).7] k (4.2) Note that the electrical circuit is stable. Thereore, û(t) reach its maximal value or t t (Fig. 2).

9 Circuits Syst Signal Process (26) 35: Fig. 2 Input signal and state variable or t 5[s] Step 4 From (3.6), we have the minimal value o the perormance index I (û) x T W (t )x (4.3) where W (t ) is given by (4.). 5 Extension to Fractional Positive Electrical Circuits with Dierent Orders Consider an electrical circuit composed o resistors, n capacitors and m voltage (current) sources. Using the Kirchho s laws, we may describe the transient states in the electrical circuit by the ractional dierential equation

10 824 Circuits Syst Signal Process (26) 35: d α x(t) dt α Ax(t) + Bu(t), <α<, (5.) where x(t) R n, u(t) R m, A R n n, B R n m. The components o the state vector x(t) and input vector u(t) are the voltages on the capacitors and source voltages, respectively. Similarly, using the Kirchho s laws, we may describe the transient states in the electrical circuit composed o resistances, inductances and voltage (current) sources by the ractional dierential equation d β x(t) dt β Ax(t) + Bu(t), <β<, (5.2) where x(t) R n, u(t) R m, A R n n, B R n m. In this case, the components o the state vector x(t) are the currents in the coils. Now let us consider electrical circuit composed o resistors, capacitors, coils and voltage (current) sources. As the state variables (the components o the state vector x(t)), we choose the voltages on the capacitors and the currents in the coils. Using Eqs. (5.), (5.2) and Kirchho s laws, we may write or the ractional linear circuits in the transient states the state equation d α x C [ ][ ] [ ] dt α A A d β 2 xc B + u, <α,β<, (5.3) x L A 2 A 22 x L B 2 dt β where the components o x C R n are voltages on the capacitors, the components o x L R n 2 are currents in the coils and the components o u R m are the source voltages and A ij R n i n j, B i R n i m, i, j, 2. (5.4) Theorem 5. The solution o Eq. (5.3) or < α < ; < β < with initial conditions has the orm t x(t) Φ (t)x + x C () x and x L () x 2 (5.5) [Φ (t τ)b + Φ 2 (t τ)b ] u(τ)dτ, (5.6a) where x(t) [ ] [ ] x (t) x, x x 2 (t), B x 2 [ ] [ ] B, B, B2

11 Circuits Syst Signal Process (26) 35: Fig. 3 Fractional electrical circuit I n or k l [ ] A A 2 or k, l T kl [ ] or k, l A 2 A 22 T T k,l + T T k,l or k + l > t kα+lβ (t) T kl Ɣ(kα + lβ + ), k l (t) k l 2 (t) k l T kl T kl t (k+)α+lβ Ɣ [(k + )α + lβ], t kα+(l+)β Ɣ [kα + (l + )β]. (5.6b) Proo is given in [8,2]. The extension o Theorem 5. to systems consisting o n subsystems with dierent ractional orders is given in [8,2]. Example 5. Consider the ractional electrical circuit shown in Fig. 3 [9,3,32]with given source voltages e, e 2, ultracapacitor C o the ractional order α.7, ultracapacitor C 2 2 o the ractional order β.6, conductances G 4, G 4, G 2 3, G 2 6, G 2 and N n + n 2 2. Using the Kirchho s laws, we can write the equations d α u C dt α G (v u ), d β u 2 C 2 dt β G 2 (v 2 u 2 ) (5.7) and [ (G + G ) ][ ] [ ][ ] [ ][ ] v G (G 2 + G 2 ) u G e v 2 G +. 2 u 2 G 2 e 2 (5.8)

12 826 Circuits Syst Signal Process (26) 35: From (5.8), we obtain [ v ] [ (G + G ) ] [ ][ ] G u v 2 (G 2 + G 2 ) G 2 u 2 [ (G + G + ) ] [ ][ ] G e (G 2 + G 2 ). (5.9) G 2 e 2 Substitution o (5.9)into [ d α ] [ u G dt α ] [ ] d β C u + u 2 dt β G 2 u 2 C 2 [ G C G 2 C 2 ] [ ] v v 2 (5.) we obtain [ d α ] u [ ] [ ] dt α u e d β A + B, (5.) u 2 u 2 e 2 dt β where A B ( ) G G C (G +G ) G G C (G +G ) G 2 G 2 C 2 (G 2 +G 2 ) ( ) G 2 G 2 C 2 (G 2 +G 2 ) [ ] [ ] 2 A, A 2 [ ] [ ] 2 B. (5.2) B 2 From (5.2), it ollows that A is a diagonal Metzler matrix and the matrix B is monomial matrix with positive diagonal entries. Thereore, the ractional electrical circuit is positive or all values o the conductances and capacitances. Find the optimal input (source voltage) ê(t) R 2 +, t [, t ] satisying the condition ê(t) [ ] [ ] ê (t) < ê 2 (t) or t [, t ] (5.3) or the perormance index (3.2) with Q diag[2, 2] which steers the system rom initial state (voltage drop on capacitances) u [] T to the inite state u [23] T and minimizes the perormance index (3.2) with (5.2). Using (5.6) and (5.2), we obtain Φ (t)b + 2 (t)b [ ][ ] Φ (t) B 2 22 (t) M(t), (5.4) B 2

13 Circuits Syst Signal Process (26) 35: where (t) 2 22 (t) k l k l From (5.6a), (5.2) and (3.3), we have t (k+).7+l.6 Γ [(k + ).7 + l.6] ( 2)k, t k.7+(l+).6 Γ [k.7 + (l + ).6] ( )l. (5.5) W (t ) M(t τ)q M T (t τ)dτ [ ][ ][ ] Φ (τ) 8 Φ T (τ) Φ22 2 (τ) 2 Φ22 2 (τ) dτ. (5.6) Note that the electrical circuit is stable. Thereore, ê(t) reach its maximal value or t t. Now using (3.4) and (5.6), we obtain ê(t) Q M T (t t)w (t )u [ ][ Φ (t t).5 Φ22 2 (t t) [ Φ (t t).5φ 2 22 (t t) ] T ] T W (t ).25[Φ (t t)] T t [ (τ)[φ (τ)]t ] dτ t.5[φ22 2 (t t)] T [ 2 22 (τ)[φ2 22 (τ)]t ] dτ [ ] 2 3 [ 8Φ (τ)[φ (τ)]t ] 2Φ22 2 (τ)[ 2 22 (τ)]t [ ] 2 dτ 3, (5.7) The minimal value o t satisying the condition (3.) can be ound rom the inequality.25[φ (t t)] T t [Φ (τ)[ (τ)]t ] dτ [ ] t.5[φ22 2 (t t)] T [Φ22 2 (τ)[ 2 22 (τ)]t ] < dτ or t [, t ] (5.8) and the minimal value o the perormance index (3.2) is equal to

14 828 Circuits Syst Signal Process (26) 35: I (ê) u T W u [23].5 6 Concluding Remarks [Φ (τ)[φ (τ)]t ] dτ + 3 [ 8Φ (τ)[φ ] [ ] (τ)]t 2 2Φ22 2 dτ (τ)[ 2 22 (τ)]t 3 [Φ 2 22 (τ)[φ2 22 (τ)]t ] dτ. (5.9) Necessary and suicient conditions or the reachability o the ractional positive electrical circuits have been established (Theorem 2.3). The minimum energy control problem or the ractional positive electrical circuits with bounded inputs has been ormulated and solved. Suicient conditions or the existence o a solution to the problem have been given (Theorem 3.), and a procedure or computation o optimal input and the minimal value o perormance index has been proposed. The eectiveness o the procedure has been demonstrated on the example o ractional positive electrical circuit. The presented method can be extended to positive discrete-time linear systems and to ractional positive discrete-time linear systems with bounded inputs. Acknowledgments This work was supported under work S/WE//. Open Access This article is distributed under the terms o the Creative Commons Attribution 4. International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate i changes were made. Reerences. M. Busłowicz, Stability o linear continuous time ractional order systems with delays o the retarded type. Bull. Pol. Acad. Sci. Tech. 56(4), (28) 2. A. Dzieliński, D. Sierociuk, G. Sarwas, Ultracapacitor parameters identiication based on ractional order model, in Proceedings o ECC 9, Budapest (29) 3. L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications (Wiley, New York, 2) 4. T. Kaczorek, An extension o Klamka s method o minimum energy control to ractional positive discrete-time linear systems with bounded inputs. Bull. Pol. Acad. Sci. Tech. 62(2), (24) 5. T. Kaczorek, Asymptotic stability o positive ractional 2D linear systems. Bull. Pol. Acad. Sci. Tech. 57(3), (29) 6. T. Kaczorek, Checking o the positivity o descriptor linear systems by the use o the shule algorithm. Arch. Control Sci. 2(3), (2) 7. T. Kaczorek, Controllability and observability o linear electrical circuits. Electr. Rev. 87(9a), (2) 8. T. Kaczorek, Fractional positive continuous-time systems and their reachability. Int. J. Appl. Math. Comput. Sci. 8(2), (28) 9. T. Kaczorek, Linear Control Systems (Research Studies Press and Wiley, New York, 992). T. Kaczorek, Minimum energy control o descriptor positive discrete-time linear systems. COMPEL Int. J. Comput. Math. Electr. Electron. Eng. 33(3), (24). T. Kaczorek, Minimum energy control o ractional positive continuous-time linear systems. Bull. Pol. Acad. Sci. Tech. 6(4), (23) 2. T. Kaczorek, Minimum energy control o ractional positive discrete-time linear systems with bounded inputs. Arch. Control Sci. 23(2), 25 2 (23)

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