#### 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. Rinaldi , Positive Linear Systems: Theory and Applications (Wiley, New York, 2000 )
4. T. Kaczorek , An extension of Klamka's method of minimum energy control to fractional positive discrete-time linear systems with bounded inputs . Bull. Pol. Acad. Sci. Tech . 62 ( 2 ), 227 - 231 ( 2014 )
5. T. Kaczorek , Asymptotic stability of positive fractional 2D linear systems . Bull. Pol. Acad. Sci. Tech . 57 ( 3 ), 289 - 292 ( 2009 )
6. T. Kaczorek , Checking of the positivity of descriptor linear systems by the use of the shuffle algorithm . Arch. Control Sci . 21 ( 3 ), 287 - 298 ( 2011 )
7. T. Kaczorek , Controllability and observability of linear electrical circuits . Electr. Rev . 87 (9a), 248 - 254 ( 2011 )
8. T. Kaczorek , Fractional positive continuous-time systems and their reachability . Int. J. Appl. Math. Comput. Sci . 18 ( 2 ), 223 - 228 ( 2008 )
9. T. Kaczorek , Linear Control Systems (Research Studies Press and Wiley, New York, 1992 )
10. T. Kaczorek , Minimum energy control of descriptor positive discrete-time linear systems . COMPEL Int. J. Comput. Math. Electr. Electron. Eng . 33 ( 3 ), 976 - 988 ( 2014 )
11. T. Kaczorek , Minimum energy control of fractional positive continuous-time linear systems . Bull. Pol. Acad. Sci. Tech . 61 ( 4 ), 803 - 807 ( 2013 )
12. T. Kaczorek , Minimum energy control of fractional positive discrete-time linear systems with bounded inputs . Arch. Control Sci . 23 ( 2 ), 205 - 211 ( 2013 )
13. T. Kaczorek , Minimum energy control of fractional positive electrical circuits , in Proceedings of 22th European Signal Processing Conference ( 2014 )
14. T. Kaczorek , Minimum energy control of positive continuous-time linear systems with bounded inputs . Int. J. Appl. Math. Comput. Sci . 23 ( 2 ), 725 - 730 ( 2013 )
15. T. Kaczorek , Minimum energy control of positive electrical circuits , in Proceedings of 19th International Conference of Methods and Models in Automation and Robotics ( 2014 )
16. T. Kaczorek , Polynomial approach to fractional descriptor electrical circuits , in Proceedings of 7th International Conference on Intelligent Information and Engineering Systems ( 2014 )
17. T. Kaczorek , Positive 1D and 2D Systems (Springer, London, 2001 )
18. T. Kaczorek , Positive linear systems consisting of n subsystems with different fractional orders . IEEE Trans. Circuits Syst . 58 ( 6 ), 1203 - 1210 ( 2011 )
19. T. Kaczorek , Positivity and reachability of fractional electrical circuits . Acta Mech. Autom . 5 ( 2 ), 42 - 51 ( 2011 )
20. T. Kaczorek , Practical stability of positive fractional discrete-time linear systems . Bull. Pol. Acad. Sci. Tech . 56 ( 4 ), 313 - 317 ( 2008 )
21. T. Kaczorek , Selected Problems of Fractional Systems Theory (Springer, Berlin, 2012 )
22. T. Kaczorek , J. Klamka , Minimum energy control of 2D linear systems with variable coefficients . Int. J. Control 44 ( 3 ), 645 - 650 ( 1986 )
23. J. Klamka , Controllability and minimum energy control problem of fractional discrete-time systems , in New Trends in Nanotechnology and Fractional Calculus, ed. by D. Baleanu, Z.B. Guvenc , J.A. Tenreiro Machado (Springer, New York, 2010 ), pp. 503 - 509
24. J. Klamka , Controllability of Dynamical Systems (Kluwer Academic Press, Dordrecht, 1991 )
25. J. Klamka , Minimum energy control of 2D systems in Hilbert spaces . Syst. Sci . 9 ( 1-2 ), 33 - 42 ( 1983 )
26. J. Klamka , Relative controllability and minimum energy control of linear systems with distributed delays in control . IEEE Trans. Autom. Control 21 ( 4 ), 594 - 595 ( 1976 )
27. K.B. Oldham , J. Spanier , The Fractional Calculus (Academic Press, New York, 1974 )
28. P. Ostalczyk , Epitome of the Fractional Calculus: Theory and Its Applications in Automatics (Wydawnictwo Politechniki Łódzkiej , Łódz´, 2008 ). (in Polish)
29. I. Podlubny , Fractional Differential Equations (Academic Press, San Diego, 1999 )
30. A.G. Radwan , A.M. Soliman , A.S. Elwakil , A. Sedeek , On the stability of linear systems with fractionalorder elements . Chaos Solitons Fractals 40 ( 5 ), 2317 - 2328 ( 2009 )
31. Ł. Sajewski, Minimum energy control of fractional positive continuous-time linear systems with two different fractional orders and bounded inputs , in Advances in Modelling and Control of Non-integerOrder Systems, Lecture Notes in Electrical Engineering , vol. 320 ( 2015 ) pp. 171 - 181
32. Ł. Sajewski, Reachability, observability and minimum energy control of fractional positive continuoustime linear systems with two different fractional orders . Multidimens. Syst. Signal Process . ( 2014 ). doi:10.1007/s11045- 014 - 0287 -2
33. E.J. Solteiro Pires , J.A. Tenreiro Machado , P.B. Moura Oliveira , Fractional dynamics in genetic algorithms , in Workshop on Fractional Differentiation and Its Application vol. 2 ( 2006 ) pp. 414 - 419
34. B.M. Vinagre , C.A. Monje , A.J. Calderon , Fractional order systems and fractional order control actions , in Lecture 3 IEEE CDC'02 TW #2: Fractional calculus Applications in Automatic Control and Robotics ( 2002 )