Definition of Euler Gamma Function and Its Properties
There exist the following two definitions of the Euler gamma function. Definition 1.1.A function given by the integral [51, 154, 163] Γ(x) ∞
0 t x−1 e −t dt, (x)>0 (1.1) is called the Euler gamma function.
The Euler gamma function can be also defined by Γ(x) = lim n→∞ n!n x x(x+ 1)ã ã ã(x+n), x∈C\{0,−1,−2, }, whereCis the field of complex numbers.
We shall show thatΓ(x)satisfies the equality Γ(x+ 1) =xΓ(x) (1.2)
Example 1.1.From (1.2) we have for: x= 1 : Γ(2) = 1ãΓ(1) = 1, since Γ(1) ∞
In general case, forx∈Nwe have Γ(n+ 1) =nΓ(n) =n(n−1)Γ(n−1) =n(n−1)(n−2)ã ã ã(1) =n!.
The gamma function is also well-defined for x being any real (complex) numbers For example we have for: x= 1.5 : Γ(2.5) = 1.5ãΓ(1.5) = 1.5ã0.5Γ(0.5), x=−0.5 : Γ(0.5) =−0.5ãΓ(−0.5) =−0.5ã(−1.5)Γ(−1.5).
Mittag-Leffler Function
The Mittag-Leffler function is a generalization of the exponential functione s i t and it plays important role in solution of the fractional differential equations. Definition 1.2.A function of the complex variablezdefined by [51, 154, 163]
E α (z) ∞ k=0 z k Γ(kα+ 1) (1.3) is called the one parameter Mittag-Leffler function.
An extension of the one parameter Mittag-Leffler function is the following two parameters function.
Definition 1.3.A function of the complex variablezdefined by [51, 154, 163]
E α,β (z) ∞ k=0 z k Γ(kα+β) (1.4) is called the two parameters Mittag-Leffler function.
Definitions of Fractional Derivative-Integral
Riemann-Liouville Definition
It is well known that to reduceN-multiple integral to 1-tiple integral the fol- lowing formula
(xưu) N ư1 f(u)du (1.5) can be used, wheref(u)is a given function.
Using the equality (N−1)! =Γ(N), the formula (1.5) can be extended for anyN ∈Rand we obtain Riemann-Liouville fractional integral
(t−τ) α−1 f(τ)dτ, (1.6) whereα∈R+ \{0} is the order of integral.
Definition 1.4.The function defined by [51, 154, 163]
(1.7) where N−1≤α < N, N ∈ N is called Riemann-Liouville fractional derivative-integral.
Note, that from (1.7), forα= 0 we obtain
0 f(τ)dτ = d 0 dt 0 f(t) =f(t) and forα= 1we have
Therefore, by induction, Definition 1.4 is true forα∈N.
Example 1.3.Consider the unit-step function f(t) =1(t)
Using (1.7), we obtain d α dt α 1(t) = 1 Γ(N−α) d N dt N t
Therefore, theαorder Riemann-Liouville derivative of unit-step function is a decreasing in time function.
Theorem 1.1.The Riemann-Liouville derivative-integral operator is a lin- ear operator satisfying the relation
Theorem 1.2.The Laplace transform of the derivative-integral (1.7) for
1.3 Definitions of Fractional Derivative-Integral 5
Proof Using (1.6), (1.7) and (A.5), (A.6) (see Appendix A.1) for
Caputo Definition
Definition 1.5.The function defined by [51, 154, 163]
0 f (N) (τ) (t−τ) α+1−N dτ, f (N ) (τ) =d N f(τ) dτ N (1.9) is called the Caputo fractional derivative-integral, where N−1 ≤α < N,
Remark 1.1.From Definition 1.5 it follows that the Caputo derivative of constant is equal to zero.
Theorem 1.3.The Caputo derivative-integral operator is linear satisfying the relation
Proof The proof is similar to the proof of Theorem 1.1
Theorem 1.4.The Laplace transform of the derivative-integral (1.9) for
Proof Using Definitions 1.5 and A.2, equations (A.3), (A.5) for
Solutions of the Fractional State Equation
of Continuous-Time Linear System
Consider the continuous-time linear system described by the equations [52]
C 0 D α t x(t) = d α x(t) dt α =Ax(t) +Bu(t), 0< α 0, (3.37) whereδ (k) is thek-th derivative of the Dirac impulse.
3.1 Descriptor Linear Electrical Circuits 95 whereQ Q 1 Q 2
Definition 3.1.The descriptor electrical circuit (3.1) is called pointwise complete for t=t f if for every final statex f ∈R n there exist initial condi- tionsx 0 ∈R n satisfying (3.36a) such thatx f =x(t f )∈ImQ 1
Theorem 3.3.The descriptor electrical circuit is pointwise complete for any t=t f and every x f ∈R n satisfying the condition x f ∈ImQ 1 (3.39)
Proof Taking into account that for any A 1 , det e A 1 t
=e −A 1 t , from (3.35) and (3.37) fort=t f we obtain ¯ x 10 =e −A 1 t x¯ 1 (t f ) and ¯x 2 (t) = 0.
Therefore, from (3.38) it follows that there exist initial conditions x 0 ∈R n such thatx f =x(t f )if (3.39) holds
In this case from (3.14c) we have
The eigenvalues of the matrix A 1 (given by (3.14b)) are s 1 = 0, s 2 =− C 1 +C 2 +C 3
R 1 C 1 (C 2 +C 3 ) and using Sylvester formula [44] we obtain e A 1 t =Z 1 +Z 2 e s 2 t ⎡
Therefore, the descriptor electrical circuit shown in Figure 3.1 is pointwise complete for anyt=t f and everyx f satisfying (3.41).
Pointwise Degeneracy of Descriptor Electrical
Consider the descriptor electrical circuit described by equation (3.1) foru(t) = 0,t≥0.
Definition 3.2.The descriptor electrical circuit (3.1) is called pointwise de- generated in the directionv∈R n fort =t f if there exists nonzero vector v such that for all initial conditionsx 0 ∈ ImQ 1 the solution of (3.1) satisfies the condition v T x f = 0.
Theorem 3.4.The descriptor electrical circuit (3.1)is pointwise degenerated in the direction v defined by v T Q 1 = 0 (3.43) for any t f > 0 and all initial conditions x¯ 10 ∈ Im ¯Q 1 , where Q 1 and Q¯ 1 are determinated by (3.36b) and (3.38), respectively.
Proof Substitution of (3.35) into (3.38) yields x f =Q 1 e A 1 t f x¯ 10 and v T x f =v T Q 1 e A 1 t f x¯ 10 = 0, since (3.43) holds for all¯x 10 = ¯Q 1 x 0 ∈Im ¯Q 1
Example 3.4.(continuation of Examples 3.1 and 3.3)
Descriptor Fractional Linear Electrical Circuits
Therefore, the descriptor electrical circuit shown in Figure 3.1 is pointwise degenerated in the direction defined by (3.44) for any t f > 0 and any values of the resistanceR 1 and capacitancesC 1 ,C 2 ,C 3
3.2 Descriptor Fractional Linear Electrical Circuits
Consider the continuous-time fractional linear system described by the state equation (1.49a).
It is assumed thatdetE= 0,rankB =mand the pencil of matrices(E, A) is regular, i.e condition (1.51) is met.
Example 3.5.Consider the fractional electrical circuit shown in Figure 3.1 with given resistance R, capacitances C 1 , C 2 , C 3 and source voltages e 1 ande 2
Using Kirchhoff’s laws, for the electrical circuit we can write the equations e 1 =RC 1 d α u 1 dt α +u 1 +u 3 ,
The equations (3.45) can be written in the form
In this case we have
Note that the matrixE is singular (detE= 0) but the pencil det [Es α −A]
Therefore, the fractional electrical circuit is a descriptor fractional linear system with regular pencil.
In general case we have the following theorem.
Theorem 3.5.If the fractional electrical circuit contains at least one mesh consisting of branches with only ideal capacitors and voltage sources, then its matrixE is singular.
The row of E associated with the mesh is a zero row, which is a direct consequence of Kirchhoff's voltage law, as the equation derived from this law is algebraic in nature.
Example 3.6.Consider the fractional electrical circuit shown in Figure 3.2 with given resistancesR 1 , R 2 , R 3 inductances L 1 , L 2 , L 3 and source volt- agese 1 ande 2
Using Kirchhoff’s laws we can write the equations e 1 =R 1 i 1 +L 1 d β i 1 dt β +R 3 i 3 +L 3 d β i 3 dt β , e 2 =R 2 i 2 +L 2 d β i 2 dt β +R 3 i 3 +L 3 d β i 3 dt β ,
Equations (3.48) can be written in the form
In this case we have
Note that the matrixE is singular but the pencil det
Therefore, the fractional electrical circuit is a descriptor linear system with regular pencil.
Theorem 3.6.If the fractional electrical circuit contains at least one node with branches with coils then its matrix E is singular.
3.2 Descriptor Fractional Linear Electrical Circuits 99
Proof Note that the equation written using the current Kirchhoff’s current law for this node is an algebraic one and in the matrix E we have zero row.
In general case we have the following theorem.
Theorem 3.7 states that any fractional electrical circuit qualifies as a descriptor system if it includes at least one mesh formed by branches containing solely ideal capacitances and voltage sources, or if it features at least one node with branches that include coils.
According to Theorem 3.5, the matrix E of a fractional electrical circuit becomes singular when there is at least one mesh composed solely of ideal capacitors and voltage sources Additionally, as stated in Theorem 3.6, the matrix E is also singular if the circuit includes at least one node with branches containing coils.
By applying the solution from equation (1.60) to equation (1.49a), we can determine the voltages across supercapacitors and the currents in supercoils during transient states of descriptor fractional linear electrical circuits With this information on voltages and currents, and utilizing equation (1.61), we can also calculate any additional currents and voltages within these circuits.
Example 3.7.(continuation of Example 3.5) Using one of the well-known methods [21, 50, 188] for the pencil (3.47), we can find the matrices
⎦, (3.49b) which transform it to the canonical form (1.52) with
Using the matrixB given by (3.46), (3.49) and (1.53c) we obtain
Using (1.54) we obtainx 1 (t)of the fractional circuit for any initial condi- tionx 10 ∈R n 1 and an arbitrary inputu(t).
Taking into account (1.60) and (1.61) we may compute any currents and voltages in singular fractional linear electrical circuit shown in Figure 3.1.
In the same way we may find currents in the supercoils and of the descriptor fractional electrical circuit shown in Figure 3.1.
Polynomial Approach to Fractional Descriptor Electrical
First the essensce of the polynomial approach will be shown on the following example.
Example 3.8.Consider the fractional descriptor electrical circuit shown in Figure 3.3 with given resistances R 1 , R 1 ; inductances L 1 ,L 2 and source currenti z
Fig 3.3 Fractional electrical circuit of Example 3.8
Using Kirchhoff’s laws we can write the equations
The equations (3.50) can be written in the form (1.49a)
3.3 Polynomial Approach to Fractional Descriptor Electrical Circuits 101
, B 2 1 we can write the equation (3.51) in the form
Theβ-order fractional differentiation of (3.52b) yields
1 1 (3.55) is nonsingular and premultiplying (3.54) by its inverse we obtain d β dt β i 1 i 2 = ¯A i 1 i 2 + ¯B 0 i z + ¯B 1 d β i z dt β , (3.56a) where
The standard equation (3.56a) can be also obtained from the equation (3.54) by reducing the matrix (3.55) to the identity matrixI2 using the ele- mentary row operations (see Appendix B)
L 1 +L 2 , L[2 + 1×(−1)] (3.57) Performing the elementary row operations (3.57) on the matrix
0s β we obtain the polynomial matrix
Therefore, the reduction of the matrix (3.55) to identity matrix by the use of elementary row operations (3.57) is equivalent to premulti- plication of the equation
X=BU by the polynomial matrix of elementary row operations (3.58), whereX =L i 1 i 2 ,U =L[i z ].
3.3 Polynomial Approach to Fractional Descriptor Electrical Circuits 103
In general case let us consider the continuous-time fractional linear system described by the state equation (1.49a).
It is assumed thatdetE = 0but the pencil of matrices(E, A)is regular, i.e condition (1.51) is met.
Applying to (1.49a) the Laplace transform for zero initial conditions we ob- tain the equation
Theorem 3.8.There exists a nonsingular polynomial matrix
L(s α ) =L 0 +L 1 s α +ã ã ã+L μ s αμ , (3.60) where μ is the nilpotency index of the pair (E, A) (see Appendix C), such that
(3.61) if and only if the pencil (E, A) is regular, i.e the condition (1.51)is met. Proof The matrix
The matrix I_n - Ā is nonsingular for all matrices Ā in R^n×n Consequently, based on equations (3.61) and (1.51), the polynomial matrix represented in (3.60) is also nonsingular Additionally, through the application of elementary row operations, any singular matrix E can always be transformed into a reduced form.
0 , whereE 1 has the full row rank r 1 and L 1 is the matrix of elementary row operations.
(3.62b) Using (3.62) we can write the equation (1.49a) in the form
Theα-order fractional differentiation of (3.63b) yields
−A 2 is nonsingular then from (3.65) we have d α x dt α = ¯A 1 x+ ¯B 10 u+ ¯B 11 d α u dt α , where
−A 2 is singular then using elementary row operations we may reduce this matrix to the form
0 , wherer 2 = rankE 2 ≥rankE 1 and we repeat the procedure.
It is well known that if the condition (1.51) is satisfied then afterμsteps of the procedure we obtain the nonsingular matrix
B μ,μ d μα u dt μα (3.68) by the inverse matrix
−1 we obtain the desired equation d α x dt α = ¯A μ x+ ¯B 0 u+ ¯B 1 d α u dt α +ã ã ã+ ¯B μ d μα u dt μα , (3.69a) where
3.3 Polynomial Approach to Fractional Descriptor Electrical Circuits 105
The standard equation (3.69a) can be derived from equation (3.68) by transforming the matrix (3.67) into the identity matrix I n through elementary row operations This process is equivalent to premultiplying equation (3.68) by an appropriate matrix of elementary row operations.
The desired polynomial matrix of elementary row operations (3.60) is given by
Note that the matrixI n−r i s α corresponds to the fractional differentiation of the algebraic equations
The considerations can be easily extended to the linear electrical circuits described by the state equation with different fractional orders.
Example 3.9.Consider the fractional descriptor electrical circuit shown in Figure 3.4 with given resistances R 1 , R 2 , R 3 ; inductances L 1 , L 2 , L 3 ; capacitanceC and source voltagese 1 ,e 2
Fig 3.4 Fractional electrical circuit of Example 3.9
Using Kirchhoff’s laws we can write the equations e 1 =L 1 d β i 1 dt β +R 1 i 1 +L 3 d β i 3 dt β +R 3 i 3 , (3.70a) e 2 =L 2 d β i 2 dt β +R 2 i 2 −L 3 d β i 3 dt β −R 3 i 3 , (3.70b) i 3 =i 1 −i 2 , (3.70c) u=e 1 +e 2 (3.70d)
The equations (3.70) can be written in the form
The pencil (E, A) is regular, since det
1 1 we can write the equation (3.71a) in the form
3.3 Polynomial Approach to Fractional Descriptor Electrical Circuits 107 and
The β-order fractional differentiation of the first equation of (3.72b) andα-order fractional differentiation of the second equation of (3.72b) yield
⎦ (3.75) is nonsingular and premultiplying (3.74) by its inverse we obtain
The standard equation (3.76a) can be also obtained from the equation (3.74) by reducing the matrix (3.75) to the identity matrixI4using elemen- tary row operations
Using the elementary row operations (3.77) on the matrix
⎦ we obtain the polynomial matrix
Positive Descriptor Fractional Electrical Circuits
Pointwise Completeness and Pointwise Degeneracy
Degeneracy of Positive Fractional Descriptor Electrical Circuits
Definition 3.4.The positive fractional descriptor electrical circuit described by (3.78) is called pointwise complete for t = t f if for every final state x f ∈ R n + there exists a vector of initial conditions x 0 ∈ Im%E¯E¯ D &
Theorem 3.15.The positive fractional descriptor electrical circuit (3.78)is pointwise complete for any t = t f and every final state x f ∈ R n + belonging to the set x f ∈Im [Φ 0 (t f )x 0 ]⊂R n + if and only if Φ 0 (t f )∈R n×n + is a monomial matrix.
Proof Substituting in (3.80a)t=t f we obtain x f =x(t f ) =Φ 0 (t f )x 0 (3.89) and x 0 = [Φ 0 (t f )] −1 x f ∈R n + , since the matrixΦ 0 (t)is monomial and[Φ 0 (t f )] −1 ∈R n×n +
Definition 3.5.The positive fractional descriptor electrical circuit (3.78) is called pointwise degenerated in the direction v for t = t f if there exists a nonzero vector v ∈ R n such that for all initial conditions x 0 ∈Im%E¯E¯ D &
Theorem 3.16.The positive fractional descriptor electrical circuit (3.78)is pointwise degenerated in the directionv defined by v T E¯ = 0 (3.91) for any t f ≥0and all initial conditions x 0 ∈Im%E¯E¯ D &
Proof Postmultiplying (3.91) by E¯ D w and using x 0 = ¯EE¯ D w and (3.90) we obtain v T E¯E¯ D w=v T x 0 = 0.
Taking into account (3.80b), (3.89) and (3.91) we obtain
3.4 Positive Descriptor Fractional Electrical Circuits 115 v T x f =v T Φ 0 (t f )x 0 ∞ k=0 v T %E¯ D A¯& k t kα f Γ(kα+ 1) x 0
In this case the set of admissible (consistent) initial conditions has the form x 0 = ¯EE¯ D w RC 1 0
Therefore, the fractional descriptor electrical circuit is pointwise complete for anyt f and every final state (3.93) for all values ofRandC 1
The electrical circuit is pointwise degenerated in the directionv T 0v 2 (v 2 - arbitrary), since from (3.91) we have v T E¯ v 1 v 2 RC 1 0
0 0 0 0 forv 1 = 0and arbitraryv 2 (3.94) Using (3.93) and (3.94) we obtain v T x f 0 v 2 x f1
Therefore, the electrical circuit is degenerated in the direction v T 0v 2 for all values ofRandC 1
Stability of Positive Standard LinearElectrical Circuits
Stability of Positive Electrical Circuits
Linear electrical circuits, consisting of resistors, coils, capacitors, and voltage or current sources, can be analyzed using Kirchhoff’s laws These laws enable us to describe the transient states of nearly all electrical circuits through state equations.
[1, 44, 47, 68, 71, 115, 172, 191] dx(t) dt =Ax(t) +Bu(t), (4.1a) y(t) =Cx(t) +Du(t), (4.1b) where x(t) ∈ R n , u(t) ∈ R m , y(t) ∈ R p are the state, input and output vectors andA∈R n×n ,B∈R n×m ,C∈R p×n ,D∈R p×m
In electrical circuit analysis, the state variables, represented as x1(t), , xn(t), correspond to the currents in the coils and the voltages across the capacitors The input vector u(t) consists of source voltages or currents, while the output vector y(t) includes the circuit's currents and voltages.
Definition 4.1.The electrical circuit described by the equations (4.1) (shortly electrical circuit (4.1) ) is called (internally) positive if for any x(0) = x 0 ∈ R n + and every u(t) ∈ R m + , t ≥ 0 we have x(t) ∈ R n + and y(t)∈R p + ,t≥0.
Theorem 4.1.[30, 47] The electrical circuit (4.1)is positive if and only if
Definition 4.2.The positive electrical circuit (4.1) is called asymptotically stable if t→∞ lim x(t) = 0 for any x 0 ∈R n +
118 4 Stability of Positive Standard Linear Electrical Circuits
The positive electrical circuit will be called unstable if it is not asymptot- ically stable.
Theorem 4.2.[30, 47] The positive electrical circuit (4.1)is asymptotically stable if and only if
Res k