9.2.1. Usual capacitor and inductor devices

Let us recall that a capacitor C is characterized by the equation (see Figure 9.1)

[9.3]

The electrostatic energy stored in the capacitor is expressed as [JOO 86, IRW 99]

[9.4]

Using the Laplace transform, equation [9.3] becomes

[9.5]

Therefore

[9.6]

where

[9.7]

is the complex impedance of the capacitor.

Similarly, for an inductor L (see Figure 9.2), we can write [IRW 99]

[9.8]

The magnetic energy stored in the inductor is expressed as

[9.9]

We can also write

[9.10]

or

[9.11]

where

[9.12]

is the complex impedance of the inductor.

Capacitor C and inductor L are the only elementary electrical devices able to store energy. On the contrary, they do not dissipate energy, at least in their idealized form.

9.2.2. Fractional capacitor and inductor

Let us now consider electrical complex devices such as RC and RL infinite length lines. The RC line has already been analyzed to investigate some properties of the fractional integrator (see Chapter 6 of Volume 1). The RL line can also be modeled as a fractional order system (see Appendix A.9.1).

These devices can be interpreted as realizations of the fractional integrator and of the fractional differentiator. However, they can also be interpreted as fractional capacitors and inductors.

9.2.2.1. Fractional capacitor

For the infinite length RC line (see Chapter 6 of Volume 1), we demonstrated that (see Figure 9.3)

[9.13]

Let us define

[9.14]

then

[9.15]

where

[9.16]

Thus

[9.17]

which is the fractional generalization of

[9.18]

9.2.2.2. Fractional inductor

It is demonstrated in Appendix A.9.1 that the infinite length RL line (or GL line as shown in Figure 9.4) is characterized by

[9.19]

Let us define

[9.20]

then

[9.21]

where

[9.22]

Thus

[9.23]

which is the fractional generalization of

[9.24]

9.2.3. Energy storage and dissipation in fractional devices

9.2.3.1. Fractional capacitor

A distributed model with the internal variable z_C(ω, t) is associated with the electrical equation i(t) = C*Dⁿ (v(t)) (see Chapter 7).

Therefore

[9.25]

The energy stored in the fractional capacitor C* is expressed (see Appendix A.9.2) as

[9.26]

In contrast to the usual capacitor, an electrical power P(t) is also dissipated in the fractional capacitor.

It is expressed as

[9.27]

The energy is stored in the distributed capacitors C and dissipated in the distributed resistors R of the infinite length RC line.

9.2.3.2. Fractional inductor

A distributed model with the internal variable z_L(ω, t) is associated with the electrical equation v(t) = L*Dⁿ(i(t)).

Therefore

[9.28]

Similarly to the fractional capacitor, the energy stored in the fractional inductor L^* is expressed as [HAR 15b]

[9.29]

and the electrical power dissipated in the fractional inductor is expressed as:

[9.30]

9.2.4. Reversibility of energy and energy balance

9.2.4.2. Energy balance

Energy balance and conservation of energy are fundamental principles of physics [JOO 86, ORT 98].

For an autonomous system, they state that, for energy transfers:

[9.31]

and for dissipated powers:

[9.32]

The term E_j(t) represents reversible energies such as kinetic, potential, magnetic or electrostatic energies for electromechanical systems. The term P_j(t) represents energy dissipation in resistors or frictions which is transformed into heat.

Equation [9.31] expresses conservation of energy or energy balance. Practically, this equation has to be adapted to systems where energy transfers are characterized by a periodical decreasing phenomenon. Thus, the derivative has to be replaced by the mean of the energy derivative, i.e. by .

9.3.1. Introduction

Although perfectly known, the energy transfers between an inductor L and a capacitor C provide the opportunity to propose a methodology to analyze the stability of fractional systems.

9.3.2. Analysis of the series RLC circuit

The series RLC circuit is presented in Figure 9.5. The components L and C are characterized by the equations

[9.33]

Moreover, R i(t) + v_L(t) + v_C(t) = 0.

The state variables are v_C(t) for C and i(t) for L. Therefore, the state space model of the circuit is

[9.34]

At t = 0, the initial conditions are v_C(0) and i(0). The energies stored in L and C are expressed as

[9.35]

Thus, the global system energy (or Lyapunov function V(t)) is the sum of these two energies

[9.36]

and its derivative is

[9.37]

therefore:

[9.38]

This derivative expresses that energy is dissipated in resistor R.

Thus:

• – for R > 0, the system is stable;
• – for R < 0, the system is unstable.

Limit of stability occurs for R = 0 and a constant amplitude oscillation appears.

9.3.3. Stability analysis

Of course, this stability analysis is trivial. However, let us consider the waveforms of E_L(t), E_C(t) and V(t) (see Figure 9.6) which are simulated with

[9.39]

The graphs of E_L(t) and E_C(t) are shifted damped sinusoids. Their sum V(t) is a monotonous decreasing function. For R = 0, these sine functions have a constant amplitude and their sum is a constant. These curves express that the energy E_L(t) with the initial value is transferred to the capacitor, so E_C(t) can increase. When E_C(t) is maximum (and E_L(t) minimum), a new cycle of energy transfer occurs, and so on. These exchanges between L and C are perfect with no loss when R = 0, i.e. when there is a perfect energy balance between the two components, with no dissipation.

When R = 0, let us define

[9.40]

Therefore

[9.41]

Thus

[9.42]

These functions have two components: a constant part, equal to their mean, and a time-varying part, the frequency of which is twice that of v_C(t) or i(t).

The two means are equal because there is no loss in the resistor.

Therefore

[9.43]

Thus, the oscillation frequency is derived from the equality of the means of the two complementary energies E_L(t) and E_C(t):

[9.44]

Let us recall that the derivative provides the stability condition: or R > 0.

This is the energy balance principle, between complementary components, which will be used to derive stability conditions for fractional order systems.

9.4.1. Introduction

The series RLC* fractional circuit is a typical non-commensurate order system, with n = 1 for L and 0 < n < 1 for C*. It allows a simple analysis of transients and energy transfers between L and C*.

9.4.2. Analysis of the series RLC* circuit

The series RLC* circuit corresponds to Figure 9.5, where capacitor C is replaced by C*. The components L and C* are characterized by the equations

[9.45]

Moreover, R i(t) + v_L(t) + v_C(t) = 0.

The state variables are i(t) for the inductor L and z_C(ω, t) for the fractional capacitor C*.

Therefore, the state space model of this circuit is

[9.46]

At t = 0, the initial conditions are z_C(ω, 0) and i(0). The energies stored in L and C* are expressed as

[9.47]

Thus, the total system energy E(t) or the Lyapunov function V(t) is the sum of these two energies

[9.48]

and its derivative is

[9.49]

Thus

[9.50]

This equation expresses that the energy V(t) is dissipated inside the fractional capacitor by internal Joule losses and outside in resistor R.

For R > 0 , the circuit is essentially dissipative and energy stored in L and C* can only decrease. Note that the damping is more important than in the previous RLC circuit thanks to internal Joule losses in C*.

However, an important difference has to be noted: for R < 0, the term –R i(t)² is now positive, whereas the second term remains negative. Negative resistors are artificially provided by positive feedback circuits [AUV 80, CHA 82] or Tunnel diodes [KHA 96].

9.4.3. Experimental stability analysis

Numerical simulations are necessary to highlight some specific features of fractional systems. The RLC* circuit is simulated with

[9.51]

For the fractional integrator, J + 1 = 21, with ω varying from ω_b = 0.001 rad/s to ω_h = 1000 rad/s; T_e = 10^–3s.

The waveforms of i(t) and v_C(t) are presented in Figure 9.7 for R = 0.1 and R = –0.1. We can note that the circuit is of course stable for R = 0.1 and also for R = –0.1. Then, the graphs of V(t), E_L(t) and E_C(t) are presented in Figure 9.8 for R = 0.1 and in Figure 9.9 for R = –0.1.

The decrease of V(t) is monotonous for R > 0, as noted for the integer order RLC circuit. On the contrary, an oscillation of V(t) appears for R < 0, i.e. we can observe a series of energy regeneration and dissipation phases.

This phenomenon is a specificity of fractional systems, and it has already been observed in the commensurate order case [TRI 13b, TRI 13c]. It is easily explained using the interpretation of .

The two components –Ri(t)², and their sum for R = –0.1 are presented in Figure 9.10.

For R > 0 (Figure 9.8), the two components are negative, so and the decrease of V(t) is monotonous.

On the contrary, for R < 0 (Figure 9.9), the two components have opposite signs and their sum is alternatively positive and negative. Consequently, V(t) (Figure 9.9) is globally decreasing, but its decrease is no longer monotonous but periodic.

For phases, V(t) is regenerated, whereas it is dissipated for phases.

This is not a new oscillating phenomenon: it is caused by the natural damped oscillation of i(t) and v_C(t), combined with the effect of the –R i(t)² term when R < 0 (see Figure 9.10).

Despite these oscillations, we can conclude that the mean of is negative when the system is stable.

Thus, ensures stability; however, this mean cannot directly be used to derive stability conditions.

9.4.4. Theoretical stability analysis

As observed with the integer order RLC circuit, the components L and C* exchange their complementary stored energies: the system remains stable if the energy is globally decreasing, i.e. if . A constant amplitude oscillation occurs at the limit of stability, and the Lyapunov function V(t) is globally constant (when all transients have vanished), i.e. its average value is constant.

We can equivalently express this property using the energy balance principle applied to the means of the complementary energies E_L(t) and E_C(t).

Thus, at the limit of stability:

[9.52]

Moreover,

[9.53]

Recall that ensures stability.

9.4.4.1. Derivation of oscillation frequency

Let us define v_C(t) = Ve^{j β t} at the limit of stability, where β is the oscillation frequency.

As i(t) = C*Dⁿ(v_C(t)), we can write

[9.54]

We have to calculate . Recall that .

Thus

[9.55]

The calculation of requires z_C(ω, t).

According to (9.46), we obtain

[9.56]

with

[9.57]

thus

[9.58]

As , we obtain

[9.59]

According to Appendix A.9.3, we can write

[9.60]

Consequently, the equality provides the value of β:

[9.61]

9.4.4.2. Stability condition

The circuit remains stable if . According to [9.50], this condition is equivalent to

[9.62]

This condition expresses that the circuit remains stable if the mean of energy regeneration is less than the mean of internal Joule losses in the fractional capacitor. Unfortunately, these means are not easily calculated in the general case. However, at the limit of stability, it is straightforward to write

[9.63]

and

[9.64]

According to Appendix A.9.3

[9.65]

thus

[9.66]

Consequently, we can deduce the limit value of R:

[9.67]

and stability is ensured if R > R_lim.

9.4.4.3. Example

The case n = 0.5 simplifies calculations. Moreover, it corresponds to the practical case of the infinite RC line (section 9.2.2.1). Note that this result can be verified using Matignon’s criterion (Appendix A.8.2).

As n = 0.5, we obtain

[9.68]

Thus

[9.69]

and

[9.70]

With L = 1 C* = 1, we obtain

[9.71]

and stability is ensured if

[9.72]

9.4.5. Conclusion

Energy balance is a fundamental principle of physics [JOO 86, ORT 98]. For an autonomous system, it states that

[9.73]

[9.74]

As demonstrated by previous numerical simulations, the application of this principle to fractional systems has required important adaptations. More specifically, because is no longer a monotonous decreasing function, it cannot be used to characterize the stability of fractional systems.

Practically, we have to replace the derivative of energy by its mean because it remains a globally decreasing function. Therefore, the stability of a fractional system is characterized by

[9.75]

However, this mean is easy to calculate only at the limit of stability.

Consequently, the energy balance principle of an electrical fractional system has been defined at the limit of stability as

[9.76]

where

[9.77]

and V(t) is the system’s Lyapunov function.

9.5.1. Circuit modeling

The previous methodology is now applied to a more complex system in order to demonstrate its generality.

Consider the RLL*C* series circuit presented in Figure 9.11, which is governed by the following equations:

[9.78]

The state variables i(t), z_C(ω, t) and z_L(ω, t) are defined as

[9.79]

with

[9.80]

The corresponding energies are

[9.81]

Thus

[9.82]

and

[9.83]

As , we obtain

[9.84]

As mentioned previously, is the sum of internal and external losses.

If R > 0, the system is naturally damped.

9.5.2. Stability analysis

9.5.2.1. Oscillation frequency

When the circuit is at the limit of stability, i(t), , and v_L(t) are sine functions.

Let us define .

As , we can write

[9.85]

Then, according to [9.79]

[9.86]

As , we obtain

[9.87]

According to [9.79]

[9.88]

The frequency β is derived from the equality of the means of magnetic and electrostatic energies

[9.89]

therefore

[9.90]

and

[9.91]

[9.92]

[9.93]

Equation [9.90] corresponds to

[9.94]

Let

[9.95]

According to [9.94], the frequency β is the solution of the equation

[9.96]

In the particular case n = 0.5 , the frequency β is the solution of

[9.97]

Let β^0.5 = X, then X is the solution of

[9.98]

9.5.2.2. Stability condition

The stability condition is derived from

[9.99]

As demonstrated previously

[9.100]

[9.101]

and

[9.102]

Then, R is derived from the inequality

[9.103]

or equivalently

[9.104]

9.6.1. Introduction

The series RL*C* fractional circuit is a typical non-commensurate order system, with two orders n₁ for L* and n₂ for C*. The particular case n₁ = n₂ = n will be used to formulate a new interpretation to the Lyapunov stability of commensurate order FDEs.

9.6.2. Analysis of the series RL*C* circuit

The series RL*C* circuit corresponds to Figure 9.12.

The components L* and C* are characterized by the equations

[9.105]

Moreover

[9.106]

The state variables are z_L(ω, t) for the inductor L* and z_c(ω,t) for the capacitor C*. Therefore, the state space model of this circuit is

[9.107]

At t = 0, the initial conditions are z_L(ω, 0) and z_C(ω, 0). The energies stored in L* and C* are expressed as

[9.108]

[9.109]

Thus

[9.110]

and its derivative is

[9.111]

The last two terms represent internal losses in L* and C*, whereas the first one corresponds to external losses in R.

Therefore, as the system is naturally damped by all these losses, instability can occur only with an artificial negative resistor used to regenerate energy.

9.6.3. Theoretical stability analysis

The components L* and C* exchange their complementary stored energies: the system remains stable if the initial energy is globally decreasing, i.e. if .

Stability conditions are derived from the energy balance principle.

9.6.3.1. Derivation of oscillation frequency

At the limit of stability, when a constant oscillation occurs, we obtain

[9.112]

Then, let us define v_C(t) = V e^{j β t}, where β is the oscillation frequency.

As i(t) = C* D^n₂ (v_c(t)), we can write

[9.113]

and as v_L(t) = L* D^n₁ (i(t))

[9.114]

According to [9.107]

[9.115]

[9.116]

with

[9.117]

We have to calculate and .

Using the same technique as mentioned previously, we obtain

[9.118]

and

[9.119]

Consequently, the equality provides the value of β

[9.120]

9.6.3.2. Stability condition

The circuit remains stable if . According to [9.111], this condition is equivalent to

[9.121]

As at the limit of stability, we obtain

[9.122]

and stability is ensured if

[9.123]

9.6.3.3. Commensurate order case

If n₁ = n₂ = n

[9.124]

Since

[9.125]

we obtain

[9.126]

9.7.1. Introduction

Previously, it has been demonstrated that the stability analysis of the commensurate order RL*C* circuit can be derived from the energy balance approach. Equivalently, the stability analysis of any two-derivative commensurate order FDE can be derived from the analysis of an arbitrary fictitious RL*C* circuit.

9.7.2. Analysis of the commensurate order FDE

The series RL*C* commensurate order circuit corresponds to the simplified equations

[9.127]

The equation R i(t) + v_L(t) + v_C(t) = 0 is equivalent to the FDE

[9.128]

i.e.

[9.129]

with and .

This commensurate order FDE is simulated with two fractional integrators according to Figure 9.13.

The two infinite state variables are z₁(ω, t) and z₂(ω, t) such as

[9.130]

Of course

[9.131]

Therefore, z₁(ω, t) = z_C(ω, t) and , where z_c(ω, t) and z_L(ω, t) have been defined previously.

Let us define

[9.132]

[9.133]

Then

[9.134]

[9.135]

Thus

[9.136]

Knowledge of R, L* and C* allows the definition of and . However, reciprocally, knowledge of a₀ and a₁ does not allow the complete definition of R, L* and C* since and R = a₁ L*.

Therefore, it is necessary to make an arbitrary choice for one of the components, for example C*.

Then, we obtain

[9.137]

Consequently, the Lyapunov function is defined arbitrarily, depending on the choice of C*.

9.7.3. Application to stability

Let C* = 1, then

[9.138]

Thus

[9.139]

At the limit of stability

[9.140]

Let us define

[9.141]

Thus

[9.142]

Therefore, a_{1 lim} corresponds to the FDE stability condition derived from Matignon’s criterion (Appendix A.8.2). However, it is important to note that the previous arbitrary choice for C* has no consequence on the stability condition of [9.129].

This result means that the stability condition of any FDE as [9.129] can be derived from the energy balance principle, with arbitrary values of the fictitious series RL*C* circuit. Oscillations can be interpreted as transfers between electrostatic energy stored in the arbitrary capacitor C* and magnetic energy stored in L*. Moreover, damping is caused by internal losses in C* and L* and external losses in R.

9.8.1. Introduction

The usual approach to stability analysis of a commensurate order system (either integer or fractional order) is to use a positive P matrix to weight the Lyapunov function (see Chapter 8). Fundamentally, it will be demonstrated that this weighting matrix technique is equivalent to the previous approach based on a fictitious arbitrary RL*C* circuit.

9.8.2. Commensurate order system

Consider the commensurate fractional order system

[9.143]

This system corresponds to the commensurate order FDE

[9.144]

which is simulated according to Figure 9.13. Its equivalence with the series RL*C* circuit has been demonstrated previously.

The eigenvalues λ₁ and λ₂ of A are the roots of the equation

[9.145]

Let a₀ > 0

• – Δ > 0 if a₁ < –2 a₀ or a₁ > 2 a₀, then λ₁ and λ₂ are real eigenvalues.
• – Δ < 0 if –2 a₀ < a₁ < 2a0, then λ₁ and λ₂ are complex eigenvalues λ_i = a ± j b with:

[9.146]

or λ_i = ρ e^{± j θ} with

[9.147]

9.8.3. Lyapunov function of a fractional differential system

As system [9.143] is a commensurate order system, a modal representation can be associated with it. Let λ_i be the N eigenvalues of A in the general case. Let M be the matrix composed of the corresponding N eigenvectors (remember that M is not unique). Recall that

[9.148]

where A_d is the modal diagonal matrix.

Consider the matrix transformation

[9.149]

N fractional integrators are associated with the modal pseudo-state variables y_i(t), where w_i(ω, t) are the internal variables (see Chapter 8):

[9.150]

The energy of each modal integrator is

[9.151]

As the modal system [9.150] is composed of N independent modes, its total energy V_d(t) is the sum of the N independent energies V_i(t).

Therefore

[9.152]

Let us define

[9.153]

therefore

[9.154]

Then, consider the inverse transformation

[9.155]

As is also defined by [9.150], we obtain

[9.156]

Therefore, we can write [9.154] as

[9.157]

Let us define (Chapter 8)

[9.158]

P is a symmetric definite positive matrix: there is an infinity of possibilities to define P depending on the choice of the eigenvectors of M.

As the matrix transformation [9.149] does not modify energy, the Lyapunov function V(t) of the N -derivative fractional system is defined as

[9.159]

9.8.4. Stability analysis

Consider the modal representation [9.150] with

[9.160]

9.8.4.1. Oscillation frequency

At the limit of stability, y₁(t) and y₂(t) are constant amplitude sine waveforms.

Let us define y₁(t) = Y e^{j β t}.

Then, y₂(t) = Y e^{–j β t}

and

[9.161]

therefore

[9.162]

and

[9.163]

Obviously, this result is the same as [9.140] derived from the energy balance approach.

9.8.4.2. Stability condition

[9.164]

with

[9.165]

At the limit of stability, .

Therefore, we have to calculate two means

[9.166]

[9.167]

Then

[9.168]

As

[9.169]

[9.170]

with

As

[9.171]

we obtain

[9.172]

Consequently

[9.173]

Thus, at the limit of stability, a = a_lim and

[9.174]

As ([9.140]), we obtain

[9.175]

Recall that .

thus

[9.176]

The system remains stable if R > R_lim or a < a_lim, i.e. if

[9.177]

This last condition is equivalent to the LMI

[9.178]

where (see Chapter 8).

9.8.5. Conclusion

The previous analysis demonstrated that the usual approach to stability based on a weighted Lyapunov function is equivalent to the energy balance principle. Note that an LMI stability condition is derived from the usual approach, whereas a classical Routh stability condition is derived from the energy balance technique [ROU 77].

The standard FDE can be interpreted as a fictitious RL*C* circuit where the Lyapunov function is weighted by an arbitrary value of C*. Equivalently, the stability of the standard FDE can be analyzed with a Lyapunov function weighted by an arbitrary symmetric positive matrix P. This P matrix is related to the A matrix eigenvalues, i.e. to the arbitrary values of the fictitious RL*C* circuit. Consequently, the weighting matrix P can be interpreted in terms of a fictitious series RL*C* circuit.

According to [9.164], the term is equivalent to Joule losses in an external resistor R, whereas the term represents internal Joule losses in the fractional integrators .

Finally, consider the general commensurate order fractional system

[9.179]

where

[9.180]

and

[9.181]

According to the previous analysis in the modal base and using the inverse transformation [9.155], the term can be interpreted as internal Joule losses in the N fractional integrators , whereas the other term can be interpreted as external Joule losses in fictitious resistors.

A.9.1. The infinite length LG line

Consider the infinite length LG line (or LR line with ) (see Figure 9.4) and its elementary cell (Figure 9.14).

L and G are inductance and conductance densities.

Then

[9.182]

If dx → 0, we obtain

[9.183]

Similarly

[9.184]

Therefore, if dx → 0

[9.185]

Differentiation of [9.183] with respect to x leads to

[9.186]

This means that the equations of the LG line

[9.187]

[9.188]

are in fact the classical diffusive equations.

Using the Laplace transform and its differentiation properties, we obtain

[9.189]

With the hypothesis i(x, 0) = 0, equation [9.187] becomes

[9.190]

Let us define

[9.191]

Then, the general solution of [9.190] is

[9.192]

For an infinite length line, we have

[9.193]

and finally

[9.194]

Then

[9.195]

Let us define the complex impedance of the LG line

[9.196]

thus

[9.197]

The complex impedance is equivalent to a fractional differentiator with n = 0.5, regardless of x.

Thus

[9.198]

and

[9.199]

or equivalently

[9.200]

This equation defines the fractional inductor with

[9.201]

A.9.2. Energy storage and dissipation in the fractional capacitor

The fractional capacitor C* is characterized by the equation

[9.202]

A distributed model is associated with this equation

[9.203]

Our objective is to express the stored energy E(t) and its dissipation P(t) with the analog distributed network {R(ω), C(ω)} (see Chapter 7).

The input of this network is the current i(t), and its output is the voltage v(t).

Equation [9.203] gives

[9.204]

Moreover, for the elementary cell {R(ω), C(ω)}, we can write

[9.205]

Finally, the total voltage v(t) is expressed as

[9.206]

therefore we obtain

[9.207]

This equality is verified with the constraints

[9.208]

Thus, we can express the stored energy

[9.209]

i.e.

[9.210]

For the dissipation of energy, we obtain

[9.211]

thus

[9.212]

i.e.

[9.213]

A.9.3. Some integrals

Consider the distributed model of the fractional integrator. If the input is a Dirac impulse v(t) = δ(t), then z(ω, t) = e^–ωt and

[9.214]

With

The Laplace transform of [9.214] provides

[9.215]

if s is real, such as s = β, then

[9.216]

and

[9.217]

Let us define

[9.218]

Consider the new variables ω² = u β² = α, so and .

Then

[9.219]

Define .

Therefore

[9.220]

and

[9.221]

Thus

[9.222]

Define

[9.223]

Using the previous variables u and α, we obtain

[9.224]

Consequently

[9.225]