10.2.1. Introduction

The indirect Lyapunov method is based on a local approximation of the nonlinear system around its equilibrium point [KHA 96]. It makes it possible to conclude on the local stability of this equilibrium point. We propose an adaptation of this methodology to nonlinear commensurate order fractional systems.

10.2.2. Linearization

Consider the nonlinear commensurate order fractional system

[10.2]

Let be the equilibrium point.

Therefore, the nonlinear system is linearized around using the development of

[10.3]

Since the terms linearly depending on are in , decays more than (see Figure 10.1), i.e.

[10.4]

This means that for any , we have

[10.5]

10.2.3. Nonlinear system analysis

System [10.2] is equivalent to

[10.6]

corresponding to the linearized model

[10.7]

In the vicinity of , system [10.6] behaves as [10.7].

System [10.6] can be expressed using a distributed model with internal variables Ẕ(ω, t):

[10.8]

Let V(t) be the Lyapunov function of a commensurate order fractional system (see Chapter 8):

[10.9]

Then

[10.10]

Therefore

[10.11]

Note that is a scalar.

Therefore

[10.12]

Then

[10.13]

Note that

[10.14]

Therefore, using the Schwarz inequality [KHA 96, SCH 98]

[10.15]

Thus, we can write

[10.16]

Since (see Figure 10.1)

[10.17]

we obtain

[10.18]

Therefore, we can write

[10.19]

If the eigenvalues of A are negative or with a negative real part, then (A^TP + PA) < 0 with P = P^T > 0, which means that the linearized system [10.7] is stable ∀n 0 < n < 1.

The term (or ) limits the negative value of in [10.13] (or in [10.19]). Thus, there is a limit value of γ > 0 such that is no longer negative and the nonlinear system [10.2] is unstable.

Let us recall that for N = 2 (see Chapter 8), we have demonstrated that can be positive (when the real parts of the eigenvalues of A are positive) but remains negative thanks to the term (caused by dissipation in the fractional integrators – see Chapter 9). This means that there is a limit value of γ such that the nonlinear system remains stable, with a damped oscillating behavior, depending on the value of the fractional order n (0 < n < 1).

10.2.4. Local stability of a one-derivative nonlinear fractional system

Consider the nonlinear system [TRI 11b]:

[10.20]

with

[10.21]

Therefore

[10.22]

and

[10.23]

i.e.

[10.24]

which is presented in the graphs of Figure 10.2.

Note that , where x = 0 is the equilibrium point.

Therefore, we can write

[10.25]

The linearized system corresponds to

[10.26]

The nonlinear system [10.20] corresponds to the distributed model

[10.27]

Consider the Lyapunov function

[10.28]

Thus

[10.29]

and the linearized system corresponds to

[10.30]

This linearized system is stable since b < 0.

The nonlinear system remains stable as long as

[10.31]

or

[10.32]

ie. if

[10.33]

which corresponds to

[10.34]

REMARK 1.– Using the notations of section 10.2.3, we can compare and , i.e. bx(t) and ax³ (t) (see Figure 10.1).

Thus, we can write

[10.35]

with

[10.36]

Then

[10.37]

Thus

[10.38]

which defines the stability domain of [10.20] (see Figure 10.3).

10.3.1. Introduction

The Lyapunov direct method relies on the appropriate choice of a Lyapunov function V(t), allowing the analysis of global stability: obviously, it is a very large domain of research [LAS 61, KRA 63]. The variable gradient method, proposed by Schultz and Gibson [GIB 63, NAS 68, SLO 91], provides a response to this problem. After a recall of this methodology in the integer order case, we propose to adapt it to the fractional order case, using two numerical examples.

10.3.2. The variable gradient method

A scalar function is connected to its gradient through the integral

[10.39]

where

[10.40]

The reconstruction of from its gradient requires that its components verify the curl condition [KOR 68]

[10.41]

The principle of the variable gradient method is to assume a specific form for the gradient instead of assuming a form for the Lyapunov function .

A simple solution is to assume that the gradient is of the form

[10.42]

where the coefficients a_i,j have to be determined.

This leads to the following procedure:

• – assume that is given by [10.42];
• – solve for the coefficients a_i,j, with constraint [10.41];
• – restrict the coefficients in [10.42] so that is negative definite;
• – compute V from by integration [10.39];
• – check whether V is positive definite.

10.3.3. Nonlinear system with one derivative

Consider again the example of section 10.2.4:

[10.43]

with

[10.44]

In the monovariable case, has only one component.

The distributed model is

[10.45]

We have to separate the monochromatic Lyapunov function v(ω, t) [TRI 11b] and the global one V(t) such that

[10.46]

Let be the gradient of v(ω, t) and assume that

[10.47]

Then

[10.48]

Finally, using [10.46], we obtain

[10.49]

Thus

[10.50]

The first term is always negative for α > 0. We have previously demonstrated that a linear system with one fractional derivative (Chapter 8) is stable if the second term is itself negative.

Thus, it is necessary to check whether

[10.51]

or equivalently

[10.52]

i.e.

[10.53]

In addition, we have to determine v(ω, t) and V(t).

Then

[10.54]

Therefore

[10.55]

The function V(t) is positive definite if α > 0 (with α arbitrary).

The nonlinear system [10.43] is globally stable if x(t) verifies the condition

[10.56]

This condition is the same as the condition in example 10.2.4.

REMARK 2.– It is important to note that the stability condition [10.56] is conservative because it does not take into account the influence of z(ω, t) in the negative term of equation [10.50].

In order to remove this conservatism, it would be necessary to analyze the influence of initial conditions on stability, i.e. z(ω, 0) with this example.

However, it is another fundamental issue, which is beyond the limited objectives of Chapter 10.

10.3.4. Nonlinear system with two fractional derivatives

Consider the commensurate order nonlinear system

[10.57]

This model is derived from the model proposed by Shultz and Gibson [GIB 63].

Its distributed model, with internal variables z₁(ω, t) and z₂(ω, t), is

[10.58]

Let v(ω, t) be the monochromatic quadratic function, depending on the frequency ω, and V(t) be the global Lyapunov function

[10.59]

Let us define

[10.60]

We know that

[10.61]

Let us define

[10.62]

Assume that is of the form

[10.63]

with

[10.64]

It is possible to impose a₂₂ = 2 (see [GIB 63, SLO 91]).

Then, [10.61] can be written as

[10.65]

Since

[10.66]

we obtain

[10.67]

Let us define

[10.68]

Then

[10.69]

Note that (it will verify a posteriori):

[10.70]

Therefore

[10.71]

Let us return to .

According to [10.63], we obtain

[10.72]

Let us define the Jacobian of

[10.73]

Ψ has to verify the curl condition [10.41], i.e. .

Since and , we have to impose

[10.74]

Let us define a priori a₁₂ = a₂₁ =1, then

[10.75]

We obtain v(ω, t) using equation [10.61]

[10.76]

REMARK 3.– x₁(t) is computed from z₁(ω, t) using the weighted integral of [10.58], idem for x₂(t).

Therefore, x₁(t) is a “constant” (with respect to ω) considering z₁(ω, t), idem for x₂(t) considering z₂(ω, t).

We calculate the integral [10.76] defining a path from {0,0} to {z₁(ω, t), z₂(ω, t)}.

Let

[10.77]

Therefore

[10.78]

Finally

[10.79]

Let us note that

[10.80]

Thus

[10.81]

It is easy to verify that for P > 0, V(t) is a positive definite quadratic form.

We can verify the previous calculation.

With and a₁₂ = a₂₁ = 1, we obtain

[10.82]

i.e.

[10.83]

This means that

[10.84]

Thus, the nonlinear system [10.57] is globally stable.

10.4.1. Electrical nonlinear system

The electrical nonlinear system is presented in Figure 10.4. It corresponds to a fractional differential system, where L_f is a fractional inductor

[10.85]

It is a negative resistance oscillator [KHA 96, AUV 80, CHA 82], which is characterized by the nonlinear equation i = f(v)

[10.86]

[10.87]

, where –R is a negative resistor.

10.4.2. Van der Pol oscillator

The nonlinear system [10.85] is modeled with two variables i_L(t) and v(t). The elimination of i_L(t) leads to the classical Van der Pol equation [KHA 96, PET 11] in connection with chaotic systems [HAR 95, STR 15].

[10.88]

since δi = G(v) δv, we obtain

[10.89]

Using

[10.90]

we obtain the standard fractional Van der Pol equation [BAR 07]

[10.91]

10.4.3. Simulation of the nonlinear system

System [10.85] is more general than equation [10.88], and better suited for its analysis and simulation by the energy balance approach (Chapter 9). A first objective is to simulate its dynamical behavior. This system can be written as

[10.92]

Then, using the distributed variable z(ω, t) associated with the fractional integrator , the first equation of [10.92] is equivalent to

[10.93]

This system is frequency discretized using the technique presented in (see Chapter 2 of Volume 1). Thus, we can simulate the nonlinear system.

10.4.4. Limit cycle

Simulation is performed with the following values [MAA 16]:

[10.94]

Thus, we obtain the different phase portraits of the Van Der Pol oscillator.

In Figure 10.5, the limit cycles are plotted with increasing values of order n, from 0.5 to 1, with α = 1 and the initial conditions v(0) = 0.5 z(ω, 0) =0 ∀ω. In Figure 10.6, for a fixed value of the order (n = 0.5), the same limit cycles are plotted with increasing values of α, from 0.9 to 5.

For n close to 1, and large values of α, the limit cycles correspond to the usual Van der Pol ones (see [BAR 07]). On the contrary, for lower values of α, the limit cycles are approximately ellipsoidal and require α > α_lim.

Therefore, the objective of the next sections is to analyze and predict these results.

10.5.1. Linearization

The nonlinear system [10.85] can be expressed as

[10.95]

The equilibrium point of the nonlinear system is .

Let us define

[10.96]

[10.97]

Thus, the linearized system corresponds to

[10.98]

10.5.2. Local stability

Let us define the Lyapunov function (see Chapter 9)

V(t)=E_L(t) + E_C(t) with

[10.99]

[10.100]

Therefore, the derivative of the Lyapunov function is

[10.101]

Let us recall (Chapter 9) that the first term represents the distributed internal Joule losses of the fractional inductor, whereas the second term corresponds to external Joule loss in the negative resistor G(0) = –α = –1/R.

This Joule loss compensates the internal losses in L_f when

[10.102]

Thus, a constant amplitude oscillation v(t) = Ve^jγt appears, where γ is the oscillation frequency.

10.5.2.1. Derivation of frequency oscillation

At the stability limit (Chapter 9)

[10.103]

and v(t) = Ve^jγt.

According to Chapter 9, we obtain

[10.104]

Therefore

[10.105]

thus

[10.106]

Since (Appendix A.9.3.)

[10.107]

and

we obtain the oscillation frequency

[10.108]

10.5.2.2. Stability condition

The system remains stable if . This condition is equivalent to

[10.109]

α_lim is obtained at the stability limit, using equation [10.105].

Since (Appendix A.9.3.):

[10.110]

we obtain

[10.111]

The system remains stable if α < α_lim.

REMARK 4.– If n = 1, then α_lim = 0. Thus, in the integer order case, the Van der Pol oscillator is unstable ∀ α ≥ 0. On the contrary, the instability of the fractional Van der Pol oscillator requires the condition α ≥ α_lim.

10.5.3. Validation of stability results

Consider the numerical values

[10.112]

Then, equations [10.108, 10.111] provide

[10.113]

In Figure 10.7, the limit cycles {i_L(t),v(t)} for α = { 0.4; 0.8;1 } are plotted with the initial conditions v(0) = 0.25; z(ω, 0) = 0 ∀ ω.

For α = 0.4 < α_lim, we observe that the nonlinear system is stable.

For α = 0.8 ≈ α_lim, the system is at the stability limit; therefore, we obtain a constant amplitude sinusoid v(t)=Ve^jγt characterized by an ellipsoidal phase portrait. Note that the amplitude of this ellipse depends on the initial conditions.

Then, for α = 1 > α_lim, the system is unstable, and the amplitude of the limit cycle is independent of the initial conditions.

10.6.1. Introduction

Small signal analysis has demonstrated that the Van der Pol oscillator provides a constant amplitude oscillation v(t) = Ve^jγt for α = α_lim. For α ≥ α_lim, this oscillation is nonlinear and characterized by a limit cycle. Consequently, our objective is to determine the amplitude V_lc of this oscillation when v(t) remains approximately sinusoidal, i.e. for the small values of α.

10.6.2. Approximation of the first harmonic [MUL 09]

When v(t) remains approximately sinusoidal, we can still use the expression v(t)=Ve^jγt. For α = α_lim, the amplitude V depends on the initial conditions v(0) and z(ω, 0). On the contrary, for α > α_lim, the amplitude depends on the characteristics of i = f(v), independently of the initial conditions.

Let

[10.114]

Since

[10.115]

we obtain

[10.116]

The nonlinear term V³ sin³(γt) can be expressed in terms of harmonics γ and 3γ:

[10.117]

The first harmonic hypothesis leads to

[10.118]

10.6.3. Lyapunov function and oscillation frequency

As described previously, V(t) = E_L(t) + E_C(t).

Since , we obtain

[10.119]

with

[10.120]

The oscillation frequency is derived from the equality . Since the amplitude of v(t) has no influence on the calculus of γ (in the first harmonic hypothesis), the oscillation frequency remains the same as in the linear case, thus

[10.121]

On the contrary, the amplitude V acts in the expression of Joule losses corresponding to .

10.6.4. Amplitude of the limit cycle

Equality of Joule losses corresponds to

[10.122]

Since [10.120]

[10.123]

In the hypothesis

[10.124]

we obtain

[10.125]

The first harmonic hypothesis leads to

[10.126]

Thus

[10.127]

Therefore, condition [10.122] leads to

[10.128]

Then, the amplitude V_lc of the limit cycle is given by

[10.129]

Finally, since v(t) = L_f Dⁿ(i_L(t)) and v(t) = Ve^jγt, we obtain

[10.130]

Consequently, the phase portrait {i_L(t),v(t)} is exactly an ellipse in the first harmonic hypothesis.

10.6.5. Prediction of the limit cycle

In Figure 10.8, the graphs of {i(t),v(t)} for n = 0.5 and v(0) = 0.5; z(ω, 0) = 0 ∀ ω are plotted with increasing values of α. Thus, we obtain the graphs of the function i = f(v) indexed by α. The first harmonic approach remains valid if v(t) does not exceed the limits of the negative resistor zone characterized by , i.e. by

[10.131]

Then, using equations [10.129, 10.130] with L_f= 1; C=1; n=0.5; β=1 and α=1, we obtain V_lc=0.525 < 0.577 and I_Llc=0.589.

Experimentally (see Figure 10.6), we obtain V_lc=0.527 and I_Llc =0.58. Note that the limit cycle is quasi-ellipsoidal; therefore, the first harmonic approach provides a good approximation of the dynamical behavior.

On the contrary, for α = 1.5, the theoretical values are V_lc = 0.792 > 0.577 and I_Llc=0.889, and the experimental values are V_lc=0.79 and I_Llc=0.85.

Since V_lc > 0.577, the limit cycle is no longer ellipsoidal, and the predicted values are only approximate, as demonstrated in Figure 10.6.

Note that better approximations of the oscillation frequency and the limit cycle characteristics can be obtained by association of the energy balance and harmonic balance approaches [KUN 86].