6.2.1. Simulation of an FDE

Consider the general linear FDE whose transfer function is

[6.1]

The fractional differentiation orders

[6.2]

are real positive numbers; they are called external or explicit orders. It is necessary to define internal or implicit derivation orders such as (see Chapter 1 of Volume 1):

[6.3]

The classical controller canonical state space form [KAI 80] is presented in Figure 6.1 with

[6.4]

and

[6.5]

6.2.2. Stability of the simulation scheme

The simulation diagram in Figure 6.1, which exhibits a direct transfer composed of integration operators Iⁿⁱ (s) and a feed-back signal r(t) composed of the sum of the weighted states a_{i – 1}x_i(t), with the error signal ε(t) = u(t) – r(t), explicitly refers to a closed-loop system, which is necessarily subject to a stability condition.

Of course, the stability condition of this feedback system corresponds to the stability of the differential system characterized by the transfer function i.e. to the stability condition of the considered FDE.

These considerations are not new: analog computer users were perfectly aware of this stability problem (see Chapter 1 of Volume 1) which became implicit with digital computers.

Consequently, the explicit formulation of this simulation stability problem, with an open-loop transfer function H_OL(s), provides a method for the stability analysis of H(s) using the Nyquist criterion [NYQ 32, GIL 90].

Let us define H_OL(s) as

[6.6]

where

[6.7]

Some elementary calculations provide

[6.8]

It is easy to verify that

[6.9]

and thus

[6.10]

The well-known term 1+ H_OL (s) explicitly refers to the stability of a closed-loop system, although H(s) corresponds to an open-loop FDE (or ODE).

6.2.3. Stability analysis of FDEs using the Nyquist criterion

As the previous simulation scheme refers to a closed-loop system with an open-loop transfer function H_OL (s), it is possible to analyze the stability of the considered FDE (or ODE) using the Nyquist criterion [NYQ 32, GIL 90]. Strict application of the Nyquist criterion is based on the definition of a D contour in the RHP. First of all, it is important to note that H_OL(s) has only poles in s = 0; the particular form of H_OL (s) excludes unstable poles. Thus, the contour has to be modified to exclude the poles in s=0.

Then, with this modified D contour (see Figure 6.2), the net number of rotations of H_OL(s) around the (–1 + j 0) point (critical point) is strictly equal to the number of unstable poles of H(s) in the RHP.

If the FDE or ODE is stable (see Figure 6.3), this net number of rotations is equal to zero, i.e. if H_OL(s) does not encircle the critical point (see [BEN 08a] for more details with FDEs).

The application of the generalized Nyquist criterion provides the number of unstable poles (see Chapter 7 of Volume 1).

Nevertheless, if we are only interested in the stability of the FDE, a simplified version of the Nyquist criterion can be used. It can be formulated as: if the Nyquist diagram H_OL (jω), plotted from ω = 0 to ω →+∞, is situated on the “right side” of the critical point (–1 + j 0), then the system H(s) (or the FDE) is stable; else it is unstable.

Moreover, the distance between the Nyquist diagram and the critical point provides information on the stability degree (phase and gain margins) of the FDE (ODE).

6.3.1. Introduction

The principle of the stability analysis based on the Nyquist criterion is exposed thanks to a third-order ODE example. First, it is necessary to define the general open-loop transfer function and to present the technique used to draw its approximate Nyquist graph

6.3.2. Open-loop transfer function

Let us define

[6.11]

which is the transfer function of a N^th -order linear ODE.

It is interesting to note that we obtain this transfer function from the model of the general FDE with N derivatives [6.1, 6.3] simply by imposing

[6.12]

According to Lord Kelvin’s principle [THO 76], the simulation of this ODE requires N integrators .

As with an FDE, we obtain a closed-loop system where the active elements are the N integrators and the a_n parameters. The b_m parameters have no effect in this closed-loop: as it is well known, this means that the stability of H(s) depends only on the transfer .

Thanks to the principle previously presented, H_OL(s) transfer function can be expressed as

[6.13]

It is straightforward to verify that

[6.14]

and

[6.15]

6.3.3. Drawing of H_OL(jω) graph in the complex plane

The objective is to present a technique to draw an approximate graph of H_OL(jω).

H_OL(s) is composed of N terms such as , i.e. of N directions in the complex plane.

Let us define

[6.16]

then

[6.17]

and

[6.18]

where

[6.19]

The corresponding graph is a ray centered at the origin and characterized by the polar angle θ_i. Since θ_i is an integer multiple of , these rays coincide with the real or imaginary coordinate axes.

The graph of H_OL(jω) is the combination of these N elementary graphs.

It is important to note that two of these directions are asymptotes:

• – if ω → 0, then H_OL (jω) behaves like the higher power element in ω, i.e.

[6.20]

which is the low frequency asymptote.

• – if ω → ∞, then H_OL (jω) behaves like the lower power element in ω, i.e.

[6.21]

which is the high frequency asymptote.

Thus, the graph of H_OL(jω) is constrained by these two asymptotes: this is a simple way to draw an approximate graph of H_OL (jω).

6.3.4. Stability of the third-order ODE

6.3.4.1. Definition

Let

[6.22]

Then

[6.23]

The two asymptotes are

[6.24]

and

[6.25]

If a₀, a₁ and a₂ are positive coefficients, the corresponding approximate graph of H_OL(jω) is presented in Figure 6.4.

6.3.4.2. Stability condition

The stability condition can be deduced from the simplified Nyquist criterion, based on the position of H_OL(jω) graph relatively to the critical point (–1 + 0j).

Let us define

[6.26]

with

[6.27]

The ODE is at the limit of stability if the H_OL(jω) graph is on the critical point.

The condition X = –1 provides

[6.28]

(oscillation frequency) whereas Y = 0 provides the parameter condition for oscillation

[6.29]

According to Figure 6.4, the system is stable if

[6.30]

This leads to

[6.31]

which is of course the stability condition provided by the Routh-Hurwitz condition [ROU 77].

Moreover, it is interesting to note that the graph of H_OL(s) in the Nichols chart provides the stability margins, i.e. information on the dominant poles of H(s) (see Figure 6.5).

6.3.4.3. Complete stability criterion

The Routh-Hurwitz criterion indicates that H(s) is stable if all the a_n parameters are strictly positive with a₀ < a₁a₂.

Is it possible to obtain all these conditions with the proposed approach?

Of course, the Nyquist criterion is able to give a satisfactory answer to this question.

For example, consider the a₁ parameter and the graphs in Figure 6.4. If a₁ = 0, the graph of H_OL (jω) is situated on the imaginary axis, and H(s) is unstable.

If and H_OL(jω) graph are in the RHP; thus, H(s) is unstable.

The case of the a₀ parameter is interesting.

Consider a₀ < 0 : it is necessary to apply the complete Nyquist procedure in this case to obtain all the pertinent information.

The complete graph of H_OL(s) in the complex plane is obtained using the excluding contour in the RHP (see Figure 6.2). With this contour, we obtain the graph presented in Figure 6.6. The closed-loop curve representing H_OL(s) encircles the critical point only once: this means that H(s) has only one unstable pole. This pole is necessary real; thus, the corresponding impulse response is divergent.

6.3.5. Conclusion

The application of the proposed approach to ODEs provides the same results as the Routh–Hurwitz criterion. A particular feature of this methodology is that it is mainly graphical: we exhibited, with the third-order ODE example, the importance of the H_OL(jω) approximate graph constructed on the basis of directions in the complex plane. Thus, the proofs of stability are obviously more simple to demonstrate than with the Routh-Hurwitz technique [HAN 96].

However this simplicity does not exclude rigor if necessary: the application of the complete Nyquist procedure provides all necessary information, for instance the number of unstable poles.

Moreover, the graph of H_OL(jω) in the Nichols chart makes it possible to quantify the stability degree of the ODE, which is a supplementary information compared to the simple Routh–Hurwitz approach.

6.4.1. Introduction

The stability of FDEs with one and two derivatives is investigated. Some analytic results for systems with N fractional derivatives are given, and a general numerical approach is proposed.

6.4.2. Drawing of H_OL(jω) graph in the complex plane

H_OL(jω) given in [6.8] is the sum of N elementary terms such as

[6.32]

Let us define

[6.33]

Then

[6.34]

and

[6.35]

As described previously, the corresponding graph is a ray centered at the origin, characterized by the polar angle θ_i. The difference with the ODE case is that the values of θ_i are fractional multiples of .

The graph of H_OL(jω) is the combination of these N elementary graphs.

Two directions are also asymptotes

• – If ω → 0, then H_OL(jω) behaves like

[6.36]

which is the low frequency asymptote.

• – If ω → ∞, then H_OL(jω) behaves like

[6.37]

which is the high frequency asymptote

As in the ODE case, H_OL(jω) is constrained by these two asymptotes: this is a simple way to draw an approximate graph.

Consider for example the graph of Figure 6.7 where all the a_i parameters are positive.

6.4.3. Stability of the one-derivative FDE

Let us consider the system

[6.38]

with one fractional derivative.

The stability results for this simple system are well known [OUS 95a]; therefore, it is interesting to compare them with our approach. In this case

[6.39]

and

[6.40]

with

[6.41]

As N = 1, H_OL(jω) is composed of only one ray characterized by the polar angle

[6.42]

The graph of H_OL(jω) is presented in Figure 6.8 for different values of n with a₀ > 0.

The simplified Nyquist criterion indicates that the system is unstable for n ≥ 2. Obviously, this result can be established analytically. The stability limit corresponds to

[6.43]

The condition Y = 0 implies

[6.44]

which is obtained with n = 2.

The condition X = –1 provides the oscillation frequency

[6.45]

The system remains stable if

[6.46]

Thus,

[6.47]

which leads to n < 2.

When a₀ is negative, it is preferable to use the complete Nyquist criterion. Using the excluding contour of Figure 6.2, the corresponding graph of H_OL(s) is presented in Figure 6.9 for 0 < n < 1.

As H_OL(s) encircles the critical point only once, H(s) has a real pole in the RHP, and the system is unstable by divergence (see Chapter 7 of Volume 1).

Thus, we can conclude as follows.

If a₀ > 0:

• – for 0 < n < 1, the system is stable and aperiodic;
• – for 1 < n < 2 , the system is stable and underdamped oscillatory;
• – for n = 2, the system is unstable and has two poles on the imaginary axis;
• – for 2 < n < 3, the system is unstable and has two poles in the RHP.

If a₀ < 0, the system is unstable and has one real pole in the RHP.

6.4.4. Stability of the two-derivative FDE

6.4.4.1. Definitions

Let us consider

[6.48]

with

[6.49]

In this case

[6.50]

The graph of H_OL(jω) is the combination of two directions:

• – a low frequency asymptote

[6.51]

• – a high frequency asymptote

[6.52]

6.4.4.2. Graphical analysis of stability

Consider a₀ > 0 and a₁ > 0 with 0 < n₁ < 1, 0 < n₂ < 1 and n₁+n₂ > 1, we obtain the graph presented in Figure 6.10.

Of course, we can conclude that H(s) is stable.

With 0 < n₂ < 1 and 2 < n₁ + n₂ < 3, we obtain the different graphs presented in Figure 6.11.

In this case, the system is conditionally stable.

REMARK 1.– It is important to note that these results have been obtained with an elementary graphical approach.

6.4.4.3. Stability condition

Let

[6.53]

with

[6.54]

The stability limit corresponds to X = –1 and Y = 0. The condition Y = 0 provides the oscillation frequency

[6.55]

Since is necessarily positive, it implies that

[6.56]

Finally, putting ω_osc in X = –1, we obtain the parametric condition between, a₀, a₁, n₁ and n₂, i.e.

[6.57]

The system (with a₀ >0, a₁ >0) remains stable if Y<0 when X = –1 or equivalently

[6.58]

Using one of these two conditions, we can analytically calculate the inequality that a₀, a₁, n₁, n₂ have to verify.

REMARK 2.– Note that when a₀ > 0 and a₁ < 0, the condition is fulfilled with 0 < n₁ + n₂ < 2.

REMARK 3.– With ODEs, only the a_i parameters have an influence on stability. On the contrary, with FDEs, the fractional orders n_i also have an influence on stability, which can be discussed graphically.

As an example, consider the case

[6.59]

When n₁ =2, the low frequency asymptote is opposite to the high frequency one, as presented in Figure 6.12. The graph of H_OL(jω) is situated on this common line: the system is unstable.

When n₁ > 2, the graph of H_OL(jω) is situated in the upper sector corresponding to the two asymptotes (see Figure 6.13) and the system is unstable.

This analysis can be generalized to n₂, and thus we obtain a fundamental result stated as follows: the fractional system H(s) is unstable ∀n_i ≥ 2.

6.4.4.4. Example

Consider the case a₀ > 0 and a₁ < 0: with a second-order ODE, this value of a₁ leads to instability. However, with an FDE, the stability depends on the respective values of n₁ and n₂.

Consider 0 < n₁ < 1 and 0 < n₂ < 1: as a₁ < 0, the direction reverses and the corresponding graph is presented in Figure 6.14.

Obviously, the stability is conditional in this case.

Consider n₁ = n₂ = 0.5, the oscillation frequency is provided by:

[6.60]

We deduce that

[6.61]

when H _OL(jω) = –1.

Moreover, stability is ensured if

[6.62]

Consider a₁ = –1, the stability is ensured by

[6.63]

For a₀ = 0.7, we present the corresponding Nichols chart in Figure 6.15 and the step response in Figure 6.16.

We can note that it is possible to quantify the damping of the step response, thanks to the stability margins in the Nichols chart.

Finally, for a₀ = 0.4 , we verify that the system is unstable: see the step response in Figure 6.17.

6.4.5. Stability of the N-derivative FDE

Finally, we consider the general transfer function [6.1]. It is difficult to provide general results due to the difficulty of the analytical calculations. However, it is possible to give some general rules, as the following one obtained graphically. The graph of H _OL(jω) is constrained by the two asymptotes already presented in Figure 6.7. Consider the case

[6.64]

then the low frequency asymptote is situated under the real axis, and the system is unconditionally stable.

Therefore, we can formulate some simple rules:

• – if a₀ < 0, the system is unstable;
• – if a_i > 0 ∀ i and , the system is unconditionally stable;
• – if a_i > 0 ∀ i and n_i > 2 ∀ i, the system is unstable.

However, despite the lack of analytical results, it is always possible to numerically analyze the stability with the proposed frequency approach. The drawing of H_OL(jω) in the Nichols chart gives essential information on system stability and on its stability degree thanks to the stability margins.

6.4.6. Conclusion

We demonstrated that the proposed approach to system stability applies to non-commensurate order FDEs, generalizing the Routh-Hurwitz criterion. Based on the Nyquist criterion approach, it provides information concerning stability, number of unstable poles and degree of system stability.

Another important feature of this method is its graphical interpretations, which provide very simple demonstrations.

Despite the complexity of analytical formulations in the general case, this method can be numerically implemented with any linear fractional system, providing information on its stability and its degree of stability.

6.5.1. Introduction

The previous technique is applied to time-delayed systems. Therefore, it is necessary to test its ability to analyze the stability of time-delayed ODEs before considering the fractional order case.

The general problem is formulated, and an example is investigated for comparison with well-known results.

6.5.2. Definitions

Many definitions are related to the different systems with time delays [DUG 97, NIC 97, RIC 03]. In this chapter, we are only interested in systems described by the following transfer function:

[6.65]

with

[6.66]

This system is simulated using a controller canonical form as shown in Figure 6.1, where and where each weighted state a_i–1x_i(t) is replaced by a_i–1x_i (t) + ad_i–1 x_i(t – τ_{i – 1}).

Thus, we obtain

[6.67]

6.5.3. Stability analysis

As described previously, H_OL(s) has only poles in s = 0. Thus, because in a first step we are only interested in stability, we will apply the simplified Nyquist criterion. H_OL(jω) is composed of N terms such as

[6.68]

H_i(jω) is the sum of two basic elements

1. 1) with modulus and polar angle . Its graph is a ray centered at the origin and characterized by its polar angle.
2. 2) with modulus and polar angle .

For ω = 0, its graph belongs to a ray similar to the previous one; on the contrary, for ω → +∞, its graph is a spiral centered at the origin caused by the rotation of the polar angle due to the time delay.

The graph of H_i(jω) is the composition of these two basic elements. It depends on:

• – the respective signs of a_i and ad_i;
• – the absolute values of a_i and ad_i.

Specifically, if |a_i| ≫ |ad_i|, then the graph is close to that of the first term, and if |a_i| ≪ |ad_i|, then the graph is close to that of the second term.

Finally, the graph of H_OL(jω) results from the composition of the N elementary graphs.

REMARK 4.– The interest of this analysis of elementary graphs is to investigate the influence of each component H_i(jω) in order to formulate analytical conclusions.

If the only objective is to test the stability of a time-delayed ODE, then it is only necessary to compute numerically H_OL(jω) for an appropriate choice of frequency values ranging from ω = 0 to ω → +∞ and to plot the corresponding curve: thus, this method can be applied easily to a large variety of systems.

6.5.4. Application to an example

Our objective is to demonstrate that this method gives the same analytical results as other approaches with a reference example.

Consider the well-known system [NIC 97, RIC 03]:

[6.69]

where a₀ = a and ad₀ = b.

Stability conditions are presented using a {a, b} plane representation. In this example

[6.70]

with

[6.71]

At the limit of stability, the H _OL (jω) Nyquist diagram goes through the critical point and X(ω) = –1; Y(ω) = 0.

The condition Y(ω) = 0 imposes which has a solution only if .

This condition defines two straight lines in the {a,b} plane: a = b and a = –b which define eight sectors numbered 1–8 (see Figure 6.18).

It is necessary to determine H _OL(jω) in each sector in order to analyze stability. We will investigate only the stability in sectors 1 and 2 in order to demonstrate the application the method, and present the global results for the {a,b} plane.

Sector 1 is characterized by b > a. Thus, the graph of H _OL(jω) is close to that of .

Equation [6.71] has a solution:

[6.72]

For this value of ω, the system is stable if: X(ω) > –1 when Y(ω) = 0.

Simple calculations give

[6.73]

Thus, the system is conditionally stable.

Sector 2 is characterized by b < a. Thus, the graph of H _OL(jω) is close to that of .

The condition Y(ω) = 0 can be fulfilled only when X(ω) → 0 (with ω → +∞).

Thus, it means that the Nyquist diagram of H _OL(jω)is always situated on the “right side” of the critical point: the system is always stable, independently of the value of the time delay τ.

Applying this method to each sector, we finally obtain the results expressed in Figure 6.19.

Of course, this stability result is identical to the results presented in [NIC 97, RIC 03].

6.6.1. Definitions

We are interested in linear FDEs with time delays whose transfer function is a generalization of the integer order case:

[6.74]

with

[6.75]

and where the internal derivative orders n_i correspond to the definition [6.3]. Then, the open-loop transfer function H _OL(s) is expressed by

[6.76]

6.6.2. Stability

As in the integer order case, we will consider the position of the Nyquist diagram H _OL(jω) with respect to the critical point.

H _OL (jω) is composed of N terms such as

[6.77]

H_i(jω) is the sum of two basic elements:

1. 1) with modulus and polar angle . Its graph is a ray centered at the origin and characterized by its polar angle;
2. 2) with modulus and polar angle .

For ω = 0, its graph belongs to a ray similar to the previous one; on the contrary, for ω → +∞, its graph is a spiral centered at the origin.

The graph of H(jω) is the composition of these elementary graphs, as in the integer order case.

Finally, the graph of H _OL(jω) results from the composition of the N elementary graphs: the system is stable if the Nyquist diagram lays on the “right side” of the critical point.

6.6.3. Application to an example

In order to test this method, we will apply it to a system where the stability condition is already known, thanks to another approach. This example is the following, given by [HWA 06]. Its characteristic polynomial A(s) is

[6.78]

This example deals with the commensurate order hypothesis: then, n₁ = n₂ = 0,5. Moreover, the time delay is also fractional of the form . However, this non-usual form is not an obstacle to the use of the method (with the values a₀ = a₁ = 1).

Then

[6.79]

which can be expressed as

[6.80]

with

[6.81]

and

[6.82]

The graph of H₂ (jω) lies between two asymptotes:

• – a ray centered at the origin, with polar angle equal to , corresponding to ω =0;
• – a ray centered at the origin, with polar angle equal to , corresponding to ω → +∞.

The transfer function is characterized by: H₁(jω) = with and .

⍴₁ decreases from K to 0, whereas θ₁ varies from 0 to –∞, when ω varies from 0 to +∞.

The graph of H₁(jω) is a spiral centered at the origin, and the Nyquist diagram of H_OL(jω) is the composition of the two previous graphs.

Stability is conditional, according to the value of K. Let us determine the value of K (K_lim) corresponding to the situation where the Nyquist diagram goes through the critical point.

Therefore, H_OL (jω) = ⍴e^jθ with ⍴ = 1 and θ = – π.

is a solution of the equation

[6.83]

whereas K is obtained from the condition ⍴ = 1 (with the previous value of ).

For τ = 1s, we obtain and K_lim = 21,51 (of course, the same value as provided by [HWA 06]).

The diagrams corresponding to K = {5;10;15;K_lim} are plotted in the Nyquist plane (see Figure 6.20).

We can verify that the stability is ensured for K < 21,51 . Moreover, the distance between H_OL (jω) and the critical point makes it possible to quantify the stability degree of this time-delayed FDE.