3.3.1. Fractional observer definition

There are many papers that deal with the Luenberger observer in the fractional order case [MAT 97, DZI 06, DAD 11, NDO 11]. In fact, these papers correspond to the estimation of the pseudo-state , not to the estimation of the internal state Ẕ(ω , t). Consequently, these fractional observers are only the generalization of integer order observers. Their requirements are the same as in the integer order case, except for stability.

Thus, it is essential to note that in this section, we really analyze the observation of the internal state Ẕ(ω, t).

The theoretical initialization of an FDS at t = t₀ concerns the estimation of its state z(ω, t₀) based on the measurements of {u(t), y(t)} on a history interval [t_p;t₀]. Therefore, a direct way is to use a fractional observer.

Consider the FDS [3.3]. The Luenberger fractional observer [LUE 66, KAI 80, ZAD 08] is defined by

[3.9]

where and Ḵ are respectively the observer pseudo-state and the vector gain.

As for the FDS, we define the distributed state vector of the observer

[3.10]

and its representation (similarly to [3.4])

[3.11]

3.3.2. Stability analysis

Let us define the pseudo-state error

[3.12]

Then

[3.13]

Thus, from [3.9] and [3.11], we obtain

[3.14]

The stability of the FDS observer [3.9] depends on and ṉ, i.e. the observer gain Ḵ has to respect the stability of the estimation error.

For a commensurate order system, the stability depends on the eigenvalues λ_i of the matrix and the value of the fractional order n (0 < n < 1). According to Matignon’s criterion [MAT 98], these eigenvalues have to verify the following condition (see also Appendix A.8.2):

[3.15]

For a non-commensurate order system, the stability condition is more complex to derive. For this purpose, we can use the technique proposed in Chapter 6, which is derived from the Nyquist criterion [NYQ 32, TRI 09c].

For example, consider the two-derivative system:

[3.16]

Let us define the transfer function

[3.17]

where

[3.18]

and

[3.19]

is the well-known characteristic polynomial [KAI 80, ZAD 08] corresponding to the system [3.16, 3.17] which is of the form

[3.20]

Thus, the estimation error will be stable if is stable, i.e. if α₀ > 0 and α₁ > 0 (according to the Nyquist stability criterion in Chapter 6).

3.3.3. Convergence analysis

Since is only the pseudo-state error, it is necessary to associate a distributed state variable error ξ_i(ω, t) with each component ε_{x, i}(t) of .

Thus, equivalent to [3.12] and [3.14], and taking into account [3.11], the distributed state vector error verifies the following equation (note that does not correspond to the open-loop distributed state in this section):

[3.21]

Using [3.14] and [3.21] leads to

[3.22]

or

[3.23]

where is the state vector error.

The observer starts at t = t_p, and it is supposed to be at rest at this instant because we have no information on Ẕ(ω, t_p).

Therefore

[3.24]

Thus, [3.23] leads to

[3.25]

In order to simplify the notations, let us consider the reduced time

[3.26]

Then, the Laplace transform of [3.21] leads to

[3.27]

with

[3.28]

Therefore

[3.29]

and

[3.30]

This result means that

[3.31]

Thus, we can conclude that all the components of converge to those of Ẕ(ω, t) as t → ∞.

Nevertheless, the dynamics of are imposed by 1/(s + ω) (see [3.30]). Thus, the lower frequency modes of Ẑ(ω, t) require an infinite time to converge. Consequently, the requirement for convergence ∀ ω is that t₀ – t_p → ∞.

3.3.4. Numerical example 1: one-derivative system

Consider the transfer function

[3.32]

Therefore, A = –a₀, Ḇ = 1 and .

At t = 0, the system is at rest, i.e. z(ω, 0) = 0∀ω, x(0) = 0 and y(0) = 0.

The input of the system is a unity step on [0, t₀]:

[3.33]

The past history interval corresponds to t_p = 0.5s and t₀ = 2s.

We simulate the system response y(t) with

The observer starts at t = t_p with K = 20 and ẑ(ω, t_p) = 0∀ω.

At t = t₀, the initial state is z(ω, t₀), whereas the initial state of the observer is ẑ(ω, t₀). Then, at t = t₀, we switch off the observer (K = 0). Consequently, its response represents the free response of the system initialized with the estimated initial condition ẑ(ω, t₀), which we call the initialized response y_init(t).

The input u(t), the theoretical system response y(t), the observer response ŷ(t) and the initialized response y_init(t) are plotted on [0, 3t₀] in Figure 3.1.

We can note that the observer response ŷ(t) is quickly close to y(t). However on [t₀, 3t₀], y_init(t) is progressively different from y(t). The first conclusion is that the equality of the pseudo-states (x(t) and at t = t₀ is not a guarantee for a good initialization.

The comparison between the components z_j(t₀) and ẑ_j(t₀) (for j = 0,…, 20) provides the explanation of this difference (see Figure 3.2): the high frequency modes are correctly estimated, whereas there is a poor estimation of the very low frequency modes. Consequently, the initialization of the fractional system is poor for long time behaviour, as highlighted on [t₀, 3 t₀] (Figure 3.1).

REMARK 1.– The increase in the observer gain K shows that there is some improvement of the convergence of low frequency modes. Therefore, very large values of K (because stability is not theoretically affected) would theoretically improve convergence. In fact, a practical value K_max is imposed by the numerical computation of the observer: note that the fractional integrator is approximated by a finite dimension model, and each mode is time discretized.

Thus, observer stability has to respect a more restrictive condition K < K_max. Consequently, there is no simple solution to the improvement of the convergence of low frequency modes.

3.3.5. Numerical example 2: non-commensurate order system

Consider the transfer function

[3.34]

corresponding to the observer canonical form [KAI 80]:

[3.35]

The values of the gain Ḵ have to be chosen to respect observer stability. Using the stability criterion of section 3.3.2, theoretical stability is ensured for K₁ ≥ 0 and K₂ ≥ 0.

The simulation protocol is the same as previously with

[3.36]

The graphs of u(t), y(t) and ŷ(t) for different values of Ḵ are presented in Figure 3.3.

Figures 3.4 and 3.5 present the graphs of the corresponding modes of and .

Obviously, the best estimation of Ẕ(t₀) is provided by . Nevertheless, the low frequency modes exhibit a poor convergence, as in the previous example.

3.4.1. Introduction

The previous convergence analysis demonstrated that the lower frequency modes of Ẕ(ω, t₀) are not correctly estimated because they require an infinite history interval [t_p, t₀] for complete convergence. Obviously, this requirement is not acceptable in practice. A straightforward approach to improve Ẕ(ω, t₀) estimation would be to artificially broaden the interval [t_p, t₀] by repeated forward/backward observations. This technique has been successfully used for the initialization of a PDE (see [RAM 10] and the references therein) where the direct model is used to perform forward observation, whereas the backward model is used to perform backward observation

Unfortunately, the application of this methodology to a fractional system is forbidden by numerical problems due to the fractional backward model [BOU 67]. This model is very sensitive to numerical errors that affect the high frequency modes. Apparently, this iterative procedure cannot be used with a fractional system.

However, this approach demonstrates that the improvement of Ẕ(ω, t₀) estimation depends on the quality of the initial value of the observer.

Thus, we propose a solution to estimate Ẕ(ω, t_p) using a fixed history interval [t_p, t₀] and the free response of the system starting at t = t_p. As demonstrated thereafter, this estimation makes it possible to initialize the observer and then to improve the estimation of Ẕ(ω, t₀).

The estimation of Ẕ(ω, t_p) is based on a gradient approach using the open-loop responses of the fractional integrators (closed-loop representation).

As demonstrated thereafter, is deduced from the free response of the system y_free(t). Since y(t) = y_free(t) + y_forced(t), the free response y_free(t) is calculated from the knowledge of y(t) on [t_p, t₀] and on the simulation of y_forced(t) based on the knowledge of u(t) on [t_p, t₀] and of the system parameters.

3.4.2. Non-commensurate order principle

Let us consider the Laplace transform of the system response [3.3, 3.4] (see Chapter 7 of Volume 1):

[3.37]

where the first term represents the free response of the system initialized by Ẕ(ω, 0), and the second term represents the forced response depending only on the input u(t).

Let us define

[3.38]

and

[3.39]

Then

[3.40]

Since we consider the free response initialized by Ẕ(ω, t_p), we obtain

[3.41]

As noted previously, the simplification of notations is based on the reduced time variable t – t_p = t; therefore, Ẕ(ω, t_p) = Ẕ(ω, 0).

Then

[3.42]

Practically, we use the frequency discretized model (see Chapter 2 of Volume 1); thus, we replace Ẕ(ω, 0) (in fact, Ẕ(ω, t_p)) with

[3.43]

and with

[3.44]

where

[3.45]

Therefore, is linear in the parameter vector .

Note that in [3.45], A_{i, I} is diagonal; thus, the matrix Φ(t) is easily computed using A_i,I and C_i,I.

As demonstrated thereafter, we can compute the free response of the system y_free(t) for t ∈ [t_p, t₀] (see sections 3.4.3.1 and 3.4.3.2).

Therefore, expressing in terms of y_free(t) on the history interval, we can estimate as follows.

Let us define

[3.46]

where is an estimation of . f(t) and are known functions on [t_p, t₀] (see the illustrative examples given in sections 3.4.3.1 and 3.4.3.2).

Note that the least-squares technique [EYK 74, NOR 86], based on the quadratic criterion , requires a matrix inversion that cannot be performed since the eigenvalues’of this matrix are distributed from 0 to + ∞, due to the wide range of ω_j modes (see Chapter 4 for an analysis of this problem).

Consequently, the gradient technique [RIC 71, LJU 87, TRI 88] is more appropriate as it does not require matrix inversion.

3.4.3. Gradient algorithm

Consider the quadratic criterion

[3.47]

Let t – t_p = k T_e (with T_e being the sampling time, k ∈ N) and be the estimation at t = k T_e. Thus,

[3.48]

Using the online gradient algorithm [LJU 87], the new estimation at (k + 1) T_e corresponds to

[3.49]

or

[3.50]

The gradient algorithm presents two known drawbacks [RIC 71, TRI 88]: its stability depends on λ and it is highly sensitive to measurement noise, which imposes a low value of λ. On the contrary, in a deterministic context, convergence is relatively fast (with a high value of λ respecting λ < λ_max). Moreover, it provides a confident estimation of Ẕ(ω, t_p) without performing matrix inversion [TAR 16c, MAA 17].

Practically, several sequences of the gradient algorithm are necessary (one sequence corresponding to t_p → t₀), initialized at the first sequence by (see Appendix A.3. for convergence and stability of the gradient algorithm).

3.4.3.1. Example 1

Consider the transfer function system

[3.51]

Thus, .

Relation [3.51] leads to

[3.52]

Let us define

[3.53]

Then

[3.54]

and using [3.44]

[3.55]

Then, we can express

[3.56]

with

[3.57]

Iⁿ(y_free(t)) corresponds to the fractional integration of y_free(t), which can be easily computed (see Chapter 2 of Volume 1).

Then, using the gradient algorithm [3.42], we can estimate .

3.4.3.2. Example 2

Consider the transfer function system

[3.58]

Thus, .

The calculation of y_free(t) and is based on the observer canonical form [KAI 80]:

[3.59]

After simple calculations, relation [3.40] provides

[3.60]

which leads, in the time domain, to

[3.61]

Thus, from [3.44], we obtain

[3.62]

or

[3.63]

where

[3.64]

corresponds to the first fractional integrator (with order n₁), and corresponds to the second integrator (order n₂).

Then, we obtain

[3.65]

The fractional integrals I^n₂(.) and I^n₁(I^n₂(.)) are easily computed using the frequency discretized model of the fractional integrators (see Chapter 2 of Volume 1). Thus, can be estimated by the gradient algorithm [3.49].

Of course, the regressor is more complex than in the first example, but it does not introduce numerical difficulties.

3.4.4. One-derivative FDE example

3.4.4.1. Introduction

Let us consider the system [3.51]:

[3.66]

Since the direct observation of the system state does not provide a confident estimation ẑ(ω, t₀) (see section 3.3.4), we use the previous gradient technique to estimate z(ω, t_p).

In Figure 3.6, we plot the step response y(t) starting at t = 0 and the corresponding free response y_free(t) starting at t_p = 0.5 s. This simulation makes it possible to provide the required data {f_k} of [3.55].

3.4.4.2. Estimation tests of z(ω, t_p)

The previous gradient algorithm is used to estimate z(ω, t_p) with (which ensures algorithm stability).

In Figure 3.7, we plot the frequency discretized components z_j(t_p) and ẑ_j(t_p) obtained after several sequences. After one sequence, the estimation is poor, particularly for the higher modes. Therefore, it is necessary to perform several sequences of the gradient algorithm to improve this estimation. We note an improvement for five sequences, and an important one at the low and medium frequencies for 10 sequences.

3.4.4.3. Estimation of z(ω, t₀)

The estimation ẑ(ω, t_p) obtained after 10 sequences is selected to initialize the observer, and we keep the same gain K = 20 as with the previous direct approach.

In Figure 3.8, we plot z_j(t₀) (exact), ẑ_j(t₀) (direct) and ẑ_j(t₀) initialized by ẑ_j(t_p) (improved).

The improvement of z(ω, t₀) estimation is now significant, whereas ẑ(ω, t₀) (direct) is far from z(ω, t₀) at low frequencies, ẑ(ω, t₀) (initialized by ẑ(ω, t_p)) is excellent at all frequencies. Thus, the initialization of the one-derivative fractional order model is now excellent (Figure 3.9).

As we cannot objectively appreciate the initialization improvement, we compute the difference between the true response y(t) and its initialization y_init(t), i.e.

[3.67]

In Figure 3.10, we plot the direct initialization error and the improved initialization error. Obviously, the proposed methodology provides an important improvement to the initialization problem.

3.4.5. Two-derivative FDE example

Let us consider again the non-commensurate example [3.58]:

[3.68]

In Figure 3.11, we present the step response y(t) starting at t = 0 and the corresponding forced response y_forced(t) and free response y_free(t) starting at t_p = 0.4s. This simulation makes it possible to provide the required data {f_k} of [3.61].

3.4.5.1. Estimation tests of z(ω, t_p)

The previous gradient algorithm was used to estimate with and (which ensure algorithm stability).

In Figures 3.12 and 3.13, we respectively plot z_{1, j}(t_p) and z_{2, j}(t_p) estimates for several sequences.

3.4.5.2. Estimation of Ẕ(ω, t₀)

The observer is initialized with the estimation obtained after 10 sequences; moreover, it operates with the gain on the history interval [t_p , t ₀].

In Figures 3.14 and 3.15, we plot Ẕ(t₀) exact, direct and initialized by respectively for Ẕ₁(t₀) and Ẕ₂(t₀).

As shown in these figures, improved by the gradient technique is now closer to Ẕ(ω, t₀) exact, as in the previous example.

Finally, in Figure 3.16, we compare the system response initialized by improved and by direct. As shown by the comparison of initialization errors in Figure 3.17, the improved initialized response fits very well with the exact system response, and there is no longer the difference caused by the low frequency modes that characterize the direct initialization.

A.3.1. Convergence of gradient algorithm

A.3.1.1. Asymptotic convergence

The gradient algorithm is expressed as

[3.69]

where .

Therefore

[3.70]

Assume that the algorithm converges to a value , then .

Therefore

[3.71]

Since , the algorithm converges to

[3.72]

Of course, this property is classical: the least-squares technique is characterized by because the model is linear in the parameters [LJU 87].

A.3.1.2. Convergence rate

In fact, convergence is not sufficient, and the convergence rate is more important in our case.

First, consider a one-parameter θ_j algorithm, with

As J(t) = e²(t), we obtain:

[3.73]

Therefore

[3.74]

This algorithm is stable if

[3.75]

[3.76]

First, consider the case . Then, remains close to 1 even for large values of k.

Therefore

[3.77]

Then, if λ is chosen as , convergence is fast and θ_{j, k} = θ_j is obtained with a few iterations.

Then, consider the case . Then, decays quickly and with a few iterations. Therefore, θ_j,k = θ_j,k+1, and convergence is stopped far from θ_j.

Consequently, convergence requires several sequences of the gradient algorithm, because the variation of θ_{j, k} is possible only for the low values of k.

Then, consider the two-parameter case.

φ₁(t) = c₁ e^–ω₁t and φ₂(t) = c₂ e^–ω₂t with ω₂ ≫ ω₁.

T_e is chosen close to , whereas .

Therefore, φ₂(t) decays quickly with k, whereas φ₁(t) remains close to 1.

With the approximation φ₁ ≈ 1, after a few iterations, the algorithm is equivalent to

[3.78]

This means that θ_{1, k} converges to θ₁, whereas θ_{2, k} is blocked to a value different from θ₂.

Consequently, θ_{1, k} converges immediately, whereas several sequences are necessary for θ_{2, k} convergence, as mentioned previously.

This result is easily generalized to a large number of parameters .

Parameters characterized by T_e close to (high frequency modes) converge slowly, whereas low frequency modes converge quickly.

A.3.2. Stability and limit value of λ

Equations [3.69, 3.70] can be written as

[3.79]

Stability of [3.79] is conditioned by the eigenvalues µ_j(k) of A(k), with the requirement

[3.80]

for N = 1, we obtain ,

therefore

[3.81]

Practically, λ_max decays with .

However, since λ is weighted by (i.e. ), the algorithm becomes increasingly stable as k increases (because λ(k) → 0).