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General conformable estimators with finitetime stability
Advances in Difference Equations volume 2020, Article number: 551 (2020)
Abstract
In this paper, some estimators are proposed for nonlinear dynamical systems with the general conformable derivative. In order to analyze the stability of these estimators, some Lyapunovlike theorems are presented, taking into account finitetime stability. Thus, to prove these theorems, a stability function is defined based on the general conformable operator, which implies exponential stability. The performance of the estimators is assessed by means of numerical simulations. Furthermore, a comparison is made between the results obtained with the integer, fractional, and general conformable derivatives.
Introduction
Fractional calculus, the generalization of calculus to noninteger orders, besides looking to extend the classical mathematical results, has had many applications to physical systems since the 1970s of the twentieth century. The first results in this area involved the Riemann–Liouville integral and the Riemann–Liouville and Caputo derivatives, which are still widely studied and used [1–8]. Nevertheless, other definitions of noninteger operators have been developed; some recent classifications of the main operators and their properties appear in [9, 10].
Given that the Riemann–Liouville and Caputo derivatives may deal with singularity issues in the kernel, some operators with nonsingular kernel have been proposed, such as Caputo–Fabrizio [11] and Atangana–Baleanu [12] derivatives; these operators are currently being studied extensively, and they have been used for both theoretical results and applications [13–17].
Furthermore, another reason for proposing other noninteger operators is that the Riemann–Liouville and Caputo derivatives do not satisfy the main results of classical calculus, such as the Leibniz product rule, the chain rule, the semigroup property, and the fundamental theorem, which would be expected to occur naturally for their use in applications. In this sense, Khalil et al. proposed the socalled conformable derivative [18]; this operator satisfies the properties mentioned and other mathematical results. Later, Abdeljawad used this definition to extend more results in calculus and linear systems [19]. After that, many papers have been focused on studying the theory and applications of this derivative [20–25].
Similarly to Khalil, Katugampola proposed some conformabletype operators, which also satisfy the classical results [26]. These operators have also been studied and applied [27–30].
Moreover, Akkurt et al. proposed the socalled generalized fractional derivative [31], which generalizes the operators defined by Khalil and Katugampola, while still satisfying the results from integer calculus. This derivative is called general due to the freedom of choice of its kernel \(k(t)\), where its adequate selection permits to obtain other operators as particular cases.
Additionally, Zhao and Luo defined another general version, called the general conformable derivative (GCD) [32], which is based on the linear extended Gâteaux derivative. This operator also generalizes the Khalil and Katugampola derivatives, but in this case the kernel not only depends on time, but also on the order α. This operator also satisfies the desired results from classical calculus and encompasses other derivatives as special cases.
On the other hand, the problem of finitetime stability becomes relevant in applications of dynamical systems, where theoretical asymptotic stability is not useful, but it is desired that the system trajectories reach the equilibrium in a determined finite time. Thus, this theme has been studied and developed formally [33–36]; some applications and specific problems can be found in [37–39]. Moreover, this subject has been addressed for fractionalorder systems [40–42] and even for systems with conformable derivatives [43–45]. However, it has been reported that, in general, nonlinear fractionalorder systems cannot have finitetime stability, but just under certain considerations [46, 47].
Given that the GCD generalizes other operators and satisfies the classical calculus rules, it is of great interest to use it to develop results in theory and applications, which would enclose existing or possible results using conformabletype derivatives. Hence, in this paper, some nonlinear estimators for dynamical systems that involve the GCD are proposed. In order to verify the stability of these estimators, an exponentiallike function is defined, based on the operator in question, which implies exponential stability. Using this function, some Lyapunovlike theorems are proven; then, some finitetime stability conditions are added to these theorems. The performance of the designed estimators is evaluated with numerical simulations; for this, the models considered are a mechanical system and a chaotic oscillator.
This work is divided as follows. In Sect. 2, the GCD is defined from the original conformable derivative; its properties and some results are presented. In Sect. 3, the socalled general conformable exponential function is defined, and then some Lyapunovlike theorems for systems with the GCD are proven. Section 4 presents some definitions and conditions required for finitetime stability, and the Lyapunovlike theorems previously defined are extended to consider them. In Sect. 5, a pair of nonlinear estimators with general conformable dynamics are designed, and they are proven to be finitetime stable in the general conformable sense. In Sect. 6, numerical simulations are used to assess the performance of the estimators; the procedure is applied to the general conformable models of the simple pendulum and the Van der Pol oscillator, and a comparison is made with their integer and fractional versions. Finally, conclusions and results are discussed in Sect. 7.
General conformable derivative
In this section, some definitions are presented in order to introduce the GCD.
Definition 1
([18])
Given a function \(f:[0,\infty )\rightarrow \mathbb{R}\), the conformable derivative (CD) of f of order α is defined by
\(\forall t>0\), \(\alpha \in (0,1)\). If f is αdifferentiable in some \((0,a)\), \(a>0\), and \(\lim_{t\rightarrow 0^{+}}f^{(\alpha )}(t)\) exists, then define
In order to extend and give a physical and geometrical interpretation to this derivative, the following definitions are required.
Definition 2
([48])
Let \(F:U\rightarrow \mathbb{R}\) be a functional, where U is a Banach space. We define the space of admissible variations for F, denoted by \(\mathscr{V}\), as follows:
Definition 3
([48])
Given \(F:U\rightarrow \mathbb{R}\), we define the Gâteaux variation of F at \(u\in U\) on the direction \(\psi \in \mathscr{V}\), denoted by \(\delta F(u,\psi )\), as follows:
if such a limit is well defined. Furthermore, if there exists \(u^{*}\in U^{*}\) such that
we say that F is Gâteaux differentiable at \(u\in U\), and \(u^{*}\in U^{*}\) is said to be the Gâteaux derivative of F at u, where \(\langle \cdot ,\cdot \rangle _{U}\) is the duality pairing between U and \(U^{*}\). Finally, we denote
Remark 1
([49])
If \(F=f\in \mathcal{C}^{1}(\mathbb{R}^{n})\) and \(u,\psi \in \mathbb{R}^{n}\), then
is just the directional derivative of f when ψ is a unit vector. Thus we have that
and this holds \(\forall \psi \in \mathbb{R}^{n}\).
Some properties of the Gâteaux derivative are as follows [32]:

1.
\(d(c)=0\) (differential of a constant).

2.
\(d(f+g)(u,\psi )=df(u,\psi )+dg(u,\psi )\) (sum rule).

3.
\(d(fg)(u,\psi )=df(u,\psi )g+dg(u,\psi )f\) (product rule).

4.
\(d(f/g)(u,\psi )=\frac{df(u,\psi )gdg(u,\psi )f}{g^{2}}\) (quotient rule).

5.
\(d(f\circ g)(u,\psi )=df(g(u),dg(u,\psi ))\) (chain rule).
Definition 4
([32])
Suppose that X and Y are locally convex topological vector spaces, \(U\subset X\) is open, \(f:X\rightarrow Y\), and \(\psi (u,\varepsilon ,\alpha ):X\times \mathbb{R}\times \mathbb{R} \rightarrow X\), where \(\alpha \in \mathbb{R}\) is a parameter. The extended Gâteaux differential (EGD) \(df(u,\psi )\) of f at \(u\in U\) is defined as
if the limit exists.
Definition 5
([32])
Suppose that X and Y are locally convex topological vector spaces, \(U\subset X\) is open, \(f:X\rightarrow Y\), and \(\psi (u,\alpha ):X\times \mathbb{R}\rightarrow X\), where \(\alpha \in \mathbb{R}\) is a parameter. The linear extended Gâteaux differential (LEGD) \(df(u,\psi )\) of f at \(u\in U\) is defined as
if the limit exists.
The LEGD satisfies the same properties of the Gâteaux derivative.
Moreover, regarding the concept of local fractional derivative (LFD), the following principles need to be considered [32]:
 P1.:

LFD should degenerate to the usual firstorder derivative when the fractional order equals one.
 P2.:

LFD should have properties consistent with the classical derivative as much as possible.
 P3.:

LFD should have clear physical or geometrical interpretations.
The LEGD satisfies P2 and P3. In order to satisfy P1, the following definition is introduced.
Definition 6
([32])
A continuous real function \(\psi (t,\alpha )\) is called a conformable function if it satisfies
Finally, the definition of the operator used in this work is presented.
Definition 7
([32])
Let \(\psi (t,\alpha )\) be a conformable function and \(\alpha \in (0,1]\). The general conformable derivative (GCD) is defined as
if the limit exists.
Remark 2
Note that

a)
when \(\psi (t,\alpha )=1\), \({}^{\psi }D^{\alpha }f(t)\) degenerates to the usual firstorder derivative.

b)
when \(\psi (t,\alpha )=t^{1\alpha }\), \({}^{\psi }D^{\alpha }f(u)\) coincides with the Khalil CD definition.
Moreover, consider the operator proposed by Katugampola.
Definition 8
([26])
Let \(f:[0,\infty )\rightarrow \mathbb{R}\) and \(t>0\). Then the derivative of f of order α is defined by
for \(t>0\), \(\alpha \in (0,1)\). If f is αdifferentiable in some \((0,a)\), \(a>0\), and \(\lim_{t\rightarrow 0^{+}}\mathcal{D}^{\alpha }(f)(t)\) exists, then define
Remark 3
When \(\psi (t,\varepsilon ,\alpha )=te^{\varepsilon t^{\alpha }}t\) in the EGD, it degenerates to the Katugampola definition. However, since \(te^{\varepsilon t^{\alpha }}t=\varepsilon t^{1\alpha }+o( \varepsilon ^{2})\), it coincides with the Khalil definition neglecting the term \(o(\varepsilon ^{2})\).
Remark 4
The GCD satisfies the same properties as the LEGD, as well as Rolle’s and mean value theorems [32].
Remark 5
Henceforth, a function is said to be αdifferentiable if it is differentiable in the sense of the GCD, with order α.
Remark 6
If f is αdifferentiable, then \({}^{\psi }D^{\alpha }f(t)=\psi (t,\alpha )\frac{d}{dt}f(t)\).
Theorem 1
([32])
If a function \(f:\mathbb{R}^{+}\rightarrow \mathbb{R}\) is αdifferentiable at \(t>0\), \(\alpha \in (0,1]\), then f is continuous at t.
Furthermore, define the inverse operator of the GCD.
Definition 9
([32])
Let \(t\geq a\geq 0\), f be a function defined on \((a,t]\). Then the αorder general conformable integral of f is defined as
if the Riemann integral exists.
Moreover, consider the following result.
Lemma 1
Let \(f:[a,\infty )\rightarrow \mathbb{R}\) be αdifferentiable on \((a,\infty )\). If \({}^{\psi }D^{\alpha }f(t)\geq 0\) (respectively ≤0) \(\forall t\in (a,\infty )\), then f is an increasing (respectively decreasing) function.
Proof
The proof of this lemma follows from the mean value theorem for the GCD [32]. □
Remark 7
Let \(f:[a,\infty )\rightarrow \mathbb{R}\) be αdifferentiable on \((a,\infty )\). Then
Remark 8
Let \(f:[a,\infty )\rightarrow \mathbb{R}^{n}\) be αdifferentiable on \((a,\infty )\). Then
Remark 9
Let \(x:[a,\infty )\rightarrow \mathbb{R}^{n}\) be αdifferentiable on \((a,\infty )\). Let P be a symmetric positive definite matrix. Then
General conformable exponential stability
Consider the following class of nonlinear systems:
where \(x\in \mathbb{R}^{n}\), \(f:\mathbb{R}^{+}\times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) is a given nonlinear function satisfying \(f(t,0)=0\) \(\forall t\geq 0\) and \(\alpha \in (0,1)\).
Definition 10
The origin of system (1) is said to be

i)
stable if, for every \(\varepsilon >0\) and \(t_{0}\in \mathbb{R}^{+}\), \(\exists \delta (\varepsilon ,t_{0})\) such that, for any \(x_{0}\in \mathbb{R}^{n}\), \(\lVert x_{0}\rVert <\delta \Longrightarrow \lVert x(t)\rVert <\varepsilon \), \(\forall t\geq t_{0}\).

ii)
attractive if, for any \(t_{0}\geq 0\), \(\exists c(t_{0})>0\) such that, for any \(x_{0}\in \mathbb{R}^{n}\), \(\lVert x_{0}\rVert < c\Longrightarrow \lim_{t \rightarrow \infty }x(t)=0\).

iii)
asymptotically stable if it is stable and attractive.

iv)
globally asymptotically stable if it is asymptotically stable for any \(x_{0}\in \mathbb{R}^{n}\).
The following definition is the essential tool to prove the stability results for this class of systems.
Definition 11
The general conformable exponential function is defined as follows:
where \(\alpha \in (0,1)\), \(\gamma \in \mathbb{R}^{+}\) and \(\psi (t,\alpha )\neq 0\) is a conformable function, \(\forall t\geq t_{0}\), \(\forall \alpha \in (0,1)\).
Remark 10
It is not difficult to verify that
Now, the notion of general conformable exponential stability is introduced.
Definition 12
The origin of system (1) is general conformable exponentially stable (GCES) if
with \(t>t_{0}\) and \(C,\gamma >0\).
Lemma 2
Let \(g:[t_{0},\infty )\rightarrow \mathbb{R}^{+}\) be an αdifferentiable function on \((t_{0},\infty )\) such that
where \(\gamma >0\) and \(\alpha \in (0,1)\). Then
Proof
Let \(h(t)=g(t)E_{\alpha }^{\psi }(\gamma ,t,t_{0})\). Using the product rule, we have
Since \({}^{\psi }D^{\alpha }h(t)\leq 0\), from Lemma 1\(h(t)\) is a decreasing function. Hence, \(h(t)\leq h(t_{0})\), which gives the result. □
The following theorems serve to prove stability in the sense of Lyapunov for the class of systems considered.
Theorem 2
Let \(x=0\) be an equilibrium point for system (1). Let \(V:\mathbb{R}^{+}\times \mathbb{R}^{n}\rightarrow R\) be an αdifferentiable function and \(a_{i}\) (\(i=1,2,3\)) be arbitrary positive constants. If the following conditions are satisfied:

(i)
\(a_{1}\lVert x\rVert ^{2}\leq V(t,x)\leq a_{2}\lVert x\rVert ^{2}\),

(ii)
\({}^{\psi }D^{\alpha }V(t,x)\leq a_{3}\lVert x\rVert ^{2}\),
then the origin of system (1) is GCES.
Proof
From conditions (i) and (ii) we have
Applying Lemma 2 to this inequality, we have
Also from (i) we have
Thus
and
with \(C=\sqrt{a_{2}/a_{1}}\) and \(\gamma =a_{3}/2a_{2}\). Therefore, the origin of system (1) is GCES. □
Definition 13
A continuous function \(k:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) is said to belong to class \(\mathcal{K}\) if it is strictly increasing and \(k(0)=0\). It belongs to class \(\mathcal{K}_{\infty }\) if \(\lim_{t\rightarrow +\infty }k(t)=+\infty \).
Theorem 3
Let \(x=0\) be an equilibrium point for system (1). Let \(V:\mathbb{R}^{+}\times \mathbb{R}^{n}\rightarrow R\) be an αdifferentiable function and \(k_{i}\) (\(i=1,2,3\)) be functions of class \(\mathcal{K}\) satisfying

(i)
\(k_{1}(\lVert x\rVert )\leq V(t,x)\leq k_{2}(\lVert x\rVert )\),

(ii)
\({}^{\psi }D^{\alpha }V(t,x)\leq k_{3}(\lVert x\rVert )\),
then the origin of system (1) is GCES.
Furthermore, if \(k_{i}\in \mathcal{K}_{\infty }\) (\(i=1,2,3\)), then the origin of system (1) is globally GCES.
Proof
From condition (i), \(V(t,x)\geq 0\); this means that \(\lim_{t\rightarrow \infty }V(t,x)=L\geq 0\). However, from condition (ii) and Lemma 1, \(V(t,x)\) is a decreasing function, so \(\lim_{t\rightarrow \infty }V(t,x)=L=0\). To prove this, assume that \(L>0\). From conditions (i) and (ii) and Lemma 1, we have
with \(\lambda =\frac{k_{3}\circ k_{2}^{1}(L)}{V(t_{0},x_{0})}\). Hence
Thus from Lemma 2 we have
which is a contradiction to the assumption for L. Therefore, \(L=0\) and from (i), \(\lim_{t\rightarrow \infty }x(t)=0\), so we have
for some \(C,\gamma >0\). Hence, the origin of system (1) is GCES.
Considering the case where \(k_{i}\in \mathcal{K}_{\infty }\) (\(i=1,2,3\)), (3) is satisfied \(\forall x_{0}\in \mathbb{R}^{n}\). In this case, the origin of system (1) is globally GCES. □
Remark 11
Note that the GCES concept implies exponential stability, taking \(\alpha =1\), which leads to \(\psi (t,\alpha )=1\).
Finitetime general conformable exponential stability
From Remark 6, dynamics with GCD can be seen as an integerorder dynamics weighted by a timedependent term
Thus, systems with GCD might be conceived as integerorder nonautonomous systems. Hence, the following definitions will be used [36].
Consider the integer order system
where \(f:\mathbb{R}^{+}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) is a continuous function. \(\phi _{t}^{x}(\tau )\) denotes a solution of (4) starting from \((t,x)\), and \(\mathcal{S}(t,x)\) represents the set of all its solutions. \(\mathcal{V}\) denotes a neighborhood of the origin in \(\mathbb{R}^{n}\), and \(\mathcal{B}_{\epsilon }\) is an open ball centered at the origin of radius ϵ.
Definition 14
([36])
A continuous function \(f:\mathbb{R}^{+}\times \mathcal{V}\rightarrow \mathbb{R}\) is decrescent if there exists a \(\mathcal{K}\) function ψ such that
Definition 15
([36])
A continuous function \(f:\mathbb{R}^{+}\times \mathbb{R}^{n}\rightarrow \mathbb{R}\) is radially unbounded if there exists a \(\mathcal{K}_{\infty }\) function φ such that
Definition 16
([36])
The origin of system (4) is weakly finitetime stable if

1.
The origin is Lyapunov stable.

2.
\(\exists T(\phi _{t}^{x})\in [0,+\infty )\) such that \(\phi _{t}^{x}(\tau )=0\), \(\forall \tau \geq t+T(\phi _{t}^{x})\). Here, the term
$$ T_{0}\bigl(\phi _{t}^{x}\bigr)=\inf \bigl\{ T \bigl(\phi _{t}^{x}\bigr)\geq 0:\phi _{t}^{x}( \tau )=0, \forall \tau \geq t+T\bigl(\phi _{t}^{x}\bigr) \bigr\} $$is called the settling time of the solution \(\phi _{t}^{x}\).

3.
Moreover, if \(T_{0}(t,x)=\sup_{\phi _{t}^{x}\in \mathcal{S}(t,x)}T_{0}(\phi _{t}^{x})<+ \infty \), then the origin of system (4) is finitetime stable.
\(T_{0}(t,x)\) is called the settling time with respect to the initial conditions of system (4).
Definition 17
([36])
The origin of system (4) is uniformly finitetime stable if

1.
The origin is uniformly asymptotically stable.

2.
The origin is finitetime stable.

3.
There exist a positive definite continuous function \(\alpha :\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) such that the settling time with respect to the initial conditions of system (4) satisfies
$$ T_{0}(t,x)\leq \alpha \bigl(\lVert x\rVert \bigr). $$
Now, consider again system (1)
Its equivalent integerorder form is
where the conformable function \(\psi (t,\alpha )\) is different from zero \(\forall t\in \mathbb{R}^{+}\), \(\forall \alpha \in (0,1)\).
Proposition 1
Let the origin be an equilibrium point for system (1), where f is continuous.

i)
If there exists a continuously differentiable Lyapunov function \(V(t,x)\) satisfying
$$ \dot{V}(t,x)\leq r\bigl(V(t,x)\bigr) $$with a positive definite continuous function \(r:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\), \(r(0)=0\), such that for some \(\epsilon >0\)
$$ \int _{0}^{\epsilon }\frac{dz}{r(z)}< +\infty , $$then the origin of system (1) is finitetime stable.

ii)
If in addition \(V(t,x)\) is decrescent, then the origin of system (1) is uniformly finitetime stable.

iii)
If in addition system (1) is globally defined and \(V(t,x)\) is radially unbounded, then the origin of system (1) is globally finitetime stable.
Proof
Considering system (1) as its integerorder version (5), the proof follows from the proof of Proposition 4.1 in [36]. □
Definition 18
Let the origin be an equilibrium point for system (1), where f is continuous. The origin is finitetime general conformable exponentially stable (FGCES) if

1.
The origin is GCES;

2.
The origin is finitetime stable.
If the origin is globally finitetime stable, then the origin of system (1) is globally FGCES.
Theorem 4
Let the origin be an equilibrium point for system (1), where f is continuous. The origin is FGCES if there exists a continuously differentiable Lyapunov function \(V(t,x)\) satisfying

(i)
\(k_{1}(\lVert x\rVert )\leq V(t,x)\leq k_{2}(\lVert x\rVert )\);

(ii)
\({}^{\psi }D^{\alpha }V(t,x)\leq k_{3}(\lVert x\rVert )\), with \(k_{i}\) (\(i=1,2,3\)) functions of class \(\mathcal{K}\).
Furthermore, if \(k_{1}\in \mathcal{K}_{\infty }\) and system (1) is globally defined, then the origin of system (1) is globally FGCES.
Proof
From i and ii, the origin of system (1) is GCES (from Theorem 3), and thus there exists a continuous and differentiable Lyapunov function \(V(t,x)\).
Moreover, from the proof of Theorem 3, we get
and, following the alternative form of the GCD, we have
where \(r(V(t,x))=\frac{\lambda V(t,x)}{\psi (t,\alpha )}\); it can be verified that \(r(0)=0\). Furthermore, by using the change of variables \([0,V(t,x)]\rightarrow [t,t+T_{0}(t,x)]\) given by \(z=V(\tau ,x(\tau ))\), we obtain
Finally, from [36] we have
Hence, the origin of system (1) is finitetime stable. Therefore, the origin of system (1) is FGCES.
Moreover, if \(k_{1}\in \mathcal{K}_{\infty }\), it means that \(V(t,x)\) is radially unbounded. If, in addition, system (1) is globally defined, the origin of system (1) is globally finitetime stable. Therefore, the origin of system (1) is globally FGCES. □
Design of nonlinear estimators
In this section, the stability results presented previously are used to design a pair of estimators for dynamical systems with the GCD. Then it is proven that the estimation error obtained is FGCES.
Consider the following class of commensurateorder nonlinear systems with single output:
where \(\alpha \in (0,1)\), \(\mathbf{x}\in \mathbb{R}^{n}\) is the state vector, \(\mathbf{u}\in \mathbb{R}^{m}\) is the control input, \(y\in \mathbb{R}\) is the output, and \(f:\mathbb{R}^{n}\times \mathbb{R}^{m}\rightarrow \mathbb{R}^{n}\) is a locally Lipschitz vector function in x and uniformly bounded in u.
Rewrite the system to its canonical form
where A is an upper shift matrix (\(A:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\), \(A_{i,j}=\delta _{i+1,j}\), with \(\delta _{i,j}\) the Kronecker delta), \(C=[1\ 0 \ 0\ \dots \ 0]\), the pair \((A,C)\) is observable, and \(\Upsilon (\mathbf{x},\mathbf{u})\) is a nonlinear vector that satisfies the Lipschitz condition
locally in a region D. The estimators will be designed based on this equivalent system; their estimation error is bounded by the general conformable exponential function (2).
LQRbased estimator
Consider the following estimator based on a linear quadratic regulator (LQR) [50]:
where \(\hat{\mathbf{x}}, \mathbf{K}_{i}\in \mathbb{R}^{n}\), \(1\leq i\leq m\). Consider the following lemma.
Lemma 3
([50])
Given a stable \(n\times m\) matrix Â and \(\gamma >0\), there exists a positive definite, symmetric matrix P such that
if and only if there exists another positive definite, symmetric matrix P̂ such that
The following LMI is equivalent to (10)
Now, for some \(\varepsilon >0\),
Let \(\upsilon =\Upsilon (\mathbf{x},\mathbf{u})\Upsilon (\hat{\mathbf{x}}, \mathbf{u})\). Considering condition (8), the estimation error \(\mathbf{e}=\mathbf{x}\hat{\mathbf{x}}\), and the solution P of (11), we have
Remark 12
An estimator is said to be finitetime general conformable exponentially stable if the estimation error e obtained with it is FGCES.
Theorem 5
Consider system (7) with pair \((A,C)\) observable. If \(\bar{A}=A\mathbf{K}_{1}C\) is stable and \(M_{i}=P^{i1}\mathbf{K}_{i}C\) is positive semidefinite for \(2\leq i\leq m\), then the LQRbased estimator (9) is a finitetime general conformable exponentially stable estimator for system (6).
Proof
From (7) and (9), the dynamics of e is
As Ā is stable and \(\varphi >0\), from Lemma 3\(\exists P>0\). Consider \(V=\lVert \mathbf{e}\rVert ^{2}_{P}=\mathbf{e}^{T}P\mathbf{e}\) a candidate Lyapunov function that satisfies the Rayleigh–Ritz inequality
From Remark 9, (12), and (13) we have
Given that \(M_{i}=P^{i1}\mathbf{K}_{i}C\geq 0\), we have
From Theorem 2, (15), and (16) it follows that the origin of system (14) is GCES, and we have
with \(C=\sqrt{\frac{\lambda _{max}(P)}{\lambda _{min}(P)}}\), \(\gamma = \frac{\varepsilon }{2\lambda _{max}(P)}\) and \(\mathbf{e}_{0}=\mathbf{e}(t_{0})\).
Furthermore, from (16) and the properties of norms we have
where \(r(V)=\frac{c_{1}\varepsilon V(t,x)}{\psi (t,\alpha )}\). We have that \(r(0)=0\) and
thus, the origin of system (14) is finitetime stable.
Therefore, the origin of system (14) is FGCES, and hence system (9) is a finitetime general conformable exponentially stable estimator for system (6). □
Highgain observer
Consider the following highgain observer (HGO) [51]:
where \(F_{\infty }=\lim_{t\rightarrow \infty }F_{\theta }(t)\), with \(F_{\theta }(t)\) positive definite solution of
Remark 13
Given that \(F_{\infty }\) is constant, \({}^{\psi }D^{\alpha }F_{\infty }=0\), and hence it may be calculated from
The coefficients of \(F_{\infty }\) are given by
where \(\alpha _{i,j}\) is symmetric positive definite, independent of θ.
Theorem 6
The highgain observer (17) is a finitetime general conformable exponentially stable estimator for system (6).
Proof
From (7) and (17), the dynamics of e is
with \(\upsilon =\Upsilon (\mathbf{x},\mathbf{u})\Upsilon (\hat{\mathbf{x}}, \mathbf{u})\). Consider \(V=\lVert \mathbf{e}\rVert ^{2}_{F_{\infty }}=\mathbf{e}^{T}F_{\infty }\mathbf{e}\) a candidate Lyapunov function that satisfies the Rayleigh–Ritz inequality
Moreover, from Remark 9 and (18) we have
Thus, from condition (8), for \(\theta >2\varphi +1\),
According to Theorem 2, from (20) and (21) it follows the origin of system (19) is GCES, and we have
with \(C=\sqrt{\frac{\lambda _{max}(F_{\infty })}{\lambda _{min}(F_{\infty })}}\), \(\gamma = \frac{ (\theta 2\varphi 1 )\lVert F_{\infty }\rVert }{2\lambda _{max}(F_{\infty })}\), and \(\mathbf{e}_{0}=\mathbf{e}(t_{0})\).
Furthermore, from (21) and the properties of norms, we have
where \(r(V)= \frac{c_{1} (\theta 2\varphi 1 )V(t,x)}{\psi (t,\alpha )}\). We have that \(r(0)=0\) and
Thus, the origin of system (19) is finitetime stable.
Therefore, the origin of system (19) is FGCES, and hence system (17) is a finitetime general conformable exponentially stable estimator for system (6). □
Numerical simulations
In this section, the results of numerical simulations of the estimation scheme proposed are presented. For this, the general conformable estimators were designed and applied based on the models of two dynamical systems with GCD. In addition, the performance of the scheme with integer, fractional, and general conformable dynamics is compared. The simulations were carried out using Simulink® from MATLAB®; moreover, to implement the fractional case, the noninteger fractional derivative from D. Valério was used [52].
Simple pendulum
Consider the extension of the simple pendulum to its noninteger order version, studied e.g. in [53–55]:
where \(x_{1}=\theta \) (angular position), \(x_{2}=\omega \) (angular velocity), \(g=9.81\mbox{ m/s}^{2}\), and \(L=1\mbox{ m}\). Rewriting it to the canonical form, we have
with $A=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$, $\mathrm{\Upsilon}(\mathbf{x})=\left[\begin{array}{c}0\\ \frac{g}{L}sin({x}_{1})\end{array}\right]$, Lipschitz constant \(\varphi =\frac{g}{L}\), and \(C=[ 1 \ 0 ]\).
A mechanical system with noninteger dynamics has the effect of additional damping on the trajectories [54, 55] which can be understood as an extra parameter that varies with the choice of the order. When using the GCD, due to the freedom to choose the order α as well as the conformable function \(\psi (t,\alpha )\), the resulting system gets some additional timevarying parameters. For instance, the graphs of the states show that the oscillation frequency varies.
From (9) with \(m=3\), the LQRbased estimator for system (22) is
where \(\mathbf{K}_{i}=[K_{i1}\ K_{i2}]^{T}\).
For these simulations \(\alpha =0.98\), the gain vectors were selected as \(\mathbf{K}_{1}=[4.0076\ 3.1305]^{T}\), \(\mathbf{K}_{2}=[4.905\ 4.905]^{T}\), \(\mathbf{K}_{3}=[5\ 2.4525]^{T}\), and the initial conditions are \(x_{1}(0)=\pi /2\), \(x_{2}(0)=0\), \(\hat{x}_{1}(0)=\pi \), \(\hat{x}_{2}(0)=0\). For implementing the GCD, the function \(\psi (t,\alpha )=0.99t^{30(1\alpha )}+0.01\) has been chosen.
Figures 1 and 2 show the state estimations; it can be seen that the estimated signals reach the equilibrium before 1 second. Figures 3 and 4 show the comparison of the estimation errors obtained from the integer, fractional, and GCD operators for the same models and parameters in logarithmic time, while Figs. 5 and 6 show in major detail these comparisons. Finally, Figs. 7 and 8 show the performance measure of the errors, obtained using the integral of the square of the error (ISE)
From these results it can be seen that the fractional case performs slightly better than the integer one, but it does not reach the equilibrium in the time scale shown (corresponding to 100 s); this can be appreciated also in a slightly increasing slope in the ISE for the fractional case. Moreover, the other cases present finitetime stabilization, but with the GCD operator, the signals converge faster and smoother.
Consider now HGO (17) for the pendulum system
Simulations were performed using the same initial conditions of the former estimator with \(\theta =25\) and the same conformable function. Figures 9 and 10 show the state estimations; it can be seen that the estimated signals reach the equilibrium around 0.25 seconds. Figures 11 and 12 show the comparison of the estimation errors obtained from the integer, fractional, and GCD operators for the same models and parameters in logarithmic time, while Figs. 13 and 14 show in major detail these comparisons. Finally, Figs. 15 and 16 show the performance measure of the errors, obtained using the ISE.
Similar to the results obtained from the LQRestimator, it can be seen that the fractional case performs slightly better than the integer one; however, it does not reach the equilibrium in the time scale shown (corresponding to 100 s); a major amplification would show that the signals still fail to reach the origin. Also, a slightly increasing slope appears in the ISE for the fractional case. Furthermore, the other cases present finitetime stabilization, but with the GCD operator, the signals converge faster and smoother.
Van der Pol oscillator
The Van der Pol oscillator is a chaotic system proposed to study oscillations in vacuum tube circuits [56]. Its noninteger counterpart has been studied e.g. in [57–59]. The extended model is
where ε is a control parameter. Rewriting it to its canonical form, we have
with $A=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$, $\mathrm{\Upsilon}(\mathbf{x})=\left[\begin{array}{c}0\\ {x}_{1}\epsilon ({x}_{1}^{2}1){x}_{2}\end{array}\right]$, Lipschitz constant \(\varphi =\max \{ 1+2\varepsilon \lvert x_{1}\rvert \lvert x_{2} \rvert +\varepsilon +\varepsilon \lvert x_{1}^{2}\rvert \} \), and \(C=[ 1 \ 0 ]\).
This case may be seen as a masterslave synchronization problem. The Van der Pol system will perform as the master system, and the estimators will serve as the slaves, so they will look for synchronizing with it.
From (9) with \(m=3\), the LQRbased estimator for system (23) is
where \(\mathbf{K}_{i}=[K_{i1}\ K_{i2}]^{T}\).
The gain vectors have been chosen as \(\mathbf{K}_{1}=[2.3094\ 1.5166]^{T}\), \(\mathbf{K}_{2}=[1.15\ 1.15]^{T}\), \(\mathbf{K}_{3}=[3\ 0.575]^{T}\). The parameters of the system are \(\alpha =0.9\), \(\varepsilon =0.1\) and the initial conditions have been chosen as \(x_{1}(0)=0.25\), \(x_{2}(0)=1.2\), \(\hat{x}_{1}(0)=3\), \(\hat{x}_{2}(0)=1.8\). With these values, the Lipschitz constant is set as \(\varphi =2.3\). For implementing the GCD, the function \(\psi (t,\alpha )=0.99t^{10(1\alpha )}+0.01\) has been chosen.
Figures 17 and 18 show the state estimations; it can be seen that the estimated signals reach the equilibrium around 2.5 seconds. Figures 19 and 20 show the comparison of the estimation errors obtained from the integer, fractional, and GCD operators for the same models and parameters in logarithmic time, while Figs. 21 and 22 show in major detail these comparisons. Finally, Figs. 23 and 24 show the performance measure of the errors obtained using the ISE.
From these results it can be seen that the fractional case performs better than the other, with a smaller overshoot, but it does not reach the equilibrium in the time scale shown (corresponding to 100 s); this can be appreciated also in a slightly increasing slope in the ISE for the fractional case. Moreover, the other cases present finitetime stabilization, but with the GCD operator, the signals converge faster and smoother. Furthermore, Fig. 25 shows the phase portrait obtained with \(\hat{x}_{1}\) and \(\hat{x}_{2}\).
Now, consider HGO (17). For the Van der Pol oscillator, the observer is
Simulations were performed using the same initial conditions of the former estimator with \(\theta =1\) and the same conformable function. Figures 26 and 27 show the state estimations; it can be seen that the estimated signals reach the equilibrium around 2.5 seconds. Figures 28 and 29 show the comparison of the estimation errors obtained from the integer, fractional, and GCD operators for the same models and parameters in logarithmic time, while Figs. 30 and 31 show in detail these comparisons. Finally, Figs. 32 and 33 show the performance measure of the errors, which was obtained using the ISE.
Similar to the results obtained from the LQRestimator, it can be seen that the fractional case performs better than the other, with a smaller overshoot, but it does not reach the equilibrium in the time scale shown (corresponding to 100 s); this can be appreciated in a slightly increasing slope in the ISE for the fractional case. Furthermore, the other cases present finitetime stabilization, but with the GCD operator, the signals converge faster and smoother. Furthermore, Fig. 34 shows the phase portrait obtained with \(\hat{x}_{1}\) and \(\hat{x}_{2}\).
Concluding remarks
In this paper, an LQRbased estimator and a highgain observer were proposed for a class of nonlinear systems with the general conformable derivative on their dynamics. By defining a general conformable exponential function, the estimators were proven to be finitetime stable in the sense of the derivative used by means of Lyapunovlike theorems. Then, to validate the proposed estimation scheme, simulations were performed on the general conformable models of the simple pendulum and the Van der Pol oscillator. Later, the results obtained with the conformable operator were compared with those obtained with the integer and fractional versions of the systems.
Regarding the results, the fractional case generally had a smaller overshoot than the integer and conformable cases; however, for the time scales shown, the signals in the fractional case did not reach the equilibrium. Furthermore, the integer and conformable cases presented finitetime stability, but the conformable case had a better performance than the integer one, showing a faster convergence though with a similar overshoot. Hence, as it was stated formerly, the fractional case did not present finitetime stability; the integer and conformable cases did, but the latter outperformed the former in the simulation results. In the graphs shown, it may be appreciated that, with the general conformable operator, the estimation error converges faster and the ISE is much smaller than with the other derivatives.
Comparing the estimators, both presented an acceptable performance with both systems, and their different outcomes depended on the choice of their parameters.
Finally, it is worth to note that these results may be improved through an adequate selection of the estimator gains and the conformable function \(\psi (t,\alpha )\), which also depends on the system to which this estimation scheme is applied.
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Acknowledgements
This work was supported by the National Council of Science and Technology (CONACYT) México as part of the Postdoctoral Project “Development of noninteger order derivatives and applications in nonlinear control systems” performed in the Universidad Iberoamericana at Mexico City.
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MeléndezVázquez, F., FernándezAnaya, G. & HernándezMartínez, E.G. General conformable estimators with finitetime stability. Adv Differ Equ 2020, 551 (2020). https://doi.org/10.1186/s13662020030032
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Keywords
 General conformable derivative
 General conformable exponential stability
 Conformable estimator
 Finitetime stability