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MittagLeffler stabilization of fractionalorder nonlinear systems with unknown control coefficients
Advances in Difference Equations volume 2018, Article number: 16 (2018)
Abstract
In this paper, we consider the problem of MittagLeffler stabilization of fractionalorder nonlinear systems with unknown control coefficients. With the help of backstepping design method, the stabilizing functions and tuning functions are constructed. The controller is designed to ensure that the pseudostate of the fractionalorder nonlinear system converges to the equilibrium. The effectiveness of the proposed method has been verified by some simulation examples.
Introduction
The concept of fractional differentiation appeared for the first time in a famous correspondence between L’Hospital and Leibniz, in 1695. Many mathematicians have further developed this area and we can mention the studies of Euler, Laplace, Abel, Liouville and Riemann. However, the fractional calculus remained for centuries a purely theoretical topic, with little if any connections to practical problems of physics and engineering. In recent years, the fractional calculus has been recognized as an effective modeling methodology for researchers [1]. As is well known, fractional calculus is a generalization of classical calculus to noninteger order. Compared with an integerorder system, a fractionalorder system is a better option for engineering physics.
Fractional systems have been paid much attention to, for example, the fractional optimal control problems [2], stability analysis of Caputolike discrete fractional systems [3, 4], fractional description of financial systems [5], fractional chaotic systems [6]. Especially, stabilization problem of fractionalorder systems is a very interesting and important research topic. In recent years, more and more researchers and scientists have begun to address this problem [7–23]. With the help of the Lyapunov direct method, MittagLeffler stability and generalized MittagLeffler stability was studied [10, 20]. The Lyapunov direct method deals with the stability problem of fractionalorder systems have been extended [11, 24]. Reference [25] studied the global MittagLeffler stability for a coupled system of fractionalorder differential equations on network with feedback controls. Robust stability and stabilization of fractionalorder interval systems with \(0<\alpha<1\) order have been studied [15]. Necessary and sufficient conditions on the asymptotic stability of the positivity continuous time fractionalorder systems with bounded timevarying delays are investigated by the monotonic and asymptotic property [14]. The stabilization problem of a class of fractionalorder chaotic systems have been addressed [12]. Pseudostate stabilization problem of fractionalorder nonlinear systems has attracted the attention of some researchers [7–9, 13, 16]. Moreover, any equilibrium of a general fractionalorder nonlinear system described by either Caputo’s or RiemannLiouville’s definition can never be finitetime stable was proved [26]. Finitetime fractionalorder adaptive intelligent backstepping sliding mode control have been proposed to deal with uncertain fractionalorder chaotic systems [27]. The timeoptimal control problem for a class of fractionalorder systems was proposed [28]. In addition, robust controller design problem for a class of fractionalorder nonlinear systems with timevarying delays was investigated [29] and state feedback \(H_{\infty}\) control of commensurate fractionalorder systems was studied [30].
Backstepping design method has been widely applied in stabilizing a general class of integerorder nonlinear systems. Backstepping design offers a choice of design tools for accommodation of uncertain nonlinearities [31]. It is well known that the backstepping design has been reported for nonlinear systems in strictfeedback form or triangular form [31–36]. Systematic design of globally stable and adaptive controllers for a class of parametric strictfeedback form are investigated by the backstepping design procedure [34]. The overparametrization and partial overparametrization problems were soon eliminated by elegantly introducing the tuning functions [33, 35]. On the other hand, with the aids of this frequency distribute model and the indirect Lyapunov method, the adaptive backstepping control of fractionalorder systems were established [37–39]. As far as we know, there are few results on the generalization of backstepping into fractionalorder systems. It was pointed out that the wellknown Leibniz rule is not satisfied for fractionalorder systems. Then an interesting question arises: when the states of system are not Leibniz rule, how to deal with the stabilization problem through design of tuning functions and adaptive feedback control law? So far, the stabilization problem of fractionalorder nonlinear systems remains an open problem.
In this paper, we investigate the MittagLeffler stabilization problem of a class of fractionalorder nonlinear systems. Compared with the existing results, the main contributions of this paper are as follows: (1) The backstepping design is extended to fractionalorder nonlinear systems with unknown control coefficients, and an adaptive control scheme with tuning functions is proposed. It is proved that the stabilization problem of fractionalorder nonlinear systems can be solved by the designed control scheme. (2) The MittagLeffler stabilization problem is achieved using a systematic design procedure and without any growth restriction on nonlinearities. (3) The controller is designed to ensure that the pseudostate of the fractionalorder system convergence to the equilibrium. (4) Successfully overcoming the difficulty of the fractionalorder system without the Leibniz rule, and the tuning function is constructed to avoid overparameterization.
The remainder of this paper is organized as follows: Section 2 the problem formulation, some necessary concepts and some necessary lemmas are given. In Section 3, as the main part of this note, an adaptive controller and tuning functions are designed by using the backstepping method for fractionalorder nonlinear systems. In Section 4, two numerical simulations are provided to illustrate the effectiveness of the proposed results. Finally, Section 5 concludes this study.
Problem formulation and preliminary results
In this paper, we consider the stabilization problem of the following nonlinear fractional systems:
where \(x=[x_{1},\ldots,x_{n}]^{T}\in R^{n}\), \(D^{q}\) is the Caputo fractional derivative of order \(0< q\leq1\), \(\theta\in R^{p}\) is an unknown constant parameter and \(b_{i},i=1,2,\ldots,n\), are unknown constants, called unknown control coefficients. \(u\in R\) is the control input, \(\varphi _{0}\), \(\beta_{0}\) and the components of \(\varphi_{i}\), \(1\leq i\leq n\) are smooth nonlinear functions in \(R^{n}\) and \(\beta_{0}(x)\neq0\) for all \(x\in R^{n}\).
Remark 1
It is worth pointing out that if let the unknown constants \(b_{i}=1\) (\(i=1,\ldots,n\)) and \(q=1\) in (1), the systems (1) reduces to the wellknown parametric strictfeedback system. Moreover, if \(b_{i}=1\) (\(i=1,\ldots,n\)) and \(\varphi_{0}(x)\) is a constant, then system (1) will become the parametric strictfeedback form of fractionalorder nonlinear system.
Definition 1
([40])
The fractionalorder derivative \(D_{t}^{q}\) (\(q>0\)) of \(g(t)\) in Caputo sense is defined as
where \(n1< q\leq n\in N\).
Remark 2
For simplicity, the symbol \(D^{q}\) is shorted for \({}^{C}_{t_{0}} D_{t}^{q}\), where t is the time.

(1)
If C is a constant, then \(D^{q} C=0\).
Similar to integerorder differentiation, fractionalorder differentiation in Caputo’s sense is a linear operation:

(2)
\(D^{q}(\mu g(t)+\omega h(t))=\mu D^{q} g(t)+\omega D^{q} h(t)\),
where μ and ω are real numbers.

(3)
Leibniz rule:
$$D^{q}\bigl( g(t)h(t)\bigr)=\sum_{r=0}^{\infty} \frac{\Gamma(q+1)}{\Gamma(r+1)\Gamma (qr+1)}D^{qr}g(t)D^{r}h(t). $$
Note that the sum is infinite and contains integrals of fractional order for \(r>[q]+1\).
Remark 3
The wellknown Leibniz rule \(D^{q}(fg) = (D^{q} f)g + f(D^{q} g)\) is not satisfied for differentiation of noninteger orders.
Lemma 1
([13])
Let \(V: D \rightarrow R\) be a continuous positive definite function defined on a domain \(D\subset R^{n}\) that contains the origin. Let \(B_{d} \subset D\) for some \(d > 0\). Then there exist classK functions \(\gamma_{1}\) and \(\gamma_{2}\) defined on \([0,d]\), such that
for all \(x\in B_{d}\). If \(D=R^{n}\), the functions \(\gamma_{1}\) and \(\gamma_{2}\) will be defined on \([0,\infty)\).
Lemma 2
([7, 11] (MittagLeffler stability))
Let \(x(t)=0\) be the equilibrium point of the fractionalorder system \(D^{q} x=f(x,t),x\in\Omega\), where Ω is neighborhood region of the origin. Assume that there exists a fractional Lyapunov function \(V(t,x(t)):[0,\infty)\times R^{n}\rightarrow R\) and classK functions \(\gamma_{i},i=1,2,3\) satisfying
Then the fractionalorder system is asymptotically MittagLeffler stable. Moreover, if \(\Omega=R^{n}\), the fractionalorder system is globally asymptotically MittagLeffler stable.
Lemma 3
([24])
Let \(x(t)\in R\) be a real continuously differentiable function. Then, for any time instant \(t\geq t_{0}\),
Remark 4
In the case when \(x(t)\in R^{n}\), Lemma 3 is still valid. That is, \(\alpha\in(0,1)\) and \(t \geq t_{0}\), \(\frac {1}{2}D^{\alpha}x^{T}(t)x(t)\leq x^{T}(t)D^{\alpha}x(t)\). In addition, let \(x(t)\in R\) be a real continuously differentiable function. Then, for any \(p = 2^{n} , n \in N\), \(D^{\alpha}x^{p}(t)\leq px^{p1}(t)D^{\alpha}x(t)\), where \(0<\alpha<1 \) (see [7]).
Backstepping design
In this section, we will give the backstepping design procedure of fractionalorder systems.
Theorem 1
The fractionalorder nonlinear system (1) can be asymptotically MittagLeffler stable by the adaptive feedback control
where \(\alpha_{1}(x_{1},\hat{\theta})=\frac{c_{1}}{b_{1}}z_{1}\frac{1}{b_{1}}\hat {\theta}^{T}\varphi_{1}(x_{1})\), and \(c_{1},c_{2},\ldots,c_{n}\) are positive constants. θ̂ is the estimate of the unknown parameter θ, \(\tilde{\theta}=\hat{\theta}\theta\) and update laws
where \(\Gamma=\operatorname{diag}[p_{1},\ldots,p_{m}]>0\) is the gain matrix of the adaptive law.
Proof
The design procedure is recursive. Its ithorder subsystem is stabilized with respect to a Lyapunov function \(V_{i}\) by the design of a stabilizing function \(\alpha_{i}\) and a tuning function \(\tau_{i}\). The update law for the parameter estimate θ̂ and the feedback control u are designed at the final step.
Step 1: Let \(z_{1} =x_{1}\) and \(z_{2}=x_{2}\alpha_{1}\), we rewrite \(D^{q} x_{1}=b_{1}x_{2} + \theta^{T}\varphi_{1}(x_{1})\) as
Choose a Lyapunov function candidate as \(V_{1}=\frac{1}{2}z_{1}^{2}+\frac {1}{2}\tilde{\theta}^{T}\Gamma^{1}\tilde{\theta}\), where \(\tilde{\theta }=\hat{\theta}\theta\) is the parameter estimate error. We have
where
To make \(D^{q} V_{1}\leq c_{1}z_{1}^{2}\), we would choose
However, we retain \(\tau_{1}\) as the first tuning function and \(\alpha_{1}\) as the first stabilizing function. We have
The second term \(b_{1}z_{1}z_{2}\) in \(D^{q}V_{1}\) will be canceled at the next step.
Substituting (11) into (8) yields
Step 2: Let \(z_{3} =x_{3} \alpha_{2}\), we rewrite \(D^{q}x_{2} =b_{2}x_{3} + \theta ^{T}\varphi_{2}(x_{1},x_{2})\) as
Choose a Lyapunov function candidate as follows: \(V_{2} = V_{1} + \frac {1}{2}z_{2}^{2}\). We have
where
Then, to make \(D^{q}V_{2} \leqc_{1} z_{1}^{2} c_{2} z_{2}^{2}\), we would choose
However, we retain \(\tau_{2}\) as the second tuning function and \(\alpha _{2}\) as the second stabilizing function. The resulting \(D^{q}V_{2}\) is
The third term in \(D^{q}V_{2}\) will be canceled at the next step.
Substituting (17) into (14) yields
Step 3: Let \(z_{4} =x_{4}\alpha_{3}\), we rewrite \(D^{q}x_{3} =b_{3}x_{4} + \theta ^{T}\varphi_{3}(x_{1},x_{2},x_{3})\) as
Choose a Lyapunov function as \(V_{3} = V_{2} + \frac{1}{2}z_{3}^{2}\). We have
where
Then, to make \(D^{q}V_{3} \leqc_{1}z_{1}^{2} c_{2}z_{2}^{2}c_{3}z_{3}^{2}\), we would choose
However, we retain \(\tau_{3}\) as the third tuning function and \(\alpha_{3}\) as the third stabilizing function. The resulting \(D^{q}V_{3}\) is
Substituting (23) into (20) yields
Step i (\(i\geq2\)): Let \(z_{i+1} =x_{i+1}\alpha_{i}\), we rewrite \(D^{q}x_{i} =b_{i}x_{i+1}+\theta^{T}\varphi_{i}(x_{1},\ldots,x_{i})\) as
Choose a Lyapunov function of the form \(V_{i}= V_{i1} + \frac {1}{2}z_{i}^{2}\). Then
where
Then, to make \(D^{q}V_{i}\leq\sum_{k=1}^{i}c_{k}z_{k}^{2}\), we would choose
However, we retain \(\tau_{i}\) as the ith tuning function and \(\alpha_{i}\) as the ith stabilizing function. The resulting \(D^{q}V_{i}\) is
Substituting (29) into (26) yields
Step n: With \(z_{n} =x_{n} \alpha_{n1}\), we rewrite \(D^{q}x_{n} =\varphi _{0}(x)+\theta^{T}\varphi_{n}(x)+b_{n}\beta_{0}(x)u\) as
and we now design the Lyapunov function as \(V_{n}= V_{n1} + \frac {1}{2}z_{n}^{2}\); we have
To eliminate \(\hat{\theta}  \theta\) from \(D^{q}V_{n}\) we choose the update law
we rewrite \(D^{q}V_{n}\) as
Finally, we choose
We have
Substituting (36) into (32) yields
According to Lemma 1, for the Lyapunov function \(V_{n}\), there exist classK functions \(\gamma_{1}\) and \(\gamma_{2}\) such that \(\gamma _{1}( \Vert \eta \Vert )\leq V_{n}(\eta)\leq\gamma_{2}( \Vert \eta \Vert )\) where \(\eta=[z_{1},\ldots,z_{n},\tilde{\theta}]\).
Unless \(z_{i}=0\), we have \(D^{q}V_{n}\leq0\), thus there exists a classK function \(\gamma_{3}\) such that \(D^{q}V_{n}\leq\gamma_{3}( \Vert \eta \Vert )\).
According to Lemma 2, the zsystem is asymptotically MittagLeffler stable. □
Remark 5
In this paper, we constructed the virtual controllers and tuning functions to deal with the fractional stabilization problem, the backstepping technique has been extended to fractionalorder systems. It should be noted that the MittagLeffler stability implies asymptotic stability [11]. Therefore, the Lyapunov direct method can be applied to obtain the asymptotical stability of the closedloop system.
Simulation results
In this section, two examples are given to verify the effectiveness of the proposed scheme.
Example 1
We consider the following fractionalorder nonlinear system:
where \(b_{1}=3\), \(\theta=2\), \(\varphi_{1}=2x_{1}^{2}\), \(\varphi _{3}=x_{1}^{2}x_{2}\sin(x_{1})\), \(\Gamma=1\) and we choose \(q=0.96\).
Step 1: Let \(z_{1} =x_{1}\) and \(z_{2}=x_{2}\alpha_{1}\), we rewrite \(D^{q} x_{1}=3x_{2}+2x_{1}^{2} \) as
choose the Lyapunov function \(V_{1}=\frac{1}{2}z_{1}^{2}+\frac{1}{2}(\hat {\theta}2)^{T}\Gamma^{1}(\hat{\theta}2)\). Then
where \(\tau_{1}=x_{1}^{3}\). Meanwhile, we choose
Then
The second term \(3z_{1}z_{2}\) in \(D^{q}V_{1}\) will be canceled at the next step.
Substituting (42) into (40) yields
Step 2: Since \(z_{2} =x_{2} \alpha_{1}\), we have
Choose the Lyapunov function \(V_{2} = V_{1} + \frac{1}{2}z_{2}^{2}\). Then
Then, to make \(D^{q}V_{3} \leqk_{1}z_{1}^{2} k_{2}z_{2}^{2}\), we would choose
In this simulation, \(k_{1}=3\), \(k_{2}=2\). The results for the initial state condition \(x_{1}(0)=1, x_{2}(0)=1\) are given in Figures 13.
Example 2
We consider the following fractionalorder nonlinear system:
where \(b_{1}=b_{2}=1\), \(\theta=2\), \(\varphi_{1}(x_{1})=x_{1}^{2}\), \(\varphi _{2}(x_{1},x_{2})=x_{1}x_{2}\) and \(\Gamma=1\), we choose \(q=0.96\).
Step 1: Let \(z_{1} =x_{1}\) and \(z_{2}=x_{2}\alpha_{1}\), we rewrite \(D^{q} x_{1}=x_{2} + 2x_{1}^{2}\) as
Choose the Lyapunov function \(V_{1}=\frac{1}{2}z_{1}^{2}+\frac{1}{2}(\hat {\theta}2)^{T}(\hat{\theta}2)\). Then
we choose
we arrive at
The second term \(z_{1}z_{2}\) in \(D^{q}V_{1}\) will be canceled at the next step. Notice that \(D^{q}z_{2} =D^{q}x_{2}  D^{q} \alpha_{1}\), that is,
Let \(z_{3}=x_{3}\alpha_{2}\), choose a Lyapunov function as \(V_{2} = V_{1} + \frac {1}{2}z_{2}^{2}\). Then
where \(D^{q}\hat{\theta}=\tau_{2}=\tau_{1}\frac{z_{2}}{\hat{\theta}2}D^{q}(\hat {\theta}^{T}x_{1}^{2})\). Then, to make \(D^{q}V_{2} \leqk_{1} z_{1}^{2} k_{2} z_{2}^{2}+z_{2}z_{3}+(\hat{\theta}2)^{T}(D^{q}\hat{\theta}\tau_{2})\), we would choose
Choose the Lyapunov candidate function \(V_{3}=V_{2}+\frac{1}{2}z_{3}^{2}\). Then
where \(D^{q}\hat{\theta}=\tau_{3}=\tau_{2}\frac{z_{3}}{\hat{\theta}2}D^{q}(\hat {\theta}^{T}x_{1}x_{2})\), we choose
we obtain
In this simulation, \(k_{1}=k_{2}=k_{3}=1\). The results for the initial state condition \(x_{1}(0)=1\), \(x_{2}(0)=2\), \(x_{3}(0)=1\) are given in Figures 46.
Conclusions
The problem of MittageLeffler stabilization has been investigated for a class of fractionalorder nonlinear systems with the unknown control coefficients. The backstepping design scheme is extended to fractionalorder systems, and an adaptive control law is proposed with fractionalorder update laws to achieve an asymptotical MittagLeffler stabilization for the closeloop system, and the tuning function is constructed to avoid overparameterization. Finally, the effectiveness of the proposed method has been verified by some simulation examples.
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Acknowledgements
The work is supported in part by the National Natural Science Foundation of China (No. 11661065) and the Scientific Research Fund of Jiangxi Provincial Education Department (GJJ171135, GJJ151264, GJJ151267, GJJ161261).
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Keywords
 fractional order
 nonlinear systems
 backstepping
 adaptive control
 tuning function