Adaptive Neural Network Sliding Mode Control for Nonlinear Singular Fractional Order Systems with Mismatched Uncertainties

: This paper focuses on the sliding mode control (SMC) problem for a class of uncertain singular fractional order systems (SFOSs). The uncertainties occur in both state and derivative matrices. A radial basis function (RBF) neural network strategy was utilized to estimate the nonlinear terms of SFOSs. Firstly, by expanding the dimension of the SFOS, a novel sliding surface was constructed. A necessary and sufﬁcient condition was given to ensure the admissibility of the SFOS while the system state moves on the sliding surface. The obtained results are linear matrix inequalities (LMIs), which are more general than the existing research. Then, the adaptive control law based on the RBF neural network was organized to guarantee that the SFOS reaches the sliding surface in a ﬁnite time. Finally, a simulation example is proposed to verify the validity of the designed procedures.


Introduction
Fractional order systems (FOSs) have been developed greatly in the past few decades. When the fractional order α is equal to the integer, the FOSs reduce to integer order systems. Therefore, FOSs have more extensive applications in real life, such as image processing [1], economics [2], and robotics [3]. Stability is fundamental to FOSs. A basic theorem of asymptotic stability for FOSs is first proposed in [4] with fractional order 0 < α < 2. But it is difficult to use this theory to design controllers to make the FOSs stable in practical application. So, many scholars have carried out further research on FOSs. In [5], the Mittag-Leffler stability definition of FOSs is introduced and the fractional order Lyapunov method is presented. Sabatier et al. propose the linear matrix inequalities (LMIs) condition of asymptotic stability of FOSs in [6]. Necessary and sufficient conditions of robust stability and stabilization of fractional order interval systems with fractional order α: 0 < α < 1 case and 1 < α < 2 case are developed in [7,8], respectively. Zhang et al. in [9] present a D-stability based LMI condition, which does not include complex variables. Liang et al. in [10] introduce the bounded real lemma of FOSs to solve H ∞ control problem. In [11], Shen et al. analyze the nonlinear FOSs and put forward some significant results.
In control systems, state variables often cannot represent physical variables in a natural way to provide a mathematical model. This leads to a singular system model [12]. Therefore, singular systems are necessary to be studied. In [13], the robust stabilization of uncertain singular time-delay systems is considered, and sufficient conditions are given to ensure the system is regular, impulse-free, and asymptotically stable. In [14], by designing the controller to convert the singular system into a normal system, Ren et al. investigate the problem of guaranteed cost control for uncertain singular

•
A new necessary and sufficient condition for admissibility of SFOSs is presented, which contains no equality constraints.

•
By expanding the dimension of the SFOS, a new sliding surface is constructed.

•
Based on RBF neural network method, f (t, x(t)) is constructed to estimate the nonlinear term f (t, x(t)). The restricted assumption that f (t, x(t)) is norm bounded in [41] is removed.

•
The adaptive control law is exploited to guarantee that the SFOS reaches the sliding surface in a finite time.
The paper is arranged as follows: The preliminaries are provided in Section 2. In Section 3, the sliding mode control scheme is presented. In Section 4, a simulation example is given to prove the validity of the proposed method. Finally, in Section 5, the conclusion is obtained.
Throughout this paper, R n denotes the n-dimensional real vectors. R n×m is the m by n real matrices. M T is the transpose of matrix M. Tr(X) denotes the trace of matrix X. X > 0(< 0) means that the matrix X is positive (negative) definite, sym(Y) denotes the expression Y + Y T , * indicates the symmetric part of a matrix, a = sin aπ 2 , b = cos aπ 2 , || · || denotes the Euclidean norm of vectors. The αth order Caputo fractional derivative of f (t) is defined as where n − 1 < α < n, n ∈ N + and Γ(·) is the Gamma function.

Preliminaries
Consider the nonlinear SFOS, where x(t) ∈ R n is the system state, u(t) ∈ R l is the control input, and α (0 < α < 1) is the fractional order, E ∈ R n×n is a singular matrix such that rank(E) = r < n. A ∈ R n×n , B ∈ R n×l are constant matrices. ∆A, ∆E ∈ R n×n are uncertain matrices, which is assumed to be of the form where U, V 1 , V 2 are known constant matrices. The uncertain matrix F(σ) satisfies F T (σ)F(σ) ≤ I, where σ ∈ Φ, Φ is a compact set in R. Besides, the unknown function f (t, x(t))∈ R l represents the nonlinear term. Let where In order to design the sliding mode controller for system (2), we introduce following facts and lemmas. Considering the unforced SFOS (3) is denoted as the triple (E, A, α).
Since rank(E) = r, it is easy to obtain that there exist nonsingular matrices M, N ∈ R n×n , such that It is noted that (3) is equivalent to where System (7) is asymptotically stable if and only if there exist two matrices X, Y ∈ R n×n , such that where Λ 1 , Λ 2 , Λ 3 , Λ 4 are real matrices such that Σ + Σ T < 0. Then, Λ 4 is nonsingular and

Lemma 3 ([7]
). There hold if and only if there exists a positive scalar such that where Ω, Γ, Θ, F are given matrices of appropriate dimension, and Ω is symmetric. (3) is admissible if and only if there exist matrices P 1 ∈ R r×r , P 2 ∈ R (n−r)×m and

Proof. [Sufficiency:] If
The 2-2 block in (13) gives hence, one has A 4 is nonsingular. According to [19], SFOS (3) is regular and impulse-free. We define It follows from Lemma 2 that We set P 1 = aX + bY, where X is an symmetric matrix and Y is an antisymmetric matrix. It is obtained that Thus, (10) is equivalent to (8). By Lemma 1,(14) together with (10) implies system (6) is asymptotically stable. We have SFOS (3) is asymptotically stable.
Then, by setting P 1 = aX + bY, we have (10) and (14) hold. Let it follows that Since A 4 is nonsingular, −A 4 A T 4 is negative definite. According to (14), one has where (11) holds.

Remark 1.
Stability conditions obtained in [19,41] involve the unknown antisymmetric matrix X 2 . In effect, these conditions contain an equality constraint that X T 2 = −X 2 . Lemma 4 obtained in this paper does not contain equality constraint. Lemma 4 is more efficient and general than other theorems [19,24,41] because fewer variables are introduced and the complex calculation is avoided successfully.

Main Results
In order to presented SMC scheme for SFOS (2), the following sliding surface is constructed where G = [G 1 G 2 ], G 1 , G 2 ∈ R l×n are given matrices. It is easy to see that G B = G 2 B. We choose the appropriate matrix G 2 so that det(G 2 B) = 0. K = [K 1 0 l×n ] is a real matrix to be designed, K 1 ∈ R l×n . When the SFOSs move on sliding surface, one hasṡ(t) = 0, this together with system (2) giveṡ So the equivalent control law is obtained By substituting (24) into system (2), we have the sliding mode dynamic (25) Letting For notational simplicity, we set G 1 = −B(G 2 B) −1 G 1 , G 2 = I n − B(G 2 B) −1 G 2 . Thus, (25) is rewritten as Remark 2. By choosing the appropriate matrix G 1 , we get that G 1 − G 2 E − G 2 ∆E is nonsingular.
The following theorem is presented to ensure that the sliding mode dynamic is admissible. (26) is admissible if and only if there exist matrices P 1 , P 2 , P 3 ∈ R n×n , Z ∈ R l×n and a scalar > 0, such that (10) and the following LMI hold.
Proof. Under the condition of Theorem 1 and using Schur complement Lemma, (27) is equivalent to Now, by Lemma 3, it is easy to see that Note that Z = K 1 P 1 , thus, (30) is obtained Therefore, by Lemma 4, system (2) is admissible.

Remark 3.
In [14,21], the admissibility problems are investigated for singular systems. However, the proportional-plus derivative state feedback controller is designed so that the system is normalizable. This control method is essentially a normal system solution approach instead of a singular system solution approach. In this paper, the novel SFOS solution approach is proposed.
In what follows, the theorem is proposed make SFOSs satisfy the reaching condition. Besides, an RBF neural network approach is employed to deal with the nonlinear function. The RBF network structure with three hidden layers is shown in the Figure 1. By the RBF neural network method, The nonlinear term f (t, x(t)) is modeled as where W = [ w 1 · · · w m ] T ∈ R m×l is the optimal weight matrix, m is the number of neuron nodes. h(x(t)) ∈ R m is the output of the Gaussian type functions, h(x(t)) = [h 1 (x(t)) h 2 (x(t)) · · · h m (x(t))] T , and where c j = [c j1 c j2 · · · c jn ] T is the vector value of the center of the jth neuron and µ j > 0 is the width of the Gaussian basis function. δ f ∈ R l is the approximation error of the network, which satisfies ||δ f || ≤ δ, δ is a known constant. We use W(t) to estimate W * , and W(t) = W(t) − W represents the estimation error. We set to approximate f (t, x(t)). The estimation error function between f (t, x(t)) and f (t, x(t)) is defined as Theorem 2. System (1) moves to the sliding surface (21) in a finite time by the following adaptive SMC law: where λ is a positive constant, σ is the bound of the norm of the uncertain matrix ∆A, which satisfies ||∆A|| ≤ σ. The adaptive law is chosen as˙ where ω is designed as a positive constant.

Simulation Example
This example is utilized to prove the validity of theorems 1 and 2. We consider uncertain SFOSs (1) with α = 0.6 and The system nonlinearity f (t, x(t)) is assumed to be x 1 sin(x 1 (t)) and G 1 , G 2 are chosen as G 1 = 1 1 1 and G 2 = 1 1 2 , respectively. Now, it is obtained that a set of solutions to the LMIs in (10) and (27) The neural network parameters are selected as m = 5, µ j = 0.2, and c ji is uniformly distributed in [−2, 2]. The state response of system (1) with adaptive SMC law u(t) is displayed in Figure 2. Figure 3 shows the state response of system (1) without SMC law. Compared Figure 2 with Figure 3, it is easy to see that the designed control scheme is effective. Figure 4 shows the surface function s(t). Figure 5 depicts the control input u(t). It is easy to see that the nonlinear term f (t, x(t)) is well estimated by f (t, x(t)) from Figure 6.

Conclusions
This paper investigated the issue of adaptive SMC for mismatched uncertain SFOSs. The new necessary and sufficient condition for the admissibility of SFOSs is developed, which is strict LMIs. The integral sliding mode surface with expanded dimension is constructed so that mismatched uncertainty does not exist in the derivative matrix of the sliding mode dynamic. By RBF neural network method, the adaptive control law is devised to make SFOSs satisfy the reaching condition. The restrictive assumption that the nonlinearity f (t, x(t)) is norm bounded is removed. In the further, the issues of SMC for SFOSs with time delay will be studied.
Author Contributions: Formal analysis, X.Z. and W.H.; Writing-review and editing, X.Z. All authors have read and agreed to the published version of the manuscript.