Indirect fuzzy sliding mode control with varying boundary layer via time-variant sliding function

Abstract

In this paper, a new indirect fuzzy sliding mode control with varying boundary layer is presented for uncertain nonlinear systems via time-variant sliding function in the presence of external disturbances. Based on rejection parameter and rejection regulator, a time-variant sliding surface as an adaptive filter is defined. This time-variant sliding surface based on an adjustable rejection parameter filters all the un-modelled frequencies. An adjustable boundary layer width, break-frequency bandwidth and the consequent parameters in fuzzy rules are tuned to attenuate the effects of the system uncertainties and un-modelled frequencies, efficiently. Also, the chattering phenomenon is completely eliminated. Three theorems and one lemma, which facilitate design of the proposed controller, are proved. Two simulation examples are presented to illustrate the performances and the advantages of the proposed method.

Keywords

Boundary layer break-frequency bandwidth fuzzy approximation rejection regulator Sliding mode control

1 Introduction

Sliding mode control has been used as an efficient method to design the robust controller for uncertain nonlinear systems in the presence of external disturbances. In the sliding mode techniques, the proper transformation of tracking errors are introduced to generalize errors. Therefore, a tracking problem can be transformed into an equivalent first order stabilization problem [11]. The main objectives in favour of sliding mode control applications are order reduction, decoupling design procedure, disturbance rejection, insensitivity to parameter variations and simple implementation by means of power converters for Electric Drives [12]. The Electro- mechanical applications of SMC, majority in control and estimation methods of automotive industry, are presented in [13]. A nonlinear sliding mode control approach, based on all aerodynamic forces and moments for a Quad rotor Vertical Take-Off and Landing (VTOL) type of Unmanned Aerial Vehicles (UAVs) in order to stabilize its vertical flight dynamics is designed [2]. The tracking performance of a helical desired trajectory is evaluated for the proposed controlled Quad rotor craft. An extended of a reliable assisted remote control for a four-rotor miniature aerial robot, guaranteeing the capability of a stable autonomous flight is presented. Many experiments on a newly implemented quad-rotor prototype have been conducted in order to validate the presented theoretical analysis [1].

The sliding mode control employs a discontinuous control to derive the system state to reach and maintain its motion on sliding surface. The discontinuity in the control action provides the chattering and this phenomenon may active the un-modelled frequencies, which are undesirable, in application. To avoid these drawbacks, the boundary layertechnique and the identification methods are usually exploited [4 , 11]. An adaptive sliding mode control approach for the synchronization of the uncertain fractional-order chaotic systems is designed [7]. A direct self-structuring adaptive fuzzy control has been presented for affine nonlinear systems [8]. The difference functions Δf and Δg must be obtained to calculate the control gain parameter [8]. In [5], an adaptive fuzzy wavelet neural sliding mode controller is proposed to design an smooth sliding mode control for a class of high-order nonlinear systems in the presence of uncertainty. The self-structuring algorithm ensures that the number of rules never exceeds a predefined upper bound. The auto-tuning neurons computation for designing the sliding mode control [3] and the fuzzy adjusting method for finding the suitable boundary layer width [6] are used. In [23], an H_∞ output feedback controller for nonlinear network systems is designed. The considered systems are represented by using T-S fuzzy models which have norm-bounded parametric uncertainties. A dynamic output feedback controller design, has been used to control the continues time semi-Markovian and discrete time Markovian jump linear systems with time-varying delay [17, 18]. Also, the output feedback controller has been used to control nonlinear systems which are represented by T-S fuzzy-affine-models [19 , 21]. To control nonlinear stochastic and discrete systems which have actuator faults, a reliable robust static output feedback controller has been designed. The plant is contained parametric uncertainties and the authors have discussed the performance of the controller by analysing and synthesising it, completely [19, 21]. An H_∞ filter design approach for a class of nonlinear time-variant delay systems has been, introduced in [20]. In the sliding mode control literature, only time-invariant sliding equation has been studied, extensively. In this paper, a time-variant sliding equation is used. For this purpose, the rejection regulator based on a parameter that is called "rejection parameter" is defined to choose objectively coefficients of error states in sliding equation. The presented method, based on adaptive laws, adaptive sliding surface and adaptive saturation function rejects the effects of all the un-modelled frequencies and eliminates chattering phenomenon with at least error and fault tolerance. Also, the sliding surface acts as a chain of (n - 1) adaptive low pass filters, in practice. Three theorems and one lemma are proved which facilitate design of an indirect fuzzy sliding mode control with varying boundary via time-variant sliding function.

In recent years, the scientific and industrial specialists have had considerable studies and researches on fuzzy model based nonlinear networked control systems [9]. For instance, design of an adaptive fuzzy controllers to guarantee the tracking performance under the effects of input dead-zone and the constraint of prescribed tracking performance functions for nonlinear networked control systems is presented in an innovation approach [16]. Therefore, a new robust fuzzy tracking control for networked industrial processes based on time-variant sliding surface and H_∞ approach will be considered as the next studies of the authors of this paper.

2 Fuzzy logic system

Before presenting and analysing of the indirect fuzzy sliding mode control with varying boundary layer via time-variant sliding function design, it is necessary to introduce the fuzzy logic system, briefly.

A fuzzy system includes a set of fuzzy rules in the following form $R^{i} : IF x_{1} is A_{1}^{i} and \dots and x_{n} is A_{n}^{i} THEN y^{i} is B^{i} .$ (1)

The fuzzy logic system introduces a nonlinear mapping from $U = U_{1} \times U_{2} \times \dots \times U_{n} \subset ℝ^{n}$ to $ℝ$ . By using of singleton fuzzification, product inference and center average defuzzification, the output function of the fuzzy system is [15] $y (x) = \frac{\sum_{1}^{c} y^{l} (\prod_{j = 1}^{n} μ_{A_{j}^{i}} (x_{j}))}{\sum_{1}^{c} \prod_{j = 1}^{n} μ_{A_{j}^{i}} (x_{j})},$ (2) where the vector $x = [x_{1}, x_{2}, \dots, x_{n}]^{T} \in ℝ^{n}$ with membership function $μ_{A_{j}^{i}} (x_{j})$ and the variable $y \in ℝ$ denote the linguistic input and output variables of the fuzzy logic system, respectively. As a common practice, we assume that μ_{B
ⁱ} (yⁱ) =1. By introducing the fuzzy basis function, the Equation (2) can be rewritten as $y (x) = θ^{T} W (x),$ (3) where θ = [y¹, …, y^c] ^T and $W (x) = \frac{\prod_{j = 1}^{n} μ_{A_{j}^{i}} (x_{j})}{\sum_{1}^{c} \prod_{j = 1}^{n} μ_{A_{j}^{i}} (x_{j})} .$ (4)

In this system, i denotes the number of the fuzzy rules and j indicates the number of input for fuzzy logic system.

3 Sliding mode control and rejection regulator

Consider the n^th order nonlinear dynamical system defined by the following state equation $x^{(n)} = f (x, \dot{x}, \dots, x^{(n - 1)}) + g (x, \dot{x}, \dots, x^{(n - 1)}) u + d .$ (5)

Where f and g are unknown bounded nonlinear functions and $u \in ℝ$ is control input of the system and d is the external bounded disturbance. Let $x = [x_{1}, x_{2}, \dots, x_{n}]^{T} = [x, \dot{x}, \dots, x^{(n - 1)}]^{T} \in ℝ^{n}$ and $x_{d} = [x_{d}, {\dot{x}}_{d}, \dots, x_{d}^{(n - 1)}]^{T} \in ℝ^{n}$ , where x and x_d are observable state vector and desired trajectory of the system, respectively. The control objective is to find a control law such that the state x can track the desired trajectory x_d. Thus, we define $e = x - x_{d} \in ℝ$ and $e = x - x_{d} = {[e, \dot{e}, \dots, e^{(n - 1)}]}^{T}$ . Also, Integral of Absolute error, $IAE = \int_{t_{0}}^{t_{f}} ∣ e (t) ∣ dt$ , is defined as a controller performance index. For achieving the control objective, the tracking error e = x - x_d must be attenuated to an arbitrary small residual tracking error independent of the uncertainties of the system. The sliding mode control is based on defining a sliding surface s = k^Te, in the error space passing. Where k = [k₀, k₁, …, k_n-1] ^T is chosen such that all the roots of the characteristic polynomial: $P (ω) = k_{0} ω^{n - 1} + k_{1} ω^{n - 2} + \dots + k_{n - 1},$ (6) corresponding to the sliding equation s = k^Te are in the open left half plane (Hurwitz polynomial). By choosing the Lyapunov function candidate $V = \frac{1}{2} s^{2}$ with V (0) =0, an equivalent control u_eq is given such that each state Lyapunov condition holds for system stability (sliding condition) [11] $s \dot{s} \leq - η ∣ s ∣, or \dot{s} sgn (s) \leq - η, η > 0 .$ (7)

An alternative definition of the sliding function is [10] $s = {(\frac{d}{dt} + λ)}^{n - 1} .$ (8)

Designing the sliding mode control has two steps. The first step is to select the parameter λ such that all the un-modelled frequencies of the system are rejected. The other step is to find the control law such that the reaching condition (7) is satisfied. By using the Laplace transformation, the linear differential equation (8) can be considered as a chain of (n - 1) first-order low-pass filters with break-frequency bandwidth λ. Such that, s and e are input and output of the filters, respectively and p is Laplace operator. The parameter λ should be selected such that all the un-modelled frequencies of the system are rejected. Therefore, if $G (p) = \frac{1}{λ + p}$ is a first order low-pass filter corresponding to the Equation (8) then clearly, in order to reject all the un-modelled frequencies, λ should satisfy the following inequality:

$\begin{matrix} λ < p, or λ < ν, \\ ν = \inf {σ_{uf} ∣ σ_{uf} are un - modelled frequencies} . \end{matrix}$ (9)

In the following, a theorem is presented to choose the coefficients of error states in sliding equation, objectively.

Theorem 1. Consider the plant (5). Let P (ω) be characteristic polynomial of sliding equation defined by Equation (8) and If ω_i, i = 1, 2, …, n - 1, are the roots of the polynomial (6),

$\begin{matrix} ν = \inf {σ_{uf} ∣ σ_{uf} are un - modelled frequencies}, \\ ω^{*} = Max {P (ω_{i}) = 0, ω_{i} < 0}, \end{matrix}$ (10)and δ is an arbitrary positive real number (rejection parameter) such that δ < ν then by choosing ω^* = ν - δ and $k_{j} = (\begin{matrix} n - 1 \\ j \end{matrix})$ (ω^*) ^j for j = 0, 1, …, n - 1, we have λ < ν, i.e., the first-order low-pass filters with break-frequency bandwidth λ, as defined in (8), rejects all the un-modelled frequencies of the system [22].

We define the rejection regulator in following form: $R_{j}^{n} (δ) = (\begin{matrix} n - 1 \\ j \end{matrix}) (ν - δ)^{j} j = 0, 1, \dots, n - 1,$ (11) where δ is named rejection parameter. Recall that δ and ν are the same notations as used in Theorem 1. It is also assumed that $R_{0}^{1} (δ) = 0$ . In most practical systems the nonlinear functions f and g are unknown. Now let $\hat{f}$ and $\hat{g}$ be the estimation of functions f and g, respectively. In order to design a controller such that $\dot{s} sgn (s) \leq - η$ , we need the following assumptions.

Assumption I. The upper bounds F, G, D and G_L, are known, such that:

$\begin{matrix} | Δ f | < F, | Δ g | < G, | d | < D, \\ 0 < G_{L} (x) \leq g (x) \end{matrix}$ (12) where $Δ f = f - \hat{f}$ and $Δ g = g - \hat{g}$ .

For notation simplicity, let $M (x, δ) = | x_{d}^{(n)} - \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j)} | .$ (13)

Theorem 2. Consider the plant (5). If the values F, G, D and M are defined as the above, and the sliding surface $s = \sum_{j = 0}^{n - 1} R_{j}^{n} (δ) e^{(n - j - 1)}$ are given, the sliding control law $u = {\hat{g}}^{- 1} [- \hat{f} + x_{d}^{(n)} - \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j)} - T sgn (s)],$ (14) where $T \geq \frac{\hat{g} (x)}{G_{L} (x)} [F + \frac{(M (x, δ) + | \hat{f} |) G}{\hat{g} (x)} + D + η],$ (15) guarantees the sliding condition $\dot{s} sgn (s) \leq - η$ .

Proof. Consider the sliding surface $s = \sum_{j = 0}^{n - 1} R_{j}^{n} (δ) e^{(n - j - 1)} = e^{(n - 1)} + \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j - 1)},$ (16) thus we have $\dot{s} = e^{(n)} + \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j)},$ (17) or

$\begin{matrix} \dot{s} & = & x^{(n)} - x_{d}^{(n)} + \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j)} \\ = & f + g [{\hat{g}}^{- 1} (- \hat{f} + x_{d}^{(n)} - \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j)})] \\ - g {\hat{g}}^{- 1} T sgn (s) + d . \end{matrix}$ (18)

In other words

$\begin{matrix} \dot{s} & = & Δ f - (g {\hat{g}}^{- 1} - 1) \hat{f} + (g {\hat{g}}^{- 1} - 1) x_{d}^{(n)} - (g {\hat{g}}^{- 1} - 1) \\ \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j)} - g {\hat{g}}^{- 1} T sgn (s) + d . \end{matrix}$ (19)

Therefore, we have

$\begin{matrix} \dot{s} & = & Δ f - (g {\hat{g}}^{- 1} - 1) \hat{f} \\ + (g {\hat{g}}^{- 1} - 1) (x_{d}^{(n)} - \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j)}) \\ - g {\hat{g}}^{- 1} T sgn (s) + d, \end{matrix}$ (20)

Now, Equations (15) and (20) yield

$\begin{matrix} \dot{s} sgn (s) \\ = [Δ f - (g {\hat{g}}^{- 1} - 1) \hat{f} \\ + (g {\hat{g}}^{- 1} - 1) (x_{d}^{(n)} - \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j)}) + d] sgn (s) \\ - g {\hat{g}}^{- 1} T \\ \leq [Δ f - (g {\hat{g}}^{- 1} - 1) \hat{f} \\ + (g {\hat{g}}^{- 1} - 1) (x_{d}^{(n)} - \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j)}) + d] sgn (s) \\ - \frac{g}{G_{L}} [F + \frac{(M + | \hat{f} |) G}{\hat{g}} + D + η] . \end{matrix}$ (21)

On the other hand, from Assumption (I) we have: $\frac{g}{G_{L}} > 1, ∣ g {\hat{g}}^{- 1} - 1 ∣ \leq G {\hat{g}}^{- 1} .$ (22)

By using the inequalities (22) and the relations (12), we have

$\begin{matrix} \dot{s} sgn (s) & \leq & | Δ f | \\ + G {\hat{g}}^{- 1} (| x_{d}^{(n)} - \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j)} | + | \hat{f} |) \\ + | d | - [F + \frac{(M + | \hat{f} |) G}{\hat{g}} + D + η] \\ \leq & - η, \end{matrix}$ (23) and this completes the proof. □

Lemma 3. If δ and $R_{j}^{n} (δ)$ are rejection parameter and rejection regulator, respectively, then [22] $\frac{\partial^{k}}{\partial δ^{k}} R_{j}^{n} (δ) = (- 1)^{k} \frac{(n - 1)!}{(n - k - 1)!} R_{j - k}^{n - k} (δ) .$ (24)

In the sliding mode control, the abrupt change of sign of the term Tsgn (s) produces chattering. To avoid this drawback, the boundary layer from both sides of the sliding surface s = 0 is usually implemented. If the state of system is out of the boundary layer, then by using the reaching condition the state reaches to the inside of the boundary layer. The fixed boundary layer usually does not guarantee the precise tracking. So, in this paper a varying boundary layer is considered. The saturation function was implemented in sliding mode control law. The tracking precision is not guaranteed by using the saturation function. In this paper, for avoiding to this drawback, an adaptive Hyperbolic Tangent function is used instead of saturation function. in the next section the tuning laws for designing the indirect fuzzy sliding mode control via time-variant sliding equation, are introduced.

4 Design of indirect fuzzy SMC with varying boundary layer

4.1 Analysis

It is very clear that the performance of SMC is sensitive to the coefficients of the sliding equation. For instance, in second order nonlinear systems, if the coefficients are assigned such that slope of sliding surface has a large value, the system gives a fast response due to the large values of the control signal. But the system may become unstable. Conversely, if the slope of the sliding surface has a small value, the system will be more stable. But the system response will become slower due to small values of the control signal. In this paper, by using the rejection parameter, the coefficients of sliding equation are replaced by the rejection regulator. For this purpose, we consider the sliding equation in the following form:

$s = \sum_{j = 0}^{n - 1} R_{j}^{n} (δ (t)) e^{(n - j - 1)}$ (25)

Where δ (t) is time-variant rejection parameter. This equation defines a time-variant sliding equation. By adjusting the rejection parameter, the performance of the overall system can be improved with respect to conventional sliding mode control. From now, the sliding equation is assumed to be time-variant as defined in the Equation (25). In this paper, stability of the closed loop system and the time variant sliding function is guaranteed via Lyapunov approach and they are facilitated by the next theorem. Since in practical systems there are always uncertainties in the dynamical equations, the free models and identification methods are used in sliding mode control approach for improving the performance and avoiding the effects of uncertainties in mathematical models. In this paper, we use the fuzzy system $\hat{f} (x | θ_{f})$ and $\hat{g} (x | θ_{g})$ for approximation of nonlinear functions f and g. The consequent parameters of fuzzy systems will be tuned by tuning laws that are presented in the next theorem. According to universal approximation theorem, the fuzzy systems can be used to approximate to any desired accuracy [15]. Consider the following sliding surface

$s = e^{(n - 1)} + \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j - 1)}$ (26) since

$\begin{matrix} \frac{\partial}{\partial t} (\sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j - 1)}) \\ = \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j)} + \frac{d δ}{dt} \sum_{n - 1}^{j = 1} \frac{\partial}{\partial δ} R_{j}^{n} (δ) e^{(n - j - 1)}, \end{matrix}$ (27) and according to the Lemma 3, we can write

$\begin{matrix} \sum_{j = 1}^{n - 1} \frac{\partial}{\partial δ} R_{j}^{n} (δ) e^{(n - j - 1)} \\ = - (n - 1) \sum_{j = 1}^{n - 1} R_{j - 1}^{n - 1} (δ) e^{(n - j - 1)}, \end{matrix}$ (28) therefore

$\begin{matrix} \dot{s} & = & e^{(n)} + \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j)} \\ - (n - 1) \dot{δ} \sum_{j = 1}^{n - 1} R_{j - 1}^{n - 1} (δ) e^{(n - j - 1)} . \end{matrix}$ (29)

Substituting Equation (5) into Equation (29) yields:

$\begin{matrix} \dot{s} & = & f (x) + g (x) u + d - x_{d}^{(n)} + \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j)} \\ - (n - 1) \dot{δ} \sum_{j = 1}^{n - 1} R_{j - 1}^{n - 1} (δ) e^{(n - j - 1)} . \end{matrix}$ (30)

Now, we define

$\begin{matrix} u & = & \frac{1}{\hat{g} (x | θ_{g})} [- \hat{f} (x | θ_{g}) + x_{d}^{(n)} \\ - \sum_{j = 1}^{n - 1} R_{j}^{n} (δ (t)) e^{(n - j)} - T Φ (σ, s)], \end{matrix}$ (31) where $Φ (σ, s) = \tanh (σ s) .$ (32)

Such that $\hat{f} (x | θ_{f}) = θ_{f}^{T} W (x)$ and $\hat{g} (x | θ_{g}) = θ_{g}^{T} W (x)$ are fuzzy approximations of f and g, respectively. Also, define the parameters $θ_{f}^{★} \in ℝ^{n}$ and $θ_{g}^{★} \in ℝ^{n}$ for the best function approximation as $\begin{matrix} θ_{f}^{★} & = & \arg \min_{θ_{f} \in Ω_{θ_{f}}} {\sup_{x \in Ω_{x}} ‖ f (x) - \hat{f} (x ∣ θ_{f}) ‖}, \\ θ_{g}^{★} & = & \arg \min_{θ_{g} \in Ω_{θ_{g}}} {\sup_{x \in Ω_{x}} ‖ g (x) - \hat{g} (x ∣ θ_{g}) ‖}, \end{matrix}$ where Ω_{θ
_f} = {θ_f : ||θ_f|| ≤ M_{θ_f,Max}} and Ω_{θ
_g} = {θ_g : ||θ_g|| ≤ M_{θ_g,Max}} are constraint sets for θ_f and θ_g, such that M_{θ_f,Max} and M_{θ_g,Max} are specified by designer. If θ_f and θ_f are estimates of $θ_{f}^{★}$ and $θ_{g}^{★}$ , respectively. We define ${\hat{f}}^{★} (x) = (θ_{f}^{★})^{T} W (x)$ and ${\hat{g}}^{★} (x) = (θ_{g}^{★})^{T} W (x)$ . Thus the Equation (30) can be rewritten as

$\begin{matrix} \dot{s} & = & ({\hat{f}}^{★} - \hat{f}) + ({\hat{g}}^{★} - \hat{g}) + ξ_{f} + ξ_{g} u + d \\ - (n - 1) \dot{δ} \sum_{j = 1}^{n - 1} R_{j - 1}^{n - 1} (δ) e^{(n - j - 1)} - T Φ (σ, s), \end{matrix}$ (33)

Where $ξ_{f} = f - {\hat{f}}^{★}$ and $ξ_{g} = f - {\hat{g}}^{★}$ . If we define $θ_{\tilde{f}} = θ_{f}^{★} - θ_{f}$ and $θ_{\tilde{g}} = θ_{g}^{★} - θ_{g}$ then we have

$\begin{matrix} \dot{s} & = & θ_{\tilde{f}}^{T} W + θ_{\tilde{g}}^{T} Wu + ɛ (t) \\ - (n - 1) \dot{δ} \sum_{j = 1}^{n - 1} R_{j - 1}^{n - 1} (δ) e^{(n - j - 1)} - T Φ (σ, s), \end{matrix}$ (34) such that ɛ (t) = ξ_f + ξ_gu + d. Now, a theorem for designing a new indirect fuzzy sliding mode control with varying boundary layer via time-variant sliding function will be presented.

Theorem 4. Consider a nonlinear system and sliding surface described by the Equations (5) and (25), respectively. For the control signal (31) if $\hat{f}$ and $\hat{g}$ are fuzzy approximations of f and g, and the parameres σ, δ, θ_f and θ_g are adjusted by using the following adaptive laws (35), (36), (39) and (40): $\begin{matrix} \dot{θ_{f}} = {\begin{matrix} η_{1} sW & if (‖ θ_{f} ‖ < M_{θ_{f}, Max}) \\ or (‖ θ_{f} ‖ = M_{θ_{f}, Max}, ({sW}^{T} θ_{f} \leq 0)) \\ P_{f} & if (‖ θ_{f} ‖ = M_{θ_{f}, Max}, ({sW}^{T} θ_{f} > 0)) \end{matrix} \end{matrix}$ (35) and $\begin{matrix} \dot{θ_{g}} = {\begin{matrix} η_{2} sWu & if (‖ θ_{g} ‖ < M_{θ_{g}, Max}) \\ or (‖ θ_{g} ‖ = M_{θ_{g}, Max}, ({usW}^{T} θ_{g} \leq 0)) \\ P_{g} & if (‖ θ_{g} ‖ = M_{θ_{g}, Max}, ({usW}^{T} θ_{g} > 0)) \end{matrix} \end{matrix}$ (36) where P_f and P_g are projection operators of θ_f and θ_g as follow $P_{f} = η_{1} sW - \frac{η_{1} {sW}^{T} θ_{f}}{‖ θ_{f} ‖^{2}} θ_{f}, η_{1} > 0,$ (37) $P_{g} = η_{2} usW - \frac{η_{2} {usW}^{T} θ_{g}}{‖ θ_{g} ‖^{2}} θ_{g}, η_{2} > 0,$ (38) and $\dot{σ} = η_{3} \frac{sT}{σ} \tanh (σ s), η_{3} > 0,$ (39) $\dot{δ} = η_{4} sgn (s) \sum_{j = 1}^{n - 1} R_{j - 1}^{n - 1} (δ) e^{(n - j - 1)}, η_{4} > 0,$ (40) then the tracking error will converge to zero asymptotically.

Proof. Consider the Lyapunov function in the following form

$\begin{matrix} V (s, θ_{\tilde{f}}, θ_{\tilde{g}}, t) \\ = \frac{1}{2} s^{2} + \frac{1}{2 η_{1}} θ_{\tilde{f}}^{T} θ_{\tilde{f}} + \frac{1}{2 η_{2}} θ_{\tilde{g}}^{T} θ_{\tilde{g}} + \frac{1}{2 η_{3}} σ^{2} . \end{matrix}$ (41)

By differentiating the Equation (41) with respect to time we have

$\begin{matrix} \dot{V} (s, θ_{\tilde{f}}, θ_{\tilde{g}}, t) \\ = s \dot{s} + \frac{1}{η_{1}} θ_{\tilde{f}}^{T} {\dot{θ}}_{\tilde{f}} + \frac{1}{η_{2}} θ_{\tilde{g}}^{T} {\dot{θ}}_{\tilde{g}} + \frac{1}{η_{3}} σ \dot{σ} . \end{matrix}$ (42)

By replacing the Equation (34) into the Equation (42) one can conclude:

$\begin{matrix} \dot{V} & = & s (θ_{\tilde{f}}^{T} W + θ_{\tilde{g}}^{T} Wu + ɛ (t) \\ - (n - 1) \dot{δ} \sum_{j = 1}^{n - 1} R_{j - 1}^{n - 1} (δ) e^{(n - j - 1)} - T Φ (σ, s)) \\ + \frac{1}{η_{1}} θ_{\tilde{f}}^{T} {\dot{θ}}_{\tilde{f}} + \frac{1}{η_{2}} θ_{\tilde{g}}^{T} {\dot{θ}}_{\tilde{g}} + \frac{1}{η_{3}} σ \dot{σ} \\ = & θ_{\tilde{f}}^{T} (sW + \frac{1}{η_{1}} {\dot{θ}}_{\tilde{f}}) + θ_{\tilde{g}}^{T} (sWu + \frac{1}{η_{2}} {\dot{θ}}_{\tilde{g}}) \\ - s (n - 1) \dot{δ} \sum_{j = 1}^{n - 1} R_{j - 1}^{n - 1} (δ) e^{(n - j - 1)} \\ + s ɛ (t) + (\frac{1}{η_{3}} σ \dot{σ} - sT Φ (σ, s)) . \end{matrix}$ (43)

Since ${\dot{θ}}_{\tilde{f}} = - {\dot{θ}}_{f}$ and ${\dot{θ}}_{\tilde{g}} = - {\dot{θ}}_{g}$ , therefore by substituting the Equations (35)–(40) into the Equation (43) we have: $\dot{V} \leq - ∣ s ∣ (n - 1) {[\sum_{j = 1}^{n - 1} R_{j - 1}^{n - 1} (δ) e^{(n - j - 1)}]}^{2} + | s | | ɛ (t) |$ (44)

If e (0) is bounded then according to the Theorem 1, $s = e^{(n - 1)} + \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j - 1)}$ with Hurwitz characteristic polynomial is bounded. So e (t) and $\sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j - 1)}$ are bounded. Therefore, s ∈ L_∞. By assuming that $∣ \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) e^{(n - j - 1)} ∣ \leq \frac{M_{δ}}{\sqrt{n - 1}}$ , the relation (44) can be written as $\dot{V} \leq - ∣ s ∣ M_{δ}^{2} - | s | | ɛ (t) |$ (45)

By integrating both sides of the Equation (45) and some manipulation, we have $\int_{0}^{t} | s (τ) | d τ \leq \frac{1}{M_{δ}^{2}} [V (0) + V (t)] + \frac{| s |}{M_{δ}^{2}} \int_{0}^{t} | ɛ (τ) | d τ$ (46)

According to the universal approximation theorem and the assumption of bounded external disturbances, x is bounded and ɛ ∈ L₁. If ɛ ∈ L₁ then from the Equation (46) we have s ∈ L₁. Moreover, the adaptive laws in the Equations (35) and (36) guarantee that θ_f ∈ Ω_{θ
_f} and θ_g ∈ Ω_{θ
_g}. This implies that all variables on the right hand side of the Equation (34) are bounded. Therefore, $\dot{s} \in L_{\infty}$ . Using the corollary of Barbalat’s Lemma [10], we have $\lim_{t \to \infty} | s | = 0$ . From the Equation (44) it follows that sɛ is of the order of the minimum approximation error ɛ. If ɛ tends to zero, form the Equation (45) it is clear that $\dot{V} \leq - ∣ s ∣ M_{δ}^{2} < 0$ . However, if ɛ ≠ 0 then $\lim_{t \to \infty} | s | = 0$ implies that $\dot{V} \leq 0$ . When |s|→0 the only possible condition for $\dot{V} = 0$ , in the Equations (25) and (44), is e = 0. In this case, the tracking error is or approach to zero once again, and this completes the proof. □

Figure 1 shows pictorially the effect of variation of σ on the boundary layer width. As σ gets larger the width of boundary layer becomes narrower.

Fig. 1

Variation of boundary layer width.

5 Simulation result

In this section a couple of examples are presented to illustrate the validity of the proposed method. The proposed method presents the better tracking performance than the conventional methods [4, 14]. Without loss of generality, we consider ν = 20 for both of examples and the performance index is the Integral of Absolute Error, $IAE = \int_{t_{0}}^{t_{f}} ∣ e (t) ∣ dt$ .

Example 1. The dynamic equation of the inverted pendulum system is

$\begin{matrix} \ddot{x} & = & \frac{g sin (x) - \frac{mL {\dot{x}}^{2} cos (x) sin (x)}{M + m}}{L (\frac{4}{3} - \frac{m cos (x)^{2}}{M + m})} \\ + [\frac{\frac{cos (x)}{M + m}}{L (\frac{4}{3} - \frac{m cos (x)^{2}}{M + m})}] u + d \end{matrix}$ (47) where, g = 9.8 m/s², M_c = 1 kg, m = 0.1 kg are the acceleration of gravity, the mass of the cart and the mass of the pole, respectively, and L = 0.5 m is half length of the pole. The control objective is to maintain the system states for tracking the reference input $x_{d} = \frac{π (sin (t) + 0.3 sin (3 t))}{10}$ .

For demonstrating the performance of the presented method, the Gaussian membership function for fuzzy approximation of f are chosen as $\begin{matrix} μ_{NE} = \exp [- ((x_{i} + \frac{π}{12}) \frac{π}{12})^{2}], \\ μ_{ZE} = \exp [- ((x_{i}) \frac{π}{12})^{2}], \end{matrix}$ (48) $μ_{PO} = \exp [- ((x_{i} - \frac{π}{12}) \frac{π}{12})^{2}],$ and $\begin{matrix} μ_{NE} = \exp [- ((x_{i} + \frac{π}{6}) \frac{π}{10})^{2}], \\ μ_{ZE} = \exp [- ((x_{i}) \frac{π}{10})^{2}], \end{matrix}$ (49) $μ_{PO} = \exp [- ((x_{i} - \frac{π}{6}) \frac{π}{10})^{2}],$ are applied for fuzzy approximation of g. The initial consequent parameters of fuzzy rules are chosen randomly in the interval [-5, 5] and [1, 2] for f and g, respectively. We select η = 0.01, M_{θ_f,Max} = 30 and M_{θ_g,Max} = 20. The initial other parameter values are selected as $x (0) = [\frac{- π}{60}, 0], δ (0) = 15 and T (0) = 8 .$

The simulation results with random disturbances in the interval [-1.5, 1.5] are shown in Figs. 2 and 3. The Fig. 2 shows that the state can trace the desired trajectory quicker than in [4] and [14], without increasing amplitudes of the control signals. Also, the control input chattering does not occur and the effects of all un-modeled frequencies are eliminated. The Fig. 3 indicates the variation of rejection and boundary layer parameters, for updating the time-variant sliding function and the boundary layer width, respectively. The results of this example are compared with those of other methodologies in Table 1 to illustrate the effectiveness of the proposed method. According to the Table 1, the designed control laws in this paper reduced the control signal amplitude and IAE significantly, in compare with [4, 14].

Fig. 2

Tracking result and control signal for Example 1.

Fig. 3

Rejection and boundary layer parameter Example 1.

Table 1

Benchmarks for controller performance in Example 1

Criterion	Number of fuzzy rule	Control signal u	IAE
IAFSMC [14]	25	-8.406 ≤ u ≤ 8.845	0.40
AFSMC [4]	36	-7.2013 ≤ u ≤ 8.011	0.2745
The proposed method	9	-3.9863 ≤ u ≤ 7.7294	0.1956

Example 2. Consider the Duffing Forced-oscillation system. The dynamic equations of such system are

$\begin{matrix} {\dot{x}}_{1} = x_{2}, \\ {\dot{x}}_{2} = - 0.1 x_{2} - x_{1}^{3} + 12 cos (t) + u (t) + d (t) . \end{matrix}$ (50)

Where the disturbance d (t) is a square wave with amplitude ±1.5. This system is chaotic without control. The control objective is to maintain the system states to track the reference input x_d = sin(t). For demonstrating the performance of the presented method, the Gaussian membership function for system states are chosen as, $\begin{matrix} μ_{NM} = \exp [- {(\frac{x_{i} + 1}{0.7})}^{2}], \\ μ_{NS} = \exp [- {(\frac{x_{i} + 0.5}{0.7})}^{2}], \\ μ_{ZE} = \exp [- {(\frac{x_{i}}{0.7})}^{2}], \\ μ_{NE} = \exp [- {(\frac{x_{i} - 1}{0.7})}^{2}], \end{matrix}$ (51) $\begin{matrix} μ_{PM} = \exp [- {(\frac{x_{i} - 1}{0.7})}^{2}], \\ μ_{PS} = \exp [- {(\frac{x_{i} - 0.5}{0.7})}^{2}] . \end{matrix}$

The initial consequent parameters of fuzzy rules are chosen randomly in the interval [-12, 12]. We select η = 0.1 and M_{θ_f,Max} = 30. The other initial parameter values are selected as $\begin{matrix} x (0) & = & [2, 2], δ (0) = 17.75, \\ σ (0) & = & 4 and T (0) = 14 . \end{matrix}$

The simulation results are shown in Figs. 4 and 5. The Fig. 4 shows that the state can trace the desired trajectory quicker than in [4] and [4] without increasing amplitudes of the control signals. Also, the control input chattering does not occur and the effects of all the un-modeled frequencies are eliminated. The Fig. 5 indicates the variation of rejection and boundary layer parameters for updating the time-variant sliding function and the boundary layer width, respectively. The results of this example are compared with those of other methodologies in Table 2 to illustrate the effectiveness of the proposed method. According to the Table 2, the designed control laws in this paper reduced the control signal amplitude and IAE significantly, in compare with [4, 14].

Fig. 4

Tracking result and control signal for Example 2.

Fig.5

Rejection and boundary layer parameter Example 2.

Table 2

Benchmarks for controller performance in Example 2

Criterion	Number of fuzzy rule	Control signal u	IAE
IAFSMC [14]	39	-15.014 ≤ u ≤ 15.17475	4.78
AFSMC [4]	36	-15 ≤ u ≤ 15	2.910
The proposed method	25	-14.2283 ≤ u ≤ 13.56454	2.7594

6 Conclusions

In this paper, an indirect fuzzy sliding mode control with varying boundary layer via time-variant sliding function is proposed for a class of uncertain nonlinear systems. The sliding surface is defined based on the adjustable terminologies rejection parameter and rejection regulator. The consequent parameters in fuzzy rules, boundary layer parameter and rejection regulator are tuned independent of a priori knowledge of the system. By using the proposed controller, based on the time-variant sliding equation and varying boundary layer, all the un-modeled frequencies are adaptively filtered. Also, the effects of the fuzzy approximation errors and the external disturbance on tracking performances are attenuated efficiently and the control input chattering does not occur. Simulation examples illustrate the superior performance and the advantages of the proposedmethod.

References

Adigbli

, Grand

, Mouret

J.B.

and Doncieux

, Nonlinear attitude and position control of a micro quadrotor using sliding mode and backstepping techniques, European Micro Air Vehicle Conference and Flight Competition (EMAV2007) (2007), 1–9.

Ammar

N.B.

, Bouallegue

and Haggege

, Modeling and sliding mode control of a quadrotor unmanned aerial vehicle, 3rd International Conference on Automation, Control Engineering Computer Science (ACECS’16), Proceedings of Engineering and Technology (PET), 2016, pp. 834–840.

Chang

W.D.

, Hwang

R.C.

and Hsieh

J.G.

, Application of an auto-tuning neuron to sliding mode control, IEEE Trancaction on System, Man and Cybernetics-Part C 32(4) (2002), 517–529.

H.F.

, Wong

Y.K.

and Rad

A.B.

, Adaptive fuzzy sliding mode control with chattering elimination for nonlinear SISI systems, Simulation Modeling Practice and Theory 17(7) (2009), 1199–1210.

Kahkeshin

M.S.

, Sheikholeslam

and Zekri

, Design of adaptive fuzzy wavelet neural sliding mode controller for uncertain nonlinear systems, ISA Transactions 52(3) (2013), 342–350.

Lee

, Kim

and Kang

H.J.

, A new sliding mode control with fuzzy boundary layer, Fuzzy Sets and Systems 120(1) (2001), 135–143.

Mohammadzadeh

and Ghaemi

, A modified sliding mode approach for synchronization of fractional-order chaotic/hyperchaotic systems by using new self-structuring hierarchical type-2 fuzzy neural network, Neurocomputing 191 (2016), 200–213.

Phan

P.A.

and Gale

T.J.

, Direct adaptive fuzzy control with a self-structuring algorithm, Fuzzy Sets and Systems 159(8) (2008), 871–899.

Qiu

, Gao

and Ding

S.X.

, Recent advances on fuzzy model-based nonlinear networked control systems: A survey, IEEE Transactions on Industrial Electronics 63(2) (2016), 1207–1217.

10.

Slotine

J.J.

and Li

, Applied Nonlinear Control, Englewood Cliffs, NJ, 1991.

11.

Tzafestas

S.G.

and Rigatos

G.G.

, Design and stability analysis of a new sliding mode fuzzy logic controller of reduced complexity, Machine Intelligence and Robotic Control 1(1) (1999), 27–41.

12.

Utkin

V.I.

, Sliding mode control design principles and applications to electric drives, IEEE Transactions on Industrial Electronics 40(1) (1993), 23–36.

13.

Utkin

V.I.

and Chang

H.C.

, Sliding mode control in electro-mechanical systems, Mathematical Problems in Engineering 8 (2002), 451–473.

14.

Wang

, Rad

A.B.

and Chan

P.T.

, Indirect adaptive fuzzy sliding mode control: Part I: Fuzzy switching, Fuzzy Sets and Systems 122 (2001), 21–30.

15.

Wang

L.X.

, A course in fuzzy systems and control, Prentice-Hall. Upper saddle River, NJ, 1997.

16.

Wang

, Qiu

, Yin

, Gao

, Fan

and Chai

, Performance-based adaptive fuzzy tracking control for networked industrial processes, IEEE Transactions on Cybernetics 46(8) (2016), 1760–1770.

17.

Wei

, Qiu

and Fu

, Mode-dependent nonrational output feedback control for continuous-time semi-Markovian jump systems with time-varying delay, Nonlinear Analysis: Hybrid Systems 16(1) (2015), 52–71.

18.

Wei

, Qiu

, Karimi

H.R.

and Wang

New results on H1 dynamic output feedback control for Markovian jump systems with time-varying delay and defective mode information, Optimal Control Applications and Methods 35(6) (2014), 656–675.

19.

Wei

, Qiu

, Lam

H.K.

and Wu

, Approaches to T-S fuzzyaffine-model-based reliable output feedback control for nonlinear Itô stochastic systems, IEEE Transactions on Fuzzy Systems PP(99) (2016), 1–14.

20.

Wei

, Qiu

, Shi

and Lam

H.K.

, A new design of H-infinity piecewise filtering for discrete-time nonlinear timevarying delay systems via T-S fuzzy affine models, IEEE Transactions on Systems, Man, and Cybernetics: Systems PP(99) (2016), 1–14.

21.

Wei

, Qiu

, Shi

and Karimi

H.R.

, Reliable output feedback control of discrete-time fuzzy affine systems with actuator faults, IEEE Transactions on Circuits and Systems-I: Regular Papers 64(1) (2017), 170–181.

22.

Yarahmadi

, Karbassi

S.M.

and Mirzaei

, Robust wavelet sliding-mode control via time-variant sliding function, IEICE Transaction Fundamentals E93(1) (2010), 1181–1189.

23.

Zhang

, Feng

, Qiu

and Zhang

W.A.

, T-S fuzzy-modelbased piecewise H1 output feedback controller design for networked nonlinear systems with medium access constraint, Fuzzy Sets and Systems 248(1) (2014), 86–105. http://www.jifs.prosemanager.com/submitpaper.asp