Robust adaptive fuzzy sliding mode controller for nonlinear uncertain hysteretic systems

Abstract

This paper deals with the problem of adaptive fuzzy control for a class of nonlinear uncertain systems with hysteresis input. Fuzzy logic systems are employed to approximate the unknown nonlinear behaviors, and the sliding mode technique is used to synthesize an adaptive fuzzy controller. A proportional integral control term is adopted to reduce the chattering phenomenon engendered by both sliding mode control technique and hysteretic characteristic of the system. The proposed control scheme ensures the boundedness of all closed-loop signals, and forces the tracking error to converge to zero. The main contribution of this work is the development of a control strategy for a class of nonlinear hysteretic systems subject to external disturbances and uncertainties. Two case studies are given to illustrate and to prove the effectiveness of the presented approach.

Keywords

Nonlinear systems hysteresis fuzzy control adaptive control sliding mode control

Introduction

The hysteresis nonlinearity is a phenomenon that exists in various fields and sciences such as physics, chemistry and mechanics (Beriozkin and Mualem, 2018; Changsh and Jianxin, 2018; Lee et al., 2018; Lim et al., 2018; Sepehri-Amin et al., 2018; Rémond et al., 2018; Weng et al., 2018; Zhang et al., 2018b). This property is undesirable in ferroelectric and ferromagnetic materials, mechanical and electrical systems, rechargeable batteries and many other systems (Agrawal and Bhattacharya, 2018; Dai et al., 2018; Lian et al., 2018; Ma et al., 2018b; Rebai et al., 2015c; Schaler et al., 2018; Zhang et al., 2018a). For this reason, the topic of modeling and control of hysteretic systems have been widely studied over the last years. Several research works have proved that the hysteresis may lead to undesirable performances and even derives the system to instability. Therefore, the investigation of modeling and control of such kind of systems is of great importance. A considerable number of techniques of modelling has been suggested to characterize the hysteresis phenomenon. These models are categorized into two groups: (i) deductive models developed based on physical laws and characterized by their complex theory (Malczyk and Izydorczyk, 2015; Padilha et al., 2017; Sima et al., 2018), and (ii) inductive models based on experimental data analysis. Examples of the second group are Preisach model (Novak et al., 2018; Nguyen et al., 2018), Bouc-Wen model (Pelliciari et al., 2018; Yu et al., 2017), Prandtl-Ishlinskii model (Ma et al., 2018a; Janaideh and Aljanaideh, 2018; Xie et al., 2018a), fuzzy model (Rebai et al., 2015d), Backlash-like model (Zhou et al., 2017) and fractional order model (Rebai et al., 2014; Zhu et al., 2018). For a detailed review on this topic see (Ikhouan, 2018; Rebai et al., 2013).

The control of nonlinear hysteretic systems has always been a great challenge. An important number of researches deal with this task using usually inverse control techniques and robust adaptive control approaches. The inverse control approaches showed acceptable performances in the elimination of hysteresis effect (Janaideh et al., 2018; Stefanski et al., 2017; Xie et al., 2018b; Zhu et al., 2017). However, the use of these techniques is limited due to the difficulty of obtaining the hysteresis inverse model.

The second category uses robust adaptive control schemes instead of the inverse model. As far as it is known, the foundational work is presented by Su et al. (2000). Recently, a considerable attention has been paid to the problem of controlling nonlinear systems with backlash-like hysteresis input, and some interesting results were reported in Su et al. (2000, 2003); Ren et al. (2009); Xiu-Yu and Yan (2010); Yue et al. (2017); Shahnazi et al. (2010); Li et al. (2012); Chen et al. (2016); Wei et al. (2017); Wu et al. (2017); and Wang et al. (2017). By combining the backlash hysteresis model with adaptive control techniques, Su et al. (2000, 2003) presented a robust adaptive control algorithm for a class of hysteretic systems with unknown backlash-like nonlinearity. The authors of Ren et al. (2009) combined the dynamic surface control strategy with neural networks to deal with output tracking control for strict-feedback non-linear systems preceded by backlash hysteresis. This control strategy is extended to uncertain disturbed strict-feedback nonlinear systems preceded by unknown backlash-like hysteresis in Xiu-Yu and Yan (2010), and to nonlinear systems with unknown distributed time delays in Yue et al. (2017). In Shahnazi et al. (2010) the authors proposed an output feedback controller for uncertain nonlinear hysteretic systems where an observer-based adaptive fuzzy control is used to estimate the system states. An extension of this control approach can be found in Li et al. (2012). Using backstepping theory, a distributed adaptive control scheme is developed in Chen et al. (2016) to solve the consensus problem of high-order nonlinear multi-agent systems with unknown backlash-like hysteresis and unknown control direction. In Wei et al. (2017), the authors propose an adaptive iterative learning control strategy for nonlinear hysteretic systems with unknown time-varying delays and control direction. In Wu et al. (2017), a practical adaptive fuzzy backstepping control scheme is presented for a class of disturbed nonlinear systems with backlash nonlinearity. By combining backstepping technique with the neural networks, an adaptive neural control algorithm is presented in Wang et al. (2017) for switched systems with unknown backlash-like hysteresis control input.

On another frontier of research, several approaches have been developed based on fuzzy control theory and neural networks (Li et al., 2016; Ma et al., 2014; Si et al., 2017; Wang et al., 2014). For example, in Ma et al. (2014), an adaptive backstepping neural networks tracking control approach for a class of pure-feedback stochastic nonlinear systems with hysteresis input is developed. This strategy is adopted for the same systems with nonstrict-feedback form in Wang et al. (2014), and extended to Multiple-Input Multiple-Output (MIMO) pure-feedback nonlinear systems in Yu et al. (2018). The authors of Li et al. (2016) present an adaptive neural prescribed performance control for a class of strict-feedback stochastic systems. In Si et al. (2017), a combination of neural networks and variable separation technique is proposed to develop an input-driven observer-based adaptive neural networks control scheme for a class of stochastic systems with both unknown hysteresis input and control directions. Although these approaches are interesting, they are limited by their inherent complexity and the multiple constraints on the system parameters, and on the backlash-like term. In addition, these approaches need to be well initialized otherwise they give poor performances.

Motivated by the aforementioned discussions and critics, we propose, in this paper, a sliding mode-based control approach for a class of nonlinear systems with unknown backlash-like hysteresis. The developed controller ensures that all signals of the closed loop system are bounded and forces the tracking error to drop asymptotically to zero. The key contributions of this work are: (i) a systematical approach to control a class of non-linear hysteretic systems; (ii) fuzzy logic systems to estimate the unknown terms of the model and hence to improve its accuracy instead of using piecewise constant functions as in Rebai et al. (2016, 2015a, 2015b); and (iii) a proportional integral control term to attenuate the chattering phenomenon and to improve the controller performances.

The organization of this paper is as follows. The problem statement is given in the next section, then the proposed adaptive fuzzy approach is described; and to validate our proposition and show its efficiency, the last section is devoted to numerical examples and simulation results.

Problem statement

Consider the class of uncertain nonlinear hysteretic SISO systems given by

S {\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = x_{3} \\ ⋮ \\ {\overset{\cdot}{x}}_{n} = [f (x) + Δ f (x)] + [g (x) + Δ g (x)] w (u) + d (t) \\ y = x_{1} \end{matrix}

(1)

where $x = {[x_{1}, x_{2}, \dots, x_{n}]}^{T} \in R^{n}$ , $u \in R$ and $y \in R$ are the state vector, control input, and system output, respectively. $f (x)$ and $g (x)$ are two unknown non-linear continuous functions. $d (t)$ represents the unknown external disturbance with $| | d | | \leq d_{m}$ . $Δ f (x)$ and $Δ g (x)$ are unknown bounded uncertainties. $w (u)$ is the backlash-like hysteresis non-linearity modeled in Zhou et al. (2004) by the following first-order differential equation

\frac{dw}{dt} = α | \frac{du}{dt} | (cu - w) + B \frac{du}{dt}

(2)

where the strictly positive parameters $α, B$ and $c$ are the switch rate, the backlash distance and the slope of the lines, respectively. The parameters $B$ and $c$ must satisfy $c > B$ . $u (t)$ is the control input.

The solution of (2) is given by

w (u) = cu + ρ (u)

(3)

where the term $ρ (u)$ is expressed by

\begin{array}{l} ρ (u) = [w_{0} - c u_{0}] e^{- α (u - u_{0}) sgn (\dot{u})} \\ + e^{- α u sgn (\dot{u})} \int_{u_{0}}^{u} [B - c] e^{- α δ sgn (\dot{u})} d δ \end{array}

(4)

The parameters $w_{0}$ and $u_{0}$ are the initial conditions. For the sake of simplicity, these parameters are considered null in this paper. $sgn (.)$ represents the signum function.

Remark 1: Using (4), the term $ρ (u (t))$ can be expressed as follows

when $\overset{\cdot}{u} \geq 0$ ,

ρ (u (t)) = \frac{c - B}{α} [1 - e^{- α u (t)}]

(5)

and when $\overset{\cdot}{u} < 0$ ,

ρ (u (t)) = \frac{c - B}{α} [1 - (2 e^{- α u_{s}} - e^{- 2 α u_{s}}) e^{α u (t)}]

(6)

where $u_{s}$ is the upper bound of $u$ .

From (5) and (6) we have

\overset{\cdot}{u} > 0 : \lim_{u \to + \infty} ρ (u) = \frac{B - c}{α}

(7)

and

\overset{\cdot}{u} < 0 : \lim_{u \to - \infty} ρ (u) = \frac{c - B}{α}

(8)

From (7) and (8), we can conclude that $ρ (u)$ is bounded, that is, $| ρ (u) | < σ$ .

To reduce the complexity of the expression (4), the term $ρ (u)$ is approximated in Rebai et al. (2015a, 2015b, 2016) by a bounded piecewise constant function that takes only two values. However, this approximation does not represent adequately this term. In this paper, fuzzy logic system is used to enhance the approximation of this term. Compared with previous results in Rebai et al. (2015a, 2015b, 2016), the features of the proposed design are listed as follows:

In Rebai et al. (2016, 2015a, 2015b), the term $ρ (u)$ is represented by a bounded piecewise constant function equal to $sgn (\overset{\cdot}{u}) \frac{B - c}{α}$ . Due to the virtue of simplicity of fuzzy logic system, their flexibility and efficiency in modeling of nonlinear complex systems, we propose to use it to estimate the term $ρ (u)$ .

In Rebai et al. (2015a, 2016), the uncertainties and external disturbances are not considered. In this paper, the adaptive fuzzy controller is designed for nonlinear systems subject to parametric uncertainties and external disturbances.

In Rebai et al. (2015a, 2015b) the undesirable chattering phenomenon is not taken into consideration. In our case, a proportional integral control term is adopted to attenuate the chattering phenomenon caused by the sliding mode technique and the hysteresis phenomenon.

The system (1) can be rewritten as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = x_{3} \\ ⋮ \\ {\overset{\cdot}{x}}_{n} = f (x) + g (x) w (u) + \bar{d} (t) \\ y = x_{1} \end{matrix}

(9)

where

{\begin{matrix} \bar{d} (t) = Δ f (x) + Δ g (x) w (u) + d (t) \\ w = cu + ρ \end{matrix}

(10)

The target of this work is the development of an adaptive fuzzy sliding mode control law that ensures the closed loop system global stability in the sense of boundedness of all signals. To this end, we suppose that the following assumptions are verified:

Assumption 1: For all $x \in R^{n}$ , we assume the existence of unknown bounded functions $g_{1} (x)$ and $g_{2} (x)$ such that

0 < g_{1} (x) < g (x) < g_{2} (x), for all x \in R^{n}

This assumption is made to ensure the controllability of system (1).

Assumption 2: The reference input and its $n$ first derivatives exist and are smooth and bounded. This assumption is employed to guarantee the existence and the boundedness of the control law.

Sliding mode controller design

Consider the sliding surface given by

\begin{matrix} s (e) = c_{1} e (t) & + c_{2} \overset{\cdot}{e} (t) + \dots + c_{n - 1} e^{(n - 2)} (t) + e^{(n - 1)} (t) \\ = c^{T} e (t) \end{matrix}

(11)

where $e$ is the error vector defined by

e (t) = y (t) - y_{r} (t) = {[e (t), \overset{\cdot}{e} (t), \dots, e^{(n - 1)} (t)]}^{T} \in R^{n}

(12)

and

c = {[c_{1}, c_{2}, \dots, c_{n - 1}, 1]}^{T} \in R^{n}

with $y (t) = {[y (t), \overset{\cdot}{y} (t), \dots, y^{(n - 1)} (t)]}^{T}$ , and $y_{r} (t) = {[y_{r} (t), {\overset{\cdot}{y}}_{r} (t), \dots, y_{r}^{(n - 1)} (t)]}^{T}$ . $y_{r} (t)$ is the reference signal.

For the initial condition $e (0) = 0$ , the tracking control problem is to keep the error on the sliding surface $s (e) = 0, t \geq 0$ . This condition can be satisfied if the control law is chosen in such manner that

s (e) \overset{\cdot}{s} (e) \leq - η | s |, η \geq 0

(13)

From (9) and (11), we have

\overset{\cdot}{s} = \sum_{i = 1}^{n - 1} c_{i} e^{(i - 1)} (t) + f (x) + g (x) [cu + ρ] - y_{r}^{(n)} (t) + \bar{d} (t)

(14)

Taking into account the condition (13), Assumption 1, Assumption 2 and if $f, g, ρ$ and $\bar{d} (t)$ are known, the proposed sliding mode control law is given by Theorem 1.

Theorem 1: Consider system (1) satisfying Assumption 1 and Assumption 2. If we consider that $f, g, ρ$ and $\bar{d} (t)$ are known, the sliding mode control law given by (15) will stabilize the closed loop system and the tracking error will converge to zero.

\begin{matrix} u = \frac{1}{cg (x)} \\ [- \sum_{i = 1}^{n - 1} c_{i} e^{(i - 1)} (t) - f (x) - ρ g (x) + y_{r}^{(n)} (t) - \bar{d} (t) - η sgn (s)] \end{matrix}

(15)

Proof: Consider the Lyapunov function given by

V = \frac{1}{2} s^{2}

(16)

From (16) we obtain

\overset{\cdot}{V} = s [\sum_{i = 1}^{n - 1} c_{i} e^{(i - 1)} (t) + f (x) + g (x) [cu + ρ] - y_{r}^{(n)} + \bar{d} (t)]

(17)

By considering (15), (17) becomes

\overset{\cdot}{V} = - η sgn (s) s \leq - η | s |

(18)

Adaptive fuzzy sliding mode control

The control law (15) ensures the stability of system (1). However, in some circumstances, the functions $f, g, ρ$ and $\bar{d} (t)$ are unknown. To solve this problem, we propose in this section the use of fuzzy systems to estimate these functions. Another major defect of control law (15) is the chattering phenomenon. To reduce this phenomenon and improve the steady-state performance, we propose the use of a Proportional Integral (PI) control term (Rebai and Guesmi, 2018).

Proposition 1: To get rid of the chattering phenomenon, a proportional integral control term is used. This controller has the following expression

u_{p} = k_{p} s (t) + k_{i} \int s (t) dt

(19)

where $s (t), k_{p}$ and $k_{i}$ are, respectively, the sliding surface, the proportional gain and the integral gain.

The expression (19) can be reformulated as

\hat{p} (s | θ_{p}) = θ_{p}^{T} ψ (s)

(20)

where $\hat{p} (s | θ_{p})$ is the estimation of the output of the Proportional Integral (PI) control term. $θ_{p} = {[k_{p}, k_{i}]}^{T} \in R^{2}$ is a parameter vector that will be also estimated using an adaptive system. $ψ (s) = {[s (t), \int sdt]}^{T} \in R^{2}$ is a regression vector.

Consider the following form of fuzzy rules

\begin{matrix} R_{i} If x_{1} is F_{1}^{j} and x_{2} is F_{2}^{j} \dots and x_{n} is F_{n}^{j}, \\ then y is b^{j} (j = 1, \dots, N) \end{matrix}

(21)

where $x = [x_{1}, x_{2}, \dots, x_{n}] \in R^{n}$ and $y \in R$ are the input and the output of the fuzzy system, respectively. $F_{j}^{i}$ and $b^{i}$ are fuzzy sets.

Using a singleton for output fuzzification, center average defuzzifier and product inference, the fuzzy system output can be expressed as

y (x) = \frac{\sum_{k = 1}^{N} {\bar{y}}^{k} (Π_{i = 1}^{n} μ_{F_{i}^{k}} (x_{i}))}{\sum_{k = 1}^{N} (Π_{i = 1}^{n} μ_{F_{i}^{k}} (x_{i}))}

(22)

where ${\bar{y}}^{k}$ stands for the center of membership functions, $μ_{F_{i}^{k}} (x_{i})$ is the membership function of the input $x_{i}$ for the fuzzy rule $k$ . The output of system (22) can be rewritten as

y (x) = θ^{T} ξ (x)

(23)

where $θ = {({\bar{y}}^{1}, {\bar{y}}^{2}, \dots, {\bar{y}}^{N})}^{T}$ and $ξ (x) = {(ξ^{1} (x), ξ^{2} (x), \dots, ξ^{N} (x))}^{T}$ are, respectively, a parameter vector to be estimated, and the fuzzy basis functions vector defined as

ξ (x) = \frac{Π_{i = 1}^{n} μ_{F_{i}^{k}} (x_{i})}{\sum_{k = 1}^{N} (Π_{i = 1}^{n} μ_{F_{i}^{k}} (x_{i}))}

(24)

The functions $f, g, ρ, \bar{d} (t)$ and the parameters vector $θ_{p}$ are replaced by their estimations in the control law (15). Hence, the control law is

\begin{matrix} u = \frac{1}{c \hat{g} (x)} \\ [- \sum_{i = 1}^{n - 1} c_{i} e^{(i - 1)} - \hat{f} (x) - \hat{ρ} \hat{g} (x) + y_{r}^{(n)} - \hat{\bar{d}} (t) - \hat{p} (s | θ_{p})] \end{matrix}

(25)

where $\hat{f} (x), \hat{g} (x), \hat{ρ}$ and $\hat{\bar{d}} (t)$ are the estimations of the functions $f, g, ρ, \bar{d} (t)$ , respectively, given by

\hat{f} (x, θ_{f}) = θ_{f}^{T} ξ_{f} (x)

(26)

\hat{g} (x, θ_{g}) = θ_{g}^{T} ξ_{g} (x)

(27)

\hat{p} (s, θ_{p}) = θ_{p}^{T} ψ (s)

(28)

\hat{\bar{d}} (t) = θ_{\bar{d}}^{T} ξ_{\bar{d}} (x)

(29)

Finally, the new adaptive fuzzy sliding mode control law is summarized in Theorem 2.

Theorem 2: Consider the system (1) satisfying Assumption 1 and Assumption 2. If we consider the sliding mode control law given by (25) with the update laws (30)–(34), then all signals of the closed loop system are bounded and the tracking error converges to zero

{\overset{\cdot}{θ}}_{f} = γ_{1} s ξ_{f} (x)

(30)

{\overset{\cdot}{θ}}_{g} = γ_{2} cs ξ_{g} (x) u

(31)

{\overset{\cdot}{θ}}_{ρ} = γ_{3} s ξ_{ρ} (x)

(32)

{\overset{\cdot}{θ}}_{p} = γ_{4} s ψ (s)

(33)

{\overset{\cdot}{θ}}_{\bar{d}} = γ_{5} s ξ_{\bar{d}} (x)

(34)

where $γ_{i}; i = 1 : 5$ are the learning parameters.

Proof: To complete the proof, we introduce the following lemmas.

Lemma 1: (Khalil, 2002) If $h, \overset{\cdot}{h} \in L_{\infty}$ and $h \in L_{q}, q \in [1, \infty [$ , then $h \to 0$ when $t \to \infty$ .

where the space $L_{\infty}$ is defined as the set of piecewise continuous bounded functions $h : [0, \infty [\to R$ such that

| | h | |_{L_{\infty}} = \sup_{t \geq 0} | | h (t) | | < \infty

(35)

The space $L_{q}$ for $1 \leq q < \infty$ is defined as the set of all piecewise continuous functions $h : [0, \infty [\to R$ such that

| | h | |_{L_{q}} = {(\int_{0}^{\infty} | | h (t) | |^{q} dt)}^{1 / q} < \infty

(36)

This lemma is the generalization of the Barbalat’s lemma introduced as follows:

Lemma 2: (Barbalat’s lemma) If $li m_{t \to \infty} \int_{0}^{\infty} h (τ) d τ$ exists and is finite, and if $h (t)$ is a uniformly continuous function, then $li m_{t \to \infty} h (t) = 0$ .

Consider the following Lyapunov function

\begin{matrix} V = \frac{1}{2} \\ [s^{2} + \frac{1}{γ_{1}} {\tilde{θ}}_{f}^{T} {\tilde{θ}}_{f} + \frac{1}{γ_{2}} {\tilde{θ}}_{g}^{T} {\tilde{θ}}_{g} + \frac{1}{γ_{3}} {\tilde{θ}}_{ρ}^{T} {\tilde{θ}}_{ρ} + \frac{1}{γ_{4}} {\tilde{θ}}_{p}^{T} {\tilde{θ}}_{p} + \frac{1}{γ_{5}} {\tilde{θ}}_{\bar{d}}^{T} {\tilde{θ}}_{\bar{d}}] \end{matrix}

(37)

where

{\tilde{θ}}_{f} = θ_{f}^{*} - θ_{f}

{\tilde{θ}}_{g} = θ_{g}^{*} - θ_{g}

{\tilde{θ}}_{p} = θ_{p}^{*} - θ_{p}

{\tilde{θ}}_{\bar{d}} = θ_{\bar{d}}^{*} - θ_{\bar{d}}

The optimal parameters vectors $θ_{f}^{*}, θ_{g}^{*}, θ_{p}^{*}$ and $θ_{\bar{d}}^{*}$ are defined, respectively, as

θ_{f}^{*} = \arg \min_{θ_{f} \in Ω_{f}} (\sup_{x \in R^{n}} | \hat{f} (x | θ_{f}) - f (x, t) |)

(38)

θ_{g}^{*} = \arg \min_{θ_{g} \in Ω_{g}} (\sup_{x \in R^{n}} | \hat{g} (x | θ_{g}) - g (x, t) |)

(39)

θ_{p}^{*} = \arg \min_{θ_{p} \in Ω_{p}} (\sup_{s \in R} | \hat{p} (s | θ_{p}) - u_{p} |)

(40)

θ_{\bar{d}}^{*} = \arg \min_{θ_{\bar{d}} \in Ω_{\bar{d}}} (\sup_{x \in R^{n}} | \hat{\bar{d}} (x | θ_{\bar{d}}) - \bar{d} (t) |)

(41)

where $Ω_{f}, Ω_{g}, Ω_{p}$ and $Ω_{\bar{d}}$ are constraint sets for $θ_{f}, θ_{g}, θ_{p}$ and $θ_{\bar{d}}$ , respectively, which have the following forms

Ω_{f} = {θ_{f} \in R^{n} : | θ_{f} | \leq M_{f}}

(42)

Ω_{g} = {θ_{g} \in R^{n} : 0 \leq | θ_{g} | \leq M_{g}}

(43)

Ω_{p} = {θ_{p} \in R^{2} : | θ_{p} | \leq M_{p}}

(44)

Ω_{d} = {θ_{d} \in R^{n} : | θ_{f} | \leq M_{d}}

(45)

where $M_{f}, M_{g}, M_{p}$ and $M_{d}$ are pre-specified positive constants for the estimation of fuzzy parameters boundaries (Wang, 1994). If we assume that the fuzzy parameters $Ω_{f}, Ω_{g}, Ω_{p}$ and $Ω_{\bar{d}}$ never reach the boundaries. Then, the time derivative of (37) is

\overset{\cdot}{V} = s \overset{\cdot}{s} - \frac{1}{γ_{1}} {\tilde{θ}}_{f}^{T} {\overset{\cdot}{θ}}_{f} - \frac{1}{γ_{2}} {\tilde{θ}}_{g}^{T} {\overset{\cdot}{θ}}_{g} - \frac{1}{γ_{3}} {\tilde{θ}}_{ρ}^{T} {\overset{\cdot}{θ}}_{ρ} - \frac{1}{γ_{4}} {\tilde{θ}}_{p}^{T} {\overset{\cdot}{θ}}_{p} - \frac{1}{γ_{5}} {\tilde{θ}}_{\bar{d}}^{T} {\overset{\cdot}{θ}}_{\bar{d}}

(46)

From (25), we have

y_{r}^{(n)} = \sum_{i = 1}^{n - 1} c_{i} e^{(i - 1)} + \hat{f} (x) + \hat{ρ} \hat{g} (x) + c \hat{g} (x) u + \hat{\bar{d}} (t) + \hat{p} (s | θ_{p})

(47)

Replacing (47) in (14) leads to

\begin{matrix} \overset{\cdot}{s} = \sum_{i = 1}^{n - 1} c_{i} e^{(i - 1)} (t) + f (x) + g (x) [cu + ρ] + \bar{d} (t) \\ + [- \sum_{i = 1}^{n - 1} c_{i} e^{(i - 1)} (t) - \hat{f} (x) - \hat{ρ} \hat{g} (x) - \hat{\bar{d}} (t) - \hat{p} (s | θ_{p}) - c \hat{g} (x) u] \end{matrix}

(48)

By adding and subtracting $\hat{p} (s, θ_{p}^{*})$ , and simplifying (48), we obtain

\begin{matrix} \overset{\cdot}{s} = & (f (x) - \hat{f} (x)) + c (g (x) - \hat{g} (x)) u \\ + (\bar{d} - \hat{\bar{d}}) + (g (x) ρ - \hat{g} (x) \hat{ρ}) \\ - \hat{p} (s, θ_{p}) + \hat{p} (s, θ_{p}^{*}) - \hat{p} (s, θ_{p}^{*}) \end{matrix}

(49)

The minimum approximation error can be defined as

ε = ε_{1} + ε_{2} + ε_{3} + ε_{4} + ε_{5}

with

ε_{1} = f (x) - \hat{f} (x, θ_{f}^{★})

(50)

ε_{2} = c (g (x) - \hat{g} (x, θ_{g}^{★})) u

(51)

ε_{3} = \bar{d} - \hat{\bar{d}} (θ_{\bar{d}}^{★})

(52)

ε_{4} = g (x) ρ - \hat{g} (x, θ_{g}^{★}) \hat{ρ} (θ_{ρ}^{★})

(53)

ε_{5} = p (s) - \hat{p} (s, θ_{p}^{*})

(54)

From (50)–(54), we have

f (x) - \hat{f} (x, θ_{f}) = {\tilde{θ}}_{f}^{T} ξ_{f} (x) + ε_{1}

(55)

c (g (x) - \hat{g} (x, θ_{g})) u = c {\tilde{θ}}_{g}^{T} ξ_{g} (x) u + ε_{2}

(56)

\bar{d} - \hat{\bar{d}} (θ_{\bar{d}}) = {\tilde{θ}}_{\bar{d}}^{T} ξ_{\bar{d}} + ε_{3}

(57)

g (x) ρ - \hat{g} (x, θ_{g}) \hat{ρ} (θ_{ρ}) = {\tilde{θ}}_{ρ}^{T} ξ_{ρ} + ε_{4}

(58)

p (s) - \hat{p} (s, θ_{p}) = {\tilde{θ}}_{p}^{T} ψ (s) + ε_{5}

(59)

Using (55)–(59) in (48), we obtain

\begin{matrix} \overset{\cdot}{s} = & {\tilde{θ}}_{f}^{T} ξ_{f} (x) + ε_{1} + c {\tilde{θ}}_{g}^{T} ξ_{g} (x) u + ε_{2} + {\tilde{θ}}_{\bar{d}}^{T} ξ_{\bar{d}} + ε_{3} + {\tilde{θ}}_{ρ}^{T} ξ_{ρ} \\ + ε_{4} + {\tilde{θ}}_{p}^{T} ψ (s) + ε_{5} - \hat{p} (s, θ_{p}^{*}) \\ = {\tilde{θ}}_{f}^{T} ξ_{f} (x) + c {\tilde{θ}}_{g}^{T} ξ_{g} (x) u + {\tilde{θ}}_{\bar{d}}^{T} ξ_{\bar{d}} + {\tilde{θ}}_{ρ}^{T} ξ_{ρ} + {\tilde{θ}}_{p}^{T} ψ (s) \\ + ε - \hat{p} (s, θ_{p}^{*}) \end{matrix}

(60)

Using (60), (46) becomes

\begin{matrix} \overset{\cdot}{V} = & s [{\tilde{θ}}_{f}^{T} ξ_{f} (x) + c {\tilde{θ}}_{g}^{T} ξ_{g} (x) u + {\tilde{θ}}_{\bar{d}}^{T} ξ_{\bar{d}} + {\tilde{θ}}_{ρ}^{T} ξ_{ρ} + {\tilde{θ}}_{p}^{T} ψ (s) + ε \\ - \hat{p} (s, θ_{p}^{*})] - \frac{1}{γ_{1}} {\tilde{θ}}_{f}^{T} {\overset{\cdot}{θ}}_{f} - \frac{1}{γ_{2}} {\tilde{θ}}_{g}^{T} {\overset{\cdot}{θ}}_{g} - \frac{1}{γ_{3}} {\tilde{θ}}_{ρ}^{T} {\overset{\cdot}{θ}}_{ρ} - \frac{1}{γ_{4}} {\tilde{θ}}_{p}^{T} {\overset{\cdot}{θ}}_{p} \\ - \frac{1}{γ_{5}} {\tilde{θ}}_{\bar{d}}^{T} {\overset{\cdot}{θ}}_{\bar{d}} \\ = & \frac{1}{γ_{1}} {\tilde{θ}}_{f}^{T} [γ_{1} s ξ_{f} (x) - {\overset{\cdot}{θ}}_{f}] + \frac{1}{γ_{2}} {\tilde{θ}}_{g}^{T} [γ_{2} cs ξ_{g} (x) u - {\overset{\cdot}{θ}}_{g}] \\ + \frac{1}{γ_{3}} {\tilde{θ}}_{ρ}^{T} [γ_{3} s ξ_{ρ} (x) - {\overset{\cdot}{θ}}_{ρ}] + \frac{1}{γ_{4}} {\tilde{θ}}_{p}^{T} [γ_{4} s ψ (s) - {\overset{\cdot}{θ}}_{p}] \\ + \frac{1}{γ_{5}} {\tilde{θ}}_{\bar{d}}^{T} [γ_{5} s ξ_{\bar{d}} - {\overset{\cdot}{θ}}_{\bar{d}}] - s \hat{p} (s, θ_{p}^{*}) + s ε \end{matrix}

(61)

The use of (30)–(34) in (61) with the assumption that $| \hat{p} (s, θ_{p}^{*}) | \leq η$ , leads to

\begin{matrix} \overset{\cdot}{V} = s ε - s \hat{p} (s, θ_{p}^{*}) \\ \leq s ε - η | s | \\ \leq 0 \end{matrix}

(62)

From the inequality (62), we can conclude the boundedness of all signals in the system. If the initial value e(0) is bounded, then $e (t), t \geq 0$ is also bounded. Due to the boundedness of the error vector e(t) and reference signal $y_{r}$ , the system states $x_{i}$ are bounded. To end up the asymptotic stability analysis, we need to prove the convergence $s \to 0$ when $t \to \infty$ . If we assume that $| s | \leq η_{s}$ , (62) can be rewritten as

\overset{\cdot}{V} \leq | s | | ε | - η | s | \leq η_{s} | ε | - η | s |

(63)

The inequality (63) can be transformed to

| s | \leq \frac{η_{s}}{η} | ε | - \frac{1}{η} \overset{\cdot}{V}

(64)

Integrating both sides of (64), we get

\int_{0}^{t} | s (τ) | d τ \leq \frac{1}{η} (| V (0) | + | V (t) |) + \frac{η_{s}}{η} \int_{0}^{t} | ε (τ) | d τ

(65)

The inequality (65) means that $s \in L_{1}$ . The inequality (62) implies that $s$ is bounded. It is easier to verify that every parameter in (60) is bounded. As a consequence, $s, \overset{\cdot}{s} \in L_{\infty}$ . According to Lemma 1, the sliding surface converges to zero when $t \to \infty$ . Then, the tracking error and its derivatives will converge to zero, that is, $e^{(i)} (t) \to 0$ when $t \to \infty$ for $i = 0, 1, \dots, n - 1$ . This concludes the stability proof.

To summarize, the overall proposed control scheme is depicted in Figure 1.

Figure 1.

Adaptive fuzzy sliding mode control for nonlinear hysteretic systems.

Simulation results

This section is devoted to the validation of the proposed approach and to the evaluation of their performances. Two case studies are considered to assess the performance of the proposed adaptive fuzzy controller. To show its efficiency, the proposed control scheme performances are compared with the adaptive backstepping and adaptive fuzzy synergetic approaches presented, respectively, in Zhou et al. (2004), Zhou et al. (2007) and Rebai et al. (2016).

Example 1

Consider the following second order nonlinear uncertain hysteretic system

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = \frac{1 - e^{x_{2}}}{1 + e^{x_{2}}} + Δ f (x) + [2 + 0.5 \sin (x_{1} x_{2}) + Δ g (x)] w (u) + d (t) \\ y = x_{1} \end{matrix}

(66)

where $x_{1}$ and $x_{2}$ denote the state variables, $y$ is the system output, $w$ is the backlash hysteresis nonlinearity described by (2) with $α = 1$ , $B = 2.356$ and $c = 0.453$ . The uncertainty terms are selected as $Δ f (x) = 0.1 \cos (x_{1})$ and $Δ g (x) = 0.3 \cos (x_{2})$ . The external disturbance is chosen as $d (t) = 1.5 \cos (15 t)$ . The control objective is to design an adaptive fuzzy control scheme such that all the signals remain bounded and the system output $y$ follows the desired trajectory $y_{d} = 0.6 \cos (2.4 t)$ . The expressions of membership functions for $f, g, ρ$ and $\bar{d} (t)$ are selected Gaussian as follows

μ_{F_{i}^{1}} (x_{i}) = {\begin{matrix} 1 if x_{i} < c_1 \\ e^{- \frac{{(x_{i} - c_1)}^{2}}{2 δ}} if x_{i} > c_1, i = 1, 2 \end{matrix}

(67)

μ_{F_{i}^{l}} (x_{i}) = e^{- \frac{{(x_{i} - c_l)}^{2}}{2 δ}}, i = 1, 2; l = 2, \dots, 4

(68)

μ_{F_{i}^{5}} (x_{i}) = {\begin{matrix} e^{- \frac{{(x_{i} - c_5)}^{2}}{2 δ}} if x_{i} < c_5, i = 1, 2 \\ 1 if x_{i} > c_5 \end{matrix}

(69)

where $[c_1, c_2, c_3, c_4, c_5]$ are selected as $[- 10, - 5, 0, 5, 10]$ for $f (x)$ , $[1.5, 1.75, 2, 2.25, 2.5]$ for $g (x)$ , $[- 2, - 1, 0, 1, 2]$ for the term $ρ$ and $[- 1.5, - 0.75, 0, 0.75, 1.5]$ for the disturbance $\bar{d} (t)$ . Each adjacent membership functions are crossing at 0.5. The chosen membership functions are shown in Figure 2.

Figure 2.

Membership functions for $f, g, ρ$ and $\bar{d} (t)$ .

The controller design parameters for this example are taken as follows: the sliding surface is chosen as $s (e) = c_{1} e (t) + \overset{\cdot}{e} (t)$ , where $c_{1} = 12$ . The PI control term is selected as $u_{p} = k_{p} s (t) + k_{i} \int s (t) dt$ , where $k_{p} = 200, k_{i} = 15$ . For the update laws (30)-(34), the learning parameters $γ_{i}; i = 1 : 5$ are taken as $γ_{1} = 0.01, γ_{2} = 0.1, γ_{3} = γ_{4} = 0.1, γ_{5} = 0.001$ . The initial conditions for the state vector $x$ are ${[x_{1} (0), x_{2} (0)]}^{T} = {[0.2, 0]}^{T}$ . The initial values $θ_{f} (0), θ_{g} (0), θ_{ρ} (0)$ and $θ_{\bar{d}} (0)$ are randomly selected from the interval $[0, 2]$ . The simulation time is $T = 10 s$ with a sampling period $t_{s} = 0.001 s$ .

The simulations results are shown in Figures 3 –6. Figure 3 shows the system output $y$ under: adaptive fuzzy synergetic control approach proposed in Rebai et al. (2016), adaptive backstepping proposed in Zhou et al. (2007) and the proposed approach for a reference signal $y_{r}$ . As can be seen in Figure 3, the desired performances are achieved while keeping the robustness of the controller against external disturbances and uncertainties. Furthermore, the proposed control scheme presents better performances in terms of setting time and tracking error compared to adaptive backstepping and fuzzy synergetic approaches. Similarly, Figure 4 shows better performances which support the previous observations. Figure 5 indicates that the state variable $x_{2}$ is bounded and $\overset{\cdot}{e} (t) \to 0$ when $t \to \infty$ . Figure 6 displays the sliding surface. From Figure 6, we observe the fluctuation of the curve of the sliding surface due to: the chattering phenomenon, the choice of the initial values of the PI controller, and the hysteresis nonlinearity. Figure 7 illustrates the evolution of the estimates of functions $f (x)$ and $g (x)$ . Figure 8 shows the norm of adaptive parameter vectors $θ_{f}, θ_{g}, θ_{ρ}$ and $θ_{p}$ and it is clear that all adaptive parameters are bounded. According to the obtained results, the proposed adaptive scheme guarantees the boundedness of all the signals of the closed loop system, which validate the theoretical results. The obtained results can be improved by using optimization techniques to find the optimal values of the design parameters.

Figure 3.

Output trajectory.

Figure 4.

Tracking error.

Figure 5.

Derivative of the output trajectory.

Figure 6.

Sliding surface.

Figure 7.

Estimation of nonlinear functions $f (x)$ and $g (x)$ .

Figure 8.

Norm of $θ_{f}, θ_{g}, θ_{ρ}$ and $θ_{p}$ .

Example 2

In this example, the tracking control of the benchmark control problem of inverted pendulum, shown in Figure 9, is investigated. Using the proposed adaptive fuzzy controller, the obtained results are compared to those obtained by adaptive backstepping control approaches proposed in Zhou et al. (2004) and Zhou et al. (2007), respectively.

Figure 9.

Inverted pendulum system.

The dynamic equations of the inverted pendulum system are given by (Wang, 1996)

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = [\frac{g \sin x_{1} - \frac{{mlx}_{2}^{2} \cos x 1 \sin x_{1}}{m_{c} + m}}{l (\frac{4}{3} - \frac{m {(\cos x_{1})}^{2}}{m_{c} + m})} + Δ f (x)] + [\frac{\frac{\cos x_{1}}{m_{c} + m}}{l (\frac{4}{3} - \frac{m {(\cos x_{1})}^{2}}{m_{c} + m})} + Δ g (x)] w (u) + d (t) \\ y = x_{1} \end{matrix}

(70)

where $x_{1} = θ$ and $x_{2} = \overset{\cdot}{θ}$ represent the angular position and the angular velocity, respectively. $g = 9.8 m / s^{2}$ , $m_{c} = 1 kg$ , $m = 0.1 kg$ , $l = 0.5 m$ are the gravity, mass of the cart, mass of the pendulum and length of pendulum, respectively. $w (u)$ is the hysteresis input. The membership functions used in this example for states $x_{i}, i = 1, 2$ are chosen as

μ_{F_{i}^{l}} (x_{i}) = μ_{G_{i}^{l}} (x_{i}) = e^{- \frac{{(x_{i} - c_l)}^{2}}{2 δ}}, i = 1, 2; l = 1, \dots, 5

(71)

with $[c_1, c_2, c_3, c_4, c_5] = [- \frac{π}{6}, - \frac{π}{12}, 0, \frac{π}{12}, \frac{π}{6}]$ , and $δ = \frac{c_2 - c_1}{2 \sqrt{2 \ln 2}}$ . The uncertainty terms and the external disturbance are selected as $Δ f (x) = 0.1 \cos (x_{1}) e^{- x_{1} x_{2}}$ , $Δ g (x) = 2 \cos (x_{1} x_{2})$ and $d (t) = 1.5 \cos (3 t)$ , respectively. The objective is to force the system output $y (t)$ to track the desired reference $y_{r} (t) = \frac{π}{15} \cos (5 t)$ where the initial conditions are $x_{0} = [0.05, 0]$ . The sliding surface constant is chosen $c_{1} = 17$ . The PI control term parameters are $k_{p} = 100, k_{i} = 02$ . The learning parameters $γ_{i}; i = 1 : 5$ for the update laws we use $γ_{1} = γ_{2} = 0.1, γ_{3} = 0.02, γ_{4} = 0.1, γ_{5} = 0.001$ . The initial values $θ_{f} (0), θ_{g} (0), θ_{ρ} (0)$ and $θ_{\bar{d}} (0)$ are randomly selected from the interval $[0, 1]$ . The simulation time is $T = 6 s$ with a sampling period $t_{s} = 0.001 s$ .

The obtained simulation results are given in Figures 10 –15. From Figure 10 and Figure 11, it can be seen that the objective control performance is achieved. Furthermore, the proposed control scheme has better tracking efficiency compared to adaptive backstepping approaches. Figure 12 shows that the state variable $x_{2}$ converges to ${\overset{\cdot}{y}}_{r}$ . From Figure 13, which displays the sliding surface, we can observe that the chattering phenomenon is decreased using the PI controller. It is shown also, from Figure 14 and Figure 15, that all adaptive parameters are bounded. These simulation results demonstrate the tracking capability of the proposed controller and its effectiveness for control tracking of uncertain nonlinear hysteretic systems with the boundedness of all signals of the closed loop system.

Figure 10.

Output trajectory.

Figure 11.

Tracking error.

Figure 12.

Derivative of the output trajectory.

Figure 13.

Sliding surface.

Figure 14.

Estimation of nonlinear functions $f (x)$ and $g (x)$ .

Figure 15.

Norm of $θ_{f}, θ_{g}, θ_{ρ}$ and $θ_{p}$ .

Conclusion

In this paper, an adaptive fuzzy control approach has been developed for a class of nonlinear hysteretic systems. The proposed adaptive fuzzy control scheme guarantees that all signals of the closed-loop system remain bounded and the tracking error converges asymptotically to the origin. The major contribution of this research is the robustness of the proposed control approach against external disturbances and system uncertainties. Simulation results and a comparison with existing adaptive approaches are provided to validate and to evaluate performances the proposed approach. An interesting continuity of this work is to apply this approach to nonlinear systems with another hysteresis models such as Prandtl-Ishlinskii and Bouc-Wen models with the extension of the proposed approaches to MIMO nonlinear systems.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Aissa Rebai

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