Adaptive asymptotic tracking control of uncertain fractional-order nonlinear systems with unknown quantized input and control directions subject to actuator failures

Abstract

This article addresses an adaptive backstepping control design for uncertain fractional-order nonlinear systems in the strict-feedback form subject to unknown input quantization, unknown state-dependent control directions, and unknown actuator failure. The system order can be commensurate or noncommensurate. The total number of failures is allowed to be infinite. The Nussbaum function is used to deal with the problem of unknown control directions. Compared with the existing results, the control gains can be functions of states and the knowledge of quantization parameters and characteristics of the actuator failure are unknown. By applying the backstepping control approach based on the frequency-distributed model, it is proved that all the closed-loop signals remain bounded and the output tracking error converges to the origin asymptotically. Finally, the effectiveness of the proposed controller is demonstrated by two simulation examples.

Keywords

fractional-order systems adaptive control input quantization unknown control direction actuator failures

1. Introduction

Fractional-order calculus (FOC) is a topic with more than 300 years of history, which can be considered as an extension of ordinary integrals and derivatives to noninteger order (Petráš, 2011). However, only during the recent decades, FOC has attracted more and more engineers’ and physicists’ interest because of its potential to describe memory and hereditary properties of many processes (Caponetto et al., 2010). According to these unique features, it is not far-fetched to claim that fractional models compared with integer-order models can describe the system dynamics more accurately (Monje et al., 2010). From the practical point of view, it should be mentioned that fractional-order differential equations have been widely used in different areas such as robotics, biology, diffusion, fluid mechanics, signal processing, control theory, etc. (Axtell and Bise, 1990; Debnath, 2003; Monje et al., 2010; Muresan et al., 2016; Sabatier et al., 2007).

Stability analysis and controller design of fractional-order systems which are more complicated than integer-order systems have been considered in several works (Alaviyan Shahri et al., 2018; Lenka and Banerjee, 2018). Stability of integer-order nonlinear systems (IONS) can be analyzed using the classical Lyapunov function (Ahmad and Shafiq, 2020b; Shafiq and Ahmad, 2019), whereas stability proof of fractional-order nonlinear systems (FONS) by choosing a conventional quadratic Lyapunov function and calculating its time derivative is not straightforward (Aguila-Camacho et al., 2014). To overcome this problem, various approaches have been suggested in the literature. Li et al. (2010) extended the Lyapunov direct method to fractional-order systems by introducing the Mittag-Leffler stability notion. This method became more applicable by introducing a useful inequality by Aguila-Camacho et al. (2014). This result allows using the classic quadratic Lyapunov functions in many stability analyses of FONS. However, in some cases, choosing a simple quadratic function is not useful (Duarte-Mermoud et al., 2015). Chen et al. (2017) developed a Lyapunov candidate function in the form of a convex and positive definite function. Because of the restrictions of the direct method, the concept of the indirect Lyapunov method based on the frequency distributed model has been purposed (Wei et al., 2016). With the aid of this method, not only noncommensurate cases can be handled, but also conventional control techniques such as backstepping can be applied. According to the mentioned advantages of the frequency distributed model, this model has been used in the present work.

1.1. Challenges and motivations

In recent years, control of the fractional-order nonlinear systems has received a great deal of interest; however, designing a practical controller for FONS to handle different restrictions is still an open area. In what follows, some of the challenges in the design of a controller that motivated this present work have been reviewed:

Generally, the exact model of the system is not available. Consequently, considering the model uncertainty has been one of the crucial issues in the control theory (Yao and Tomizuka, 2001). The unknown dynamics and parametric uncertainties are often the main forms of uncertainties. In case of unknown dynamics, intelligent approximators can be used to estimate the system unknown dynamics. Most of the controllers designed based on these approximators, guarantee the stability of the closed-loop system in the sense of Lyapunov. Several works have been proposed for integer-order nonlinear systems with parametric uncertainties and asymptotic stability has been established (Ahmad et al., 2016; Ahmad and Shafiq, 2020a). In the present work, a system with parametric uncertainty has been considered to achieve the asymptotic tracking performance which is more desirable.

Although adaptive backstepping is a common technique widely used for IONS (Hua et al., 2008), it cannot be directly applied to fractional-order systems because the traditional Leibniz and chain rules are not valid for fractional derivatives (Podlubny, 1998). Some researchers have extended the integer-order adaptive backstepping technique to fractional-order ones (Nikdel and Badamchizadeh, 2019; Sheng et al., 2017; Wei et al., 2015, 2016). Because the design of the controller for the FONS is in its early stage, adaptive backstepping control of the FONS has not been widely investigated as done for IONS (Liu et al., 2017). Thus, designing an appropriate adaptive backstepping controller for FONS remains an open issue.

In many real processes, the system control directions may be unknown which means the effect of input change on the direction of output change is not known. For example, in the control of wind turbines, because the wind speed variation is unpredictable, the control direction is not known a priori (Habibi et al., 2018). In such cases, designing an efficient controller is a challenging problem. A common approach to overcome this problem is using the Nussbaum function (Nussbaum, 1983). The design of adaptive controllers for IONS with unknown control directions have been addressed by Askari et al. (2017); Wang et al. (2015c); Xudong and Ding (2001). Similarly, based on the Nussbaum gain function, several adaptive controllers for FONS with unknown control direction have been proposed (Khettab et al., 2016; Zhang and Xie, 2014; Zouari et al., 2017). To make the controller more practical, in this work, the unknown control direction restriction has been taken into account in the controller design.

In many industrial applications, such as flight control systems and nuclear power plants, actuators are subjected to faults during operation, which results in a poor control performance and even instability (Li and Yang, 2016). Actuator fault is a type of failure affecting the system inputs. Because of abnormal operation or material aging, actuator faults may occur in the system. Abnormal operating condition means a condition that indicates a malfunction of the process components or deviation from normal operations (violating process variables limits such as temperature and pressure). This failure leads to catastrophic consequences and hazards for personnel, plants, and environment (Li, 2016). To keep processes stable and compensate for the effect of failure, an effective fault-tolerant control (FTC) design is necessary. Although many actuator failure compensation schemes have been introduced for integer-order systems (Moradvandi et al., 2019; Wang et al., 2015a; Wang and Wen, 2010), a few works can be found in the literature regarding controller design for FONS subject to actuator failures. Sakthivel et al. (2018) proposed an FTC algorithm for nonlinear fractional-order systems which guarantees the robust asymptotic stability of the system. It is worth mentioning that in the aforementioned work, it has been assumed that the total number of failures is finite. Recently, an adaptive controller has been designed for a class of FONS with the infinite number of actuator failures that can only be applied to commensurate systems (Li et al., 2019). Therefore, designing a fault-tolerant controller that can handle the infinite number of failures and additional restrictions such as unknown direction for noncommensurate FONS is desired and motivated this work.

In recent years, the controller design for systems with quantized input has attracted researchers’ interest because of its practical applications such as networked control systems (Wang and Liu, 2008), control of discrete-nature systems (Lunze et al., 1999), and control of systems equipped with stepper motors (Liu et al., 2019). In controlling systems with input quantization, a continuous control signal is quantized by a quantizer before applying it to the plant which results in an unavoidable quantization error. Ignoring this error affects the control performance and leads to unexpected problems, such as oscillations or chaos and even system instability (Niu et al., 2017).

Although the study of adaptive quantized control for IONS has received a great deal of attention (Choi and Yoo, 2017; Li and Yang, 2016; Yu and Lin, 2016), there are very few control schemes for adaptive quantized tracking control for FONS (Hua et al., 2018; Song et al., 2020). To the best of authors’ knowledge, the controller design proposed for uncertain FONS with quantized input and unknown control directions subject to an infinite number of actuator failures and time-varying external disturbance has not been addressed in the literature before. Meanwhile, the designed controller can guarantee the asymptotic stability of the closed-loop system.

1.2. Contributions

The main contributions of this work are summarized below:

This work presents the controller design for FONS in the strict-feedback form subject to input quantization, unknown control direction, infinite number of actuator failures, and external disturbance that can be applied to both commensurate and noncommensurate order systems.

The proposed control scheme can be applied without the knowledge of quantization parameters.

It has been assumed that the unknown control gains are a function of states.

To reduce the computational load, only one adaptive law has been applied for estimating the unknown upper bounds of control gains. As the system dimension increases, the reduction of computational burden for the proposed control scheme becomes highlighted and the controller design becomes simpler.

The supremum of the faults uncertainty bound has been used to handle the infinite number of actuator failures.

It has been shown that all closed-loop signals remain bounded and the tracking error converges to the origin asymptotically.

This article is organized as follows. In Section 2, the problem statement and necessary preliminaries have been presented. Based on the frequency distributed model, the design of an adaptive fractional-order controller by using the backstepping technique has been addressed in Section 3. Two examples have been presented in Section 4 to show the effectiveness of the proposed control scheme. Finally, the conclusion has been drawn in Section 5.

2. Problem statement and preliminaries

Consider a class of uncertain fractional-order nonlinear systems subject to quantized input described by

{\begin{cases} D^{γ_{i}} x_{i} = g_{i} ({\underline{x}}_{i}) x_{i + 1} + φ_{i}^{T} ({\underline{x}}_{i}) θ, 1 \leq i \leq n - 1 \\ ⋮ \\ D^{γ_{n}} x_{n} = g_{n} ({\underline{x}}_{n}) k (q (u)) + φ_{n}^{T} ({\underline{x}}_{n}) θ + d (t) \\ y = x_{1} \end{cases}

(1)

where

0 < γ_{i} < 1 (i = 1,2, \dots, n)

are fractional noncommensurate orders.

{\underline{x}}_{i} = {[x_{1}, x_{2}, \dots, x_{i}]}^{T} \in R^{i}

(

i = 1, 2, \dots, n

) are the state vectors, and

y \in R

is the system output.

φ_{i} ({\underline{x}}_{i}) \in R^{p}

(

i = 1, \dots n

) are known nonlinear functions.

θ \in R^{p}

is an unknown constant vector.

g_{i} (\cdot)

is an unknown nonzero state-dependent control gain function, and

d (t) \in R

represents the unknown bounded time-varying external disturbance.

q (u) \in R

denotes the quantized input, where

u \in R

is the control signal to be designed.

k (\cdot)

denotes the actuator fault model which will be described later.

Remark 1

As stated in the Introduction section, the system model can be partially or totally unknown. System (1) has parametric uncertainty which means vector parameter $θ$ is unknown, whereas functions $φ_{i} ({\underline{x}}_{i})$ are assumed to be known. The advantage of such a model is the fact that the asymptotic stability of the closed-loop system can be established. Hence, the objective is to obtain satisfactory control performance in presence of variations of $θ$ during operation. To show that the proposed control scheme is robust against parameter uncertainties, $θ$ has been changed in the simulation study which will be presented later.

Remark 2

The three mostly used definitions for the fractional derivatives are the Riemann–Liouville (RL), the Grunwald–Letnikov (GL), and the Caputo definitions. Among these definitions, in this article, the Caputo fractional derivative operator has been considered because of its advantages such as:

The initial conditions for fractional-order systems with Caputo derivatives have clear physical interpretation and are the same as those for integer-order differential equations.

The Caputo derivative of a constant is zero.

The equilibrium points of non-integer order systems with Caputo derivatives are the same as those used for integer-order ones.

Note that $q (u (t))$ is the output of the quantizer with quantized values, which makes the stability analysis and controller design difficult. In what follows the type of input quantizer used in the present work will be described. To avoid chattering in the input, the following hysteretic quantizer has been used (Ceragioli et al., 2011)

{\begin{cases} u_{i} sgn (u), \frac{u_{i}}{1 + δ} < | u | \leq u_{i}, \dot{u} < 0, or \\ u_{i} < | u | \leq \frac{u_{i}}{1 - δ}, \dot{u} > 0 \\ u_{i} (1 + δ) sgn (u), u_{i} < | u | \leq \frac{u_{i}}{1 - δ}, \dot{u} < 0, or \\ \frac{u_{i}}{1 - δ} < | u | \leq \frac{u_{i} (1 + δ)}{1 - δ}, \dot{u} > 0 \\ 0, 0 \leq | u | < \frac{u_{\min}}{1 + δ}, \dot{u} < 0, or \\ \frac{u_{\min}}{1 + δ} \leq | u | < u_{\min}, \dot{u} > 0 \\ q (u (t^{-})), \dot{u} = 0 \end{cases}

(2)

where

u_{i} = ϑ^{1 - i} u_{min} (i = 1,2, \dots)

and

u_{m i n} > 0

donates the size of the dead-zone. The constant

ϑ \in (0,1)

is a measure of quantization density and

δ = (1 - ϑ) / (1 + ϑ)

. It is easy to show that

0 < δ < 1

. It can be concluded from equation (2) that

q (u (t))

belongs to set

U = {0, \pm u_{i}, \pm u_{i} (1 + δ), i = 1,2, \dots}

. It should be noted that the smaller

ϑ

results in the coarser of the quantizer, that is, as

ϑ

approaches zero,

δ

approaches 1 leading to the fewer quantization levels. Figure 1 demonstrates the applied input quantization. As can be seen from this figure, depending on the range of the control signal and its derivative sign,

q (u)

can vary from one value to another. The detailed discussion can be found in De Persis and Mazenc (2010).

To design an appropriate control scheme and overcome the problem of quantization, hysteresis quantizer $q (u)$ can be decomposed by the following lemma.

Lemma 1

(Liu et al., 2015). The quantized input $q (u)$ can be represented as

q (u) = ℳ (u) u (t) + ℵ (t)

(3)

where

ℳ (u) = {\begin{cases} \frac{q (u)}{u}, q (u) \neq 0 \\ 1, q (u) = 0 \end{cases}

ℵ (t) = {\begin{cases} 0, q (u) \neq 0 \\ - u (t), q (u) = 0 \end{cases}

(4)

In the above decomposition, $ℳ (u)$ and $ℵ (t)$ satisfy the following inequalities

1 - δ \leq ℳ (u) \leq 1 + δ

(5)

| ℵ (t) | \leq u_{\min}

(6)

Remark 3

According to the above inequalities, it should be noted that only the boundedness of functions $ℳ (u)$ and $ℵ (t)$ has been used, and it is assumed that they are unknown.

Note that most of the practical applications are faced with actuator fault. In this article, the infinite number of actuator failures is described by (Wang et al., 2015a)

k (q (u)) = ρ_{h} q (u) + u_{f_{h}} (t), t \in [T_{h}^{s}, T_{h}^{e}), h = 1, 2, 3, \dots

(7)

where

ρ_{h}

T_{h}^{s}

and

T_{h}^{e}

are unknown constants satisfying

0 < ρ_{h} < 1

and

0 \leq T_{1}^{s} < T_{1}^{e} \leq T_{2}^{s} < T_{2}^{e} \leq \dots \leq \infty

, and

u_{f_{h}} (t)

is an unknown piecewise continuous and bounded signal.

T_{h}^{s}

and

T_{h}^{e}

are the time instants that the

h

th failure of the actuator starts and ends, respectively. To construct a model that includes the normal operation and faulty conditions, the two following piecewise continuous functions are defined

ρ (t) = {\begin{cases} ρ_{h}, t \in [T_{h}^{s}, T_{h}^{e}) \\ 1, t \in [T_{h}^{e}, T_{h + 1}^{s}) \end{cases}

(8)

u_{f} (t) = {\begin{cases} u_{f_{h}} (t), t \in [T_{h}^{s}, T_{h}^{e}) \\ 0, t \in [T_{h}^{e}, T_{h + 1}^{s}) \end{cases}

(9)

Using the above functions, equation (7) can be written as

k (q (u)) = ρ (t) q (u) + u_{f} (t)

(10)

Using equations (3) and (10), system (1) can be rewritten as

{\begin{cases} D^{γ_{i}} x_{i} = g_{i} ({\underline{x}}_{i}) x_{i + 1} + φ_{i}^{T} ({\underline{x}}_{i}) θ, 1 \leq i \leq n - 1 \\ ⋮ \\ D^{γ_{n}} x_{n} = g_{n} ({\underline{x}}_{n}) (ρ (t) ℳ (u) u (t) + ρ (t) ℵ (t) + u_{f} (t)) + φ_{n}^{T} ({\underline{x}}_{n}) θ + d (t) \\ y = x_{1} \end{cases}

(11)

The control objective is designing an adaptive controller such that all closed-loop signals remain bounded and the output tracking error converges to the origin asymptotically in presence of input quantization, unknown control direction, infinite number of actuator faults, and external disturbance.

Assumption 1

The desired signal $y_{d}$ and its first $\sum_{i = 1}^{j} γ_{i}$ th ( $j = 1, \dots, n$ ) order derivatives are available, piecewise continuous, and bounded.

Assumption 2

The state-dependent control gain $g_{i} ({\underline{x}}_{i})$ and its sign are unknown, and there exist unknown positive constants $g_{i l}$ and $g_{i u}$ such that $0 < g_{i l} \leq | g_{i} ({\underline{x}}_{i}) | \leq g_{i u}$ .

Assumption 3

It is assumed that the external disturbance is bounded, that is, $| d (t) | \leq \bar{d}$ , where $\bar{d}$ is an unknown positive constant.

Remark 4

Assumption 1 is not restrictive and made in several works (Sheng et al., 2017; Wei et al., 2015). In most practical applications, the reference signal and its derivatives are available. Assumption 2 implies that gain functions are strictly either positive or negative, which ensures the system controllability. This assumption has been made by many researchers (Choi and Yoo, 2017; Ge et al., 2004; Lai et al., 2016; Na, 2013; Wang et al., 2008, 2018). It should also be noted that the lower and upper bounds of control gains are only required for analytical purposes and their values are assumed to be unknown. Therefore Assumption 2 is reasonable and not restrictive. Furthermore, for all real applications, the external disturbances are bounded and thus Assumption 3 is not restrictive and has been made by other researchers (Li and Yang, 2016; Wang and Yang, 2018). It is worth noting that the disturbance upper bound is only required for the theoretical analysis and assumed to be unknown.

Remark 5

In contrast to some of the existing works (Liu et al., 2015; Yu and Lin, 2016), the quantized parameter $δ$ does not appear in the control law, and therefore it can be unknown.

Definition 1

(Nussbaum, 1983). Function $N (ξ)$ which has the following properties is called the Nussbaum function

\lim_{s \to + \infty} sup \frac{1}{s} \int_{s_{0}}^{s} N (ξ) d ξ = + \infty

(12)

\lim_{s \to + \infty} inf \frac{1}{s} \int_{s_{0}}^{s} N (ξ) d ξ = - \infty

(13)

Many functions satisfy the above conditions such as $ξ^{2} cos (ξ)$ , $ξ^{2} sin (ξ)$ , $e^{ξ^{2}} \cos (π ξ / 2)$ and $e^{ξ^{2}} sin (π ξ / 2)$ (Ilchmann, 1993).

Lemma 2

(Wang et al., 2015a). For any positive uniform continuous and bounded function $σ (t)$ and any variable $x \in R$ , the following relationship holds

0 \leq | x | - \frac{x^{2}}{\sqrt{x^{2} + σ {(t)}^{2}}} \leq σ (t)

(14)

with

σ (t)

satisfying

\lim_{t \to \infty} \int_{t_{0}}^{t} σ (t) d s \leq \bar{σ} < \infty

(15)

where

\bar{σ}

denotes any positive constant.

Lemma 3

(Trigeassou et al., 2013). The fractional-order system $D^{α} y (t) = u (t)$ with $0 < α < 1$ , $y (t) \in R$ and $u (t) \in R$ , can be shown as the following frequency distributed model

{\begin{cases} \frac{\partial z (ω, t)}{\partial t} = - ω z (ω, t) + u (t) \\ y (t) = \int_{0}^{\infty} μ_{α} (ω) z (ω, t) d ω \end{cases}

(16)

with

μ_{α} (ω) = sin (α π) / π ω^{α}

and

z (ω, t) \in ℝ

μ_{α} (ω)

is the frequency weighting function with the fractional-order

α

Lemma 4

(Wei et al., 2014). Suppose $v (t)$ is the input signal to be differentiated with $| D^{α} v (t) | < \infty$ and $r_{1}$ is the maximum value of control input $u (t)$ , then the solution of the following tracking differentiator

{\begin{cases} D^{α} x_{1} (t) = x_{2} (t) \\ D^{α} x_{2} (t) = u (t) \end{cases}

(17)

is convergent in the sense that $x_{1} (t)$ is uniformly convergent to $v (t)$ on $[0, \infty)$ as $r \to \infty$ , and $x_{2} (t)$ is the general $α$ -th order differentiation of $v (t)$ , where the control input $u (t)$ is given by

\begin{array}{l} u (t) = - r_{1} tanh (x_{1} (t) - v (t) + \frac{x_{2} (t) | x_{2} (t) |}{r_{1}} [1 - \frac{Γ^{2} (α + 1)}{Γ (2 α + 1)}], r_{2}) \\ , r_{2} > 0 and \tanh (z, γ) = \frac{e^{γ z} - e^{- γ z}}{e^{γ z} + e^{- γ z}} \end{array}

(18)

Remark 6

As a matter of fact, the fractional-order tracking differentiator (FOTD) can be considered as a generalization of integer-order one (Han, 2009). With the help of FOTD, the required fractional-order derivative of a signal can be generated online by adjusting $r_{1}$ and $r_{2}$ to reach the desired speed and smoothness.

3. Controller design and stability analysis

In this section, an adaptive controller design for FONS using the backstepping technique based on the frequency distributed model has been presented.

To achieve the control objective, the following error variables have been introduced

ε_{1} = x_{1} - y_{d}

(19)

ε_{i} = x_{i} - α_{i - 1}, i = 2, \dots, n

(20)

where

α_{i}

are the virtual controllers and will be designed later.

Further, based on Assumption 2 for the unknown positive constants $g_{i u}$ $(i = 1, \dots, n)$ , $g_{u}$ has been defined as

g_{u} = \max {g_{1 u}^{2}, \dots, g_{n u}^{2}}

(21)

Denoting ${\hat{g}}_{u} \in R$ as the estimate of $g_{u}$ , the following estimation error has been defined

{\tilde{g}}_{u} = g_{u} - {\hat{g}}_{u}

(22)

Remark 7

To reduce the computational burden, the maximum squared of $g_{i u}$ $(i = 1, \dots, n)$ has been estimated instead of estimating the upper bounds of control gains individually.

Step 1

The fractional derivative of $ε_{1}$ is given by

D^{γ_{1}} ε_{1} = D^{γ_{1}} x_{1} - D^{γ_{1}} y_{d}

(23)

Substituting equation (20) into equation (23) yields

D^{γ_{1}} ε_{1} = g_{1} (x_{1}) (ε_{2} + α_{1}) + φ_{1}^{T} θ - D^{γ_{1}} y_{d}

(24)

By utilizing Lemma 3, from the frequency distributed model we have

{\begin{cases} \frac{\partial z_{1} (ω, t)}{\partial t} = - ω z_{1} (ω, t) + g_{1} (x_{1}) (ε_{2} + α_{1}) + φ_{1}^{T} θ - D^{γ_{1}} y_{d} \\ ε_{1} = \int_{0}^{\infty} μ_{γ_{1}} (ω) z_{1} (ω, t) d ω \end{cases}

(25)

where

μ_{γ_{1}} (ω) = \sin (γ_{1} π) / ω^{γ_{1}} π

Based on the indirect Lyapunov approach, the first Lyapunov function is chosen as

V_{1} = \frac{1}{2} \int_{0}^{\infty} μ_{γ_{1}} (ω) z_{1}^{2} (ω, t) d ω + \frac{1}{2 λ_{θ}} {\tilde{θ}}^{T} \tilde{θ} + \frac{1}{2 λ_{g}} {\tilde{g}}_{u}^{2}

(26)

Because $μ_{α} (ω)$ is positive for all $ω$ , $V (t)$ is also positive. $\tilde{θ} = θ - \hat{θ}$ is the estimation error of $θ$ . $λ_{θ}$ and $λ_{g}$ are the positive design constants.

Time derivative of $V_{1}$ is given by

{\dot{V}}_{1} = \int_{0}^{\infty} μ_{γ_{1}} (ω) z_{1} (ω, t) \frac{\partial z_{1} (ω, t)}{\partial t} d ω + \frac{1}{λ_{θ}} {\tilde{θ}}^{T} \dot{\tilde{θ}} + \frac{1}{λ_{g}} {\tilde{g}}_{u} {\dot{\tilde{g}}}_{u}

(27)

Using equation (25), the above equation can be written in the following form

\begin{array}{l} {\dot{V}}_{1} = \int_{0}^{\infty} μ_{γ_{1}} (ω) z_{1} (ω, t) (- ω z_{1} (ω, t) + g_{1} (x_{1}) (ε_{2} + α_{1}) + φ_{1}^{T} θ - D^{γ_{1}} y_{d}) d ω \\ + \frac{1}{λ_{θ}} {\tilde{θ}}^{T} \dot{\tilde{θ}} + \frac{1}{λ_{g}} {\tilde{g}}_{u} {\dot{\tilde{g}}}_{u} \\ = - \int_{0}^{\infty} ω μ_{γ_{1}} (ω) z_{1}^{2} (ω, t) d ω + ε_{1} (g_{1} (x_{1}) ε_{2} + g_{1} (x_{1}) α_{1} + φ_{1}^{T} θ - D^{γ_{1}} y_{d}) \\ + \frac{1}{λ_{θ}} {\tilde{θ}}^{T} \dot{\tilde{θ}} + \frac{1}{λ_{g}} {\tilde{g}}_{u} {\dot{\tilde{g}}}_{u} \end{array}

(28)

Using Assumption 2 and Young’s inequality, the following inequality can be obtained

g_{1} (x_{1}) ε_{1} ε_{2} \leq g_{1 u}^{2} ε_{2}^{2} + \frac{ε_{1}^{2}}{4} \leq g_{u} ε_{2}^{2} + \frac{ε_{1}^{2}}{4}

(29)

Utilizing the above inequality, $\dot{\tilde{θ}} = - \dot{\hat{θ}}$ and ${\dot{\tilde{g}}}_{u} = - {\dot{\hat{g}}}_{u}$ , the time derivative of Lyapunov function (28) can be expressed as

\begin{array}{l} {\dot{V}}_{1} \leq - \int_{0}^{\infty} ω μ_{γ_{1}} (ω) z_{1}^{2} (ω, t) d ω + ε_{1} (g_{1} (x_{1}) α_{1} + \frac{ε_{1}}{4} + φ_{1}^{T} θ - D^{γ_{1}} y_{d}) \\ + \frac{1}{λ_{θ}} {\tilde{θ}}^{T} \dot{\hat{θ}} - \frac{1}{λ_{g}} {\tilde{g}}_{u} {\dot{\hat{g}}}_{u} + g_{u} ε_{2}^{2} \end{array}

(30)

The following virtual control law $α_{1}$ has been proposed

α_{1} = N (ξ_{1}) (c_{1} ε_{1} + \frac{ε_{1}}{4} + φ_{1}^{T} \hat{θ} - D^{γ_{1}} y_{d})

(31)

{\dot{ξ}}_{1} = ε_{1} (c_{1} ε_{1} + \frac{ε_{1}}{4} + φ_{1}^{T} \hat{θ} - D^{γ_{1}} y_{d})

(32)

where

c_{1} > 0

is a design parameter.

Substituting $θ = \tilde{θ} + \hat{θ}$ , $g_{u} = {\tilde{g}}_{u} + {\hat{g}}_{u}$ and equation (30) into (29) yields

\begin{array}{l} {\dot{V}}_{1} \leq - \int_{0}^{\infty} ω μ_{γ_{1}} (ω) z_{1}^{2} (ω, t) d ω - c_{1} ε_{1}^{2} + (g_{1} (x_{1}) N (ξ_{1}) + 1) {\dot{ξ}}_{1} \\ + \frac{1}{λ_{θ}} {\tilde{θ}}^{T} (λ_{θ} ε_{1} φ_{1}^{T} - \dot{\hat{θ}}) + \frac{1}{λ_{g}} {\tilde{g}}_{u} (λ_{g} ε_{2}^{2} - {\dot{\hat{g}}}_{u}) + {\hat{g}}_{u} ε_{2}^{2} \end{array}

(33)

Choose the first tuning functions $τ_{1}$ and $ϱ_{1}$ as

τ_{1} = λ_{θ} ε_{1} φ_{1}^{T} - λ_{θ} σ (t) \hat{θ}

(34)

ϱ_{1} = λ_{g} ε_{2}^{2} - λ_{g} σ (t) {\hat{g}}_{u}

(35)

Terms $λ_{θ} σ (t) \hat{θ}$ and $λ_{g} σ (t) {\hat{g}}_{u}$ , called $σ$ -modification, have been used to make the control system more robust.

Using equations (34) and (35), equation (33) can be written as

\begin{array}{l} {\dot{V}}_{1} \leq - \int_{0}^{\infty} ω μ_{γ_{1}} (ω) z_{1}^{2} (ω, t) d ω - c_{1} ε_{1}^{2} + (g_{1} (x_{1}) N (ξ_{1}) + 1) {\dot{ξ}}_{1} + \frac{1}{λ_{θ}} \tilde{θ} (τ_{1} - \dot{\hat{θ}}) \\ + \frac{1}{λ_{g}} {\tilde{g}}_{u} (ϱ_{1} - {\dot{\hat{g}}}_{u}) + {\hat{g}}_{u} ε_{2}^{2} + σ (t) {\tilde{θ}}^{T} \hat{θ} + σ (t) {\tilde{g}}_{u} {\hat{g}}_{u} \end{array}

(36)

The derivation of step 2 and step $i$ can be found in Appendix 1.

Step n

The input quantization and actuator fault affect the last step of the design and therefore this step plays an important role in the controller design and has been discussed in detail.

The fractional derivative of $ε_{n}$ is as follows

\begin{array}{l} D^{γ_{n}} ε_{n} = D^{γ_{n}} x_{n} - D^{γ_{n}} α_{n - 1} \\ = g_{n} ({\underline{x}}_{n}) (ρ (t) ℳ (u) u (t) + ρ (t) ℵ (t) + u_{f} (t)) + φ_{n}^{T} θ + d (t) - D^{γ_{n}} α_{n - 1} \end{array}

(37)

Based on the frequency distributed model, equation (37) can be expressed as

{\begin{cases} \frac{\partial z_{n} (ω, t)}{\partial t} = - ω z_{n} (ω, t) + g_{n} ({\underline{x}}_{n}) (ρ (t) ℳ (u) u (t) + ρ (t) ℵ (t) + u_{f} (t)) + φ_{n}^{T} θ \\ + d (t) - D^{γ_{n}} α_{n - 1} \\ ε_{n} = \int_{0}^{\infty} μ_{γ_{n}} (ω) z_{n} (ω, t) d ω \end{cases}

(38)

where

μ_{γ_{n}} (ω) = \sin (γ_{n} π) / ω^{γ_{n}} π

To compensate for quantization error, unknown actuator failures, and external disturbance, the following variable has been defined

β = \bar{d} + \sup_{t \geq 0} | g_{n} ({\underline{x}}_{n}) u_{f} (t) | + \sup_{t \geq 0} | g_{n} ({\underline{x}}_{n}) ρ (t) ℵ (t) |

(39)

It is worth pointing out that from Assumption 3 and Lemma 1, constant $β$ is an unknown bounded parameter and can be estimated by the adaptive law to be designed later.

Consider the last Lyapunov function candidate as

V_{n} = V_{n - 1} + \frac{1}{2} \int_{0}^{\infty} μ_{γ_{n}} (ω) z_{n}^{2} (ω, t) d ω + \frac{1}{2 λ_{β}} {\tilde{β}}^{2}

(40)

In the above equation, $\tilde{β} = β - \hat{β}$ , where $\hat{β}$ is the estimate of $β$ and $λ_{β}$ is an adjustable parameter.

Remark 8

The above Lyapunov function remains continuous in case of changes in the actuator failures because $β$ is unaffected by actuator failure changes. Hence, the Lyapunov function has no jump and therefore the infinite number of faults can be handled.

Calculating time derivative of $V_{n}$ with the aid of equation (38) results in

\begin{array}{l} {\dot{V}}_{n} = {\dot{V}}_{n - 1} - \int_{0}^{\infty} ω μ_{γ_{n}} (ω) z_{n}^{2} (ω, t) d ω + ε_{n} (g_{n} ({\underline{x}}_{n}) (ρ (t) ℳ (u) u (t) \\ + g_{n} ({\underline{x}}_{n}) ρ (t) ℵ (t) + u_{f} (t) + φ_{n}^{T} θ + d (t) - D^{γ_{n}} α_{n - 1}) + \frac{1}{λ_{β}} \tilde{β} \dot{\tilde{β}} \end{array}

(41)

Using $θ = \tilde{θ} + \hat{θ}$ and time derivative of the previous step Lyapunov function, we get

\begin{array}{l} {\dot{V}}_{n} \leq - \sum_{j = 1}^{n} \int_{0}^{\infty} ω μ_{γ_{j}} (ω) z_{j}^{2} (ω, t) d ω - \sum_{j = 1}^{n - 1} c_{j} ε_{j}^{2} + \sum_{j = 1}^{n - 1} (g_{j} ({\underline{x}}_{j}) N (ξ_{j}) + 1) {\dot{ξ}}_{j} + {\hat{g}}_{u} ε_{n}^{2} \\ + ε_{n} (g_{n} ({\underline{x}}_{n}) ρ (t) ℳ (u) u (t) + φ_{n}^{T} \hat{θ} - D^{γ_{n}} α_{n - 1}) + ε_{n} g_{n} ({\underline{x}}_{n}) ρ (t) ℵ (t) \\ + ε_{n} g_{n} ({\underline{x}}_{n}) u_{f} (t) + ε_{n} d (t) + \frac{1}{λ_{θ}} {\tilde{θ}}^{T} (τ_{n - 1} + λ_{θ} ε_{n} φ_{n}^{T} - \dot{\hat{θ}}) + \frac{1}{λ_{g}} {\tilde{g}}_{u} (ϱ_{n - 1} - {\dot{\hat{g}}}_{u}) \\ + σ (t) {\tilde{θ}}^{T} \hat{θ} + σ (t) {\tilde{g}}_{u} {\hat{g}}_{u} + \frac{1}{λ_{β}} \tilde{β} \dot{\tilde{β}} \end{array}

(42)

The following control and adaptive laws have been proposed

u = N (ξ_{n}) (c_{n} ε_{n} + {\hat{g}}_{u} ε_{n} + φ_{n}^{T} \hat{θ} - D^{γ_{n}} α_{n - 1} + \frac{ε_{n} {\hat{β}}^{2}}{\sqrt{ε_{n}^{2} {\hat{β}}^{2} + σ {(t)}^{2}}})

(43)

{\dot{ξ}}_{n} = ε_{n} (c_{n} ε_{n} + {\hat{g}}_{u} ε_{n} + φ_{n}^{T} \hat{θ} - D^{γ_{n}} α_{n - 1} + \frac{ε_{n} {\hat{β}}^{2}}{\sqrt{ε_{n}^{2} {\hat{β}}^{2} + σ {(t)}^{2}}})

(44)

\dot{\hat{θ}} = τ_{n} = τ_{n - 1} + λ_{θ} ε_{n} φ_{n}^{T}

(45)

{\dot{\hat{g}}}_{u} = ϱ_{n - 1} = ϱ_{n - 2} + λ_{g} ε_{n}^{2}

(46)

Using equations (43)–(46) in equation (42) gives

\begin{array}{l} {\dot{V}}_{n} \leq - \sum_{j = 1}^{n} \int_{0}^{\infty} ω μ_{γ_{j}} (ω) z_{j}^{2} (ω, t) d ω - \sum_{j = 1}^{n} c_{j} ε_{j}^{2} + \sum_{j = 1}^{n - 1} (g_{j} ({\underline{x}}_{j}) N (ξ_{j}) + 1) {\dot{ξ}}_{j} \\ + (g_{n} ({\underline{x}}_{n}) ρ (t) ℳ (u) N (ξ_{n}) + 1) {\dot{ξ}}_{n} + ε_{n} g_{n} ({\underline{x}}_{n}) ρ (t) ℵ (t) + ε_{n} g_{n} ({\underline{x}}_{n}) u_{f} (t) \\ + ε_{n} d (t) + σ (t) {\tilde{θ}}^{T} \hat{θ} + σ (t) {\tilde{g}}_{u} {\hat{g}}_{u} + \frac{1}{λ_{β}} \tilde{β} \dot{\tilde{β}} - \frac{ε_{n}^{2} {\hat{β}}^{2}}{\sqrt{ε_{n}^{2} {\hat{β}}^{2} + σ {(t)}^{2}}} \end{array}

(47)

Utilizing equation (39) and the fact that $x y \leq | x y | \leq | x ‖ y |$ , the following inequality can be obtained

\begin{array}{l} ε_{n} g_{n} ({\underline{x}}_{n}) ρ (t) ℵ (t) + ε_{n} g_{n} ({\underline{x}}_{n}) u_{f} (t) + ε_{n} d (t) \\ \leq | ε_{n} | | g_{n} ({\underline{x}}_{n}) ρ (t) ℵ (t) | + | ε_{n} | | g_{n} ({\underline{x}}_{n}) u_{f} (t) | + | ε_{n} | | d (t) | \\ \leq | ε_{n} | β \end{array}

(48)

Substituting equation (48) into equation (47) and noting $β = \tilde{β} + \hat{β}$ and $\dot{\tilde{β}} = - \dot{\hat{β}}$ yields

\begin{array}{l} {\dot{V}}_{n} \leq - \sum_{j = 1}^{n} \int_{0}^{\infty} ω μ_{γ_{j}} (ω) z_{j}^{2} (ω, t) d ω - \sum_{j = 1}^{n} c_{j} ε_{j}^{2} + \sum_{j = 1}^{n - 1} (g_{j} ({\underline{x}}_{j}) N (ξ_{j}) + 1) {\dot{ξ}}_{j} \\ + (g_{n} ({\underline{x}}_{n}) ρ (t) ℳ (u) N (ξ_{n}) + 1) {\dot{ξ}}_{n} + | ε_{n} | \hat{β} + σ (t) {\tilde{θ}}^{T} \hat{θ} + σ (t) {\tilde{g}}_{u} {\hat{g}}_{u} \\ + \frac{1}{λ_{β}} \tilde{β} (λ_{β} | ε_{n} | - \dot{\hat{β}}) - \frac{ε_{n}^{2} {\hat{β}}^{2}}{\sqrt{ε_{n}^{2} {\hat{β}}^{2} + σ {(t)}^{2}}} \end{array}

(49)

The following adaptive law for updating $\hat{β}$ has been proposed

\dot{\hat{β}} = λ_{β} | ε_{n} | - λ_{β} σ (t) \hat{β}

(50)

where term

λ_{β} σ (t) \hat{β}

is called

σ

-modification.

According to Lemma 2, it can be verified that

| ε_{n} | \hat{β} - \frac{ε_{n}^{2} {\hat{β}}^{2}}{\sqrt{ε_{n}^{2} {\hat{β}}^{2} + σ {(t)}^{2}}} < σ (t)

(51)

Thus, inequality (49) can be rewritten in the following form

\begin{array}{l} {\dot{V}}_{n} \leq - \sum_{j = 1}^{n} \int_{0}^{\infty} ω μ_{γ_{j}} (ω) z_{j}^{2} (ω, t) d ω - \sum_{j = 1}^{n} c_{j} ε_{j}^{2} + \sum_{j = 1}^{n - 1} (g_{j} ({\underline{x}}_{j}) N (ξ_{j}) + 1) {\dot{ξ}}_{j} \\ + (g_{n} ({\underline{x}}_{n}) ρ (t) ℳ (u) N (ξ_{n}) + 1) {\dot{ξ}}_{n} + σ (t) {\tilde{θ}}^{T} \hat{θ} \\ + σ (t) {\tilde{g}}_{u} {\hat{g}}_{u} + σ (t) \tilde{β} \hat{β} + σ (t) \end{array}

(52)

Using Young’s inequality, it can be shown

\begin{array}{l} σ (t) {\tilde{θ}}^{T} \hat{θ} = σ (t) {\tilde{θ}}^{T} (θ - \tilde{θ}) \\ \leq σ (t) (- {\tilde{θ}}^{T} \tilde{θ} + {\tilde{θ}}^{T} \tilde{θ} + \frac{1}{4} θ^{T} θ) \\ \leq \frac{1}{4} σ (t) θ^{T} θ \end{array}

(53)

\begin{array}{l} σ (t) {\tilde{g}}_{u} {\hat{g}}_{u} = σ (t) {\tilde{g}}_{u} (g_{u} - {\tilde{g}}_{u}) \\ \leq σ (t) (- {\tilde{g}}_{u}^{2} + {\tilde{g}}_{u}^{2} + \frac{1}{4} g_{u}^{2}) \\ \leq \frac{1}{4} σ (t) g_{u}^{2} \end{array}

(54)

\begin{array}{l} σ (t) \tilde{β} \hat{β} = σ (t) \tilde{β} (β - \tilde{β}) \\ \leq σ (t) (- {\tilde{β}}^{2} + {\tilde{β}}^{2} + \frac{1}{4} β^{2}) \\ \leq \frac{1}{4} σ (t) β^{2} \end{array}

(55)

For the sake of simplicity, new variables $G_{i} (\cdot)$ are defined as follows

{\begin{cases} G_{i} ({\underline{x}}_{i}) = g_{i} ({\underline{x}}_{i}) i = 1, \dots, n - 1 \\ G_{n} ({\underline{x}}_{n}) = g_{n} ({\underline{x}}_{n}) ρ (t) ℳ (u) \end{cases}

(56)

Because $ℳ (u)$ is positive, $g_{i} (\cdot)$ and $G_{i} (\cdot)$ have the same sign.

Substituting inequalities (53)–(55) and equation (56) into equation (52) gives

\begin{array}{l} {\dot{V}}_{n} \leq - \sum_{i = 1}^{n} \int_{0}^{\infty} ω μ_{γ_{i}} (ω) z_{i}^{2} (ω, t) d ω - \sum_{i = 1}^{n} c_{i} ε_{i}^{2} + \sum_{i = 1}^{n} (G_{i} ({\underline{x}}_{i}) N (ξ_{i}) + 1) {\dot{ξ}}_{i} \\ + σ (t) (1 + \frac{1}{4} θ^{T} θ + \frac{1}{4} g_{u}^{2} + \frac{1}{4} β^{2}) \end{array}

(57)

Remark 9

It should be pointed out that the term $ε_{n} {\hat{β}}^{2} / \sqrt{ε_{n}^{2} {\hat{β}}^{2} + σ {(t)}^{2}}$ in equation (43), plays a key role in compensating the effects of the quantizer dead-zone, fault unknown parameters, and the external disturbance. The gains of $σ$ -modification terms that appeared in adaptive laws (34), (35), and (50) are positive time-varying and integrable functions that make the corresponding residue terms also integrable and are crucial for asymptotic regulation.

Theorem 1

Consider the fractional-order nonlinear system (1) subject to input quantization and actuator failure under Assumptions 1–3. Applying control law (43) and adaptive laws (45), (46), and (50) guarantees boundedness of all closed-loop signals and asymptotic convergence of the tracking error to the origin.

Proof

Because terms $- \sum_{i = 1}^{n} \int_{0}^{\infty} ω μ_{γ_{i}} (ω) z_{i}^{2} (ω, t) d ω$ and $- \sum_{i = 1}^{n} c_{i} ε_{i}^{2}$ are strictly negative, inequality (57) can be rewritten as

{\dot{V}}_{n} \leq \sum_{i = 1}^{n} (G_{i} ({\underline{x}}_{i}) N (ξ_{i}) + 1) {\dot{ξ}}_{i} + σ (t) (1 + \frac{1}{4} θ^{T} θ + \frac{1}{4} g_{u}^{2} + \frac{1}{4} β^{2})

(58)

Let define

η = 1 + \frac{1}{4} θ^{T} θ + \frac{1}{4} g_{u}^{2} + \frac{1}{4} β^{2}

(59)

Using equation (59) in (58) yields

{\dot{V}}_{n} \leq \sum_{i = 1}^{n} (G_{i} ({\underline{x}}_{i}) N (ξ_{i}) + 1) {\dot{ξ}}_{i} + σ (t) η

(60)

Integrating equation (60) over $[0, t)$ gives

V (t) - V (0) \leq \int_{0}^{t} \sum_{i = 1}^{n} (G_{i} ({\underline{x}}_{i}) N (ξ_{i}) + 1) {\dot{ξ}}_{i} d s + η \bar{σ}

(61)

By applying the same procedure as used in Jing and Yang (2017) and Wang and Yang (2018), it can be concluded from equation (61) that $ξ_{i} (t)$ $(i = 1, \dots, n)$ are bounded. Boundedness of $ξ_{i} (t)$ implies that $\int_{0}^{t} \sum_{i = 1}^{n} (G_{i} ({\underline{x}}_{i}) N (ξ_{i}) + 1) {\dot{ξ}}_{i} d s$ are also bounded which indicates that $V (t)$ is bounded. As a result, all the closed-loop signals, that is, $ε_{i}$ , $\tilde{θ}$ , ${\tilde{g}}_{u}$ , and $\tilde{β}$ are bounded. Furthermore, because $θ$ is a constant, $\hat{θ}$ is also bounded. From $ε_{1} = x_{1} - y_{d}$ and boundedness of $ε_{1}$ and $y_{d}$ , it is concluded that state $x_{1}$ is bounded. According to equation (31) and Assumption 1, $α_{1}$ remains bounded. From $x_{2} = ε_{2} + α_{1}$ , boundedness of state $x_{2}$ is resulted. Using definitions of virtual controllers $α_{i}$ with Assumption 1 and relation $x_{i} = ε_{i} + α_{i - 1}$ , one can show that $x_{i}$ s are bounded. From the above discussion, it is concluded that all variables appearing on the right-hand side of equation (43) are bounded, indicating boundedness of control signal $u$ . Besides, from equation (57) one has

{\dot{V}}_{n} \leq - \sum_{i = 1}^{n} c_{i} ε_{i}^{2} + \sum_{i = 1}^{n} (G_{i} ({\underline{x}}_{i}) N (ξ_{i}) + 1) {\dot{ξ}}_{i} + σ (t) η

(62)

Integrating equation (62) and noting that $V_{n} (t) \geq 0$ , we have

\sum_{i = 1}^{n} c_{i} \int_{0}^{t} ε_{i}^{2} d s \leq V_{n} (0) + \int_{0}^{t} \sum_{i = 1}^{n} (G_{i} ({\underline{x}}_{i}) N (ξ_{i}) + 1) {\dot{ξ}}_{i} d s + \bar{σ} η

(63)

Because all terms on the right-hand side of equation (63) are bounded, $ε_{2}$ belongs to $L_{2}$ . In addition, with the same reasoning, it is concluded from equation (24) that $D^{γ_{1}} ε_{1} \in L_{\infty}$ . Eventually, from Barbalat Lemma (Wang et al., 2015b), we have

\lim_{t \to \infty} ε_{1} = \lim_{t \to \infty} y - y_{d} = 0

(64)

Therefore, asymptotic tracking is fulfilled, and the proof is completed.

Remark 10

It can be seen from the quantizer map (2) that the quantization parameter $δ$ is theoretically allowed to vary between 0 and 1, but simulation results indicate that the smaller values of $δ$ and $u_{min}$ lead to a better tracking performance. $λ_{θ}$ , $λ_{g}$ , and $λ_{β}$ affect the update rate of adaptive parameters and have no significant effect on the controller performance. Increasing $c_{i}$ s leads to a faster speed of convergence and improves the control performance. However, if $c_{i}$ s are chosen too big, the control input will be increased leading to more aggressive control actions and higher input power cost. Therefore, a trade-off between better performance and the input cost should be made.

Remark 11

For simulation purpose, fractional-order derivative signals, that is, $D^{\sum_{j = 1}^{i} γ_{j}} y_{d}$ ( $i = 1, \dots, n$ ) and $D^{γ_{i}} α_{i}$ ( $i = 1, \dots, n - 1$ ), can be obtained by FOTD given in Lemma 4 or any efficient methods.

4. Simulation results

In this section, two examples are presented to illustrate the effectiveness of the proposed controller.

Example 1

Consider the following fractional-order nonlinear system with the aforementioned restrictions

{\begin{cases} D^{0.9} x_{1} = g_{1} (x_{1}) x_{2} + φ_{1} θ \\ D^{0.8} x_{2} = g_{1} (x_{2}) k (q (u)) + φ_{2} θ + d (t) \end{cases}

(65)

where

φ_{1} (x_{1}) = {(x_{1} / 3)}^{2} (1 - \sin^{4} (x_{1}))

φ_{2} (x_{2}) = x_{1}^{3} x_{2}^{3}

θ = 1

, and

d (t) = 0.1 sin (t)

. The unknown state-dependent control gains are

g_{1} (x_{1}) = 1 + 0.1 sin (x_{1})

and

g_{2} ({\underline{x}}_{2}) = 1 + 0.1 \sin (x_{2})

. The quantized parameters described in hysteresis map (2) are chosen as

δ = 0.1

and

u_{min} = 0.02

. The desired trajectory is set to

y_{d} = \sin (0.5 t) + sin (1.5 t)

The applied actuator failures are as follows

k (q (t)) = {\begin{cases} 0.5 q (u) + 0.1 e^{0.1 t}, t \in [2 k, 2 k + 1) \\ q (u), t \in [2 k + 1, 2 k + 2) \end{cases}

(66)

The design control parameters are set to $c_{1} = 5, c_{2} = 6$ , $λ_{θ} = λ_{g} = λ_{β} = 0.2$ and $σ (t) = 7 e^{- 0.01 t}$ . The initial conditions are selected as $x (0) = {[x_{1} (0), x_{2} (0)]}^{T} = {[0.1, 0.1]}^{T}$ , $\hat{θ} (0) = 1$ , $\hat{β} (0) = 0.1$ , and ${\hat{g}}_{u} (0) = 0.1$ . The simulation results of the proposed scheme have been shown in Figures 2–6. Figure 2 illustrates the tracking performance indicating that the reference signal $y_{d}$ has been tracked asymptotically in the presence of quantized input, unknown repetitive actuator failures, and state-dependent control directions. As can be seen, the system output has reached the desired signal pretty fast. The corresponding control input, $u$ , and its quantized values are depicted in Figure 3. The quantized input levels can be observed from zoomed part of Figure 3. Figure 4 shows variations of the estimated adaptive parameters. As can be seen, all estimated parameters are bounded. The Nussbaum functions and their arguments are demonstrated in Figure 5 and Figure 6. The results indicate the effectiveness of the proposed control scheme and boundedness of all closed-loop signals.

Example 2

Consider a single-link robot system in the following integer-order form (Li and Yang, 2016)

M \ddot{a} + 0.5 m g l s i n (a) = τ

(67)

where

a

\dot{a}

, and

\ddot{a}

are the angle position, angle velocity, and angle acceleration of the link, respectively.

M

is the inertia,

g

is the gravity,

l

m,

and

τ

are the length, the mass, and the control force of the link, respectively.

Let $x_{1} = a$ , $x_{2} = \dot{a}$ , and $u = τ$ and assume that the input is quantized, the control direction is unknown, the actuator is subject to failure, and there exists an external disturbance acting on the system. Under these conditions, equation (67) can be written as

{\begin{cases} {\dot{x}}_{1} = g_{1} x_{2} \\ {\dot{x}}_{2} = g_{2} k (q (u)) + φ_{2}^{T} (x_{1}, x_{2}) θ + d (t) \\ y = x_{1} \end{cases}

(68)

As stated in Ahmed et al. (2019) and Mujumdar et al. (2015), the noncommensurate fractional model provides a more accurate description of the above system and therefore the following model has been selected for describing the system dynamics

{\begin{cases} D^{γ_{1}} x_{1} = g_{1} x_{2} \\ D^{γ_{2}} x_{2} = g_{2} k (q (u)) + φ_{2}^{T} (x_{1}, x_{2}) θ + d (t) \\ y = x_{1} \end{cases}

(69)

where

γ_{1} = 0.5

γ_{2} = 0.6

φ_{2}^{T} (x_{1}, x_{2}) = {[- g \sin (x_{1}), (x_{1} / 1) + x_{2}^{4}]}^{T}

θ = {[0.5 m l / M, 1]}^{T}

g_{1} = 1

g_{2} = 1 / M

, and

d (t) = 0.5 sin (t)

. Noted that fractional-orders,

γ_{1}

and

γ_{2}

, can choose any noninteger number in the interval

(0,1)

for the simulation purpose, but in practice, they should be fixed based on the experimental data. It is assumed that

M = 1 {kgm}^{2}

. The model parameters are chosen as

m = 1 kg

l = 1 m

, and

g = 9.8 m / s^{2}

. The desired trajectory is considered to be

y_{d} = 1 + sin (2 t)

. The selected actuator fault model is given below

k (q (t)) = {\begin{cases} q (u), t < 5 \\ 0.7 q (u) + 5, t < 5 \end{cases}

(70)

The initial values are $x (0) = {[0.5, 0.1]}^{T}$ , $\hat{θ} (0) = {[0.5, 0.5]}^{T}$ , and all the other initial conditions are set to 0.1. The quantizer parameters are $δ = 0.1$ and $u_{min} = 0.01$ . The controller design parameters are selected as $c_{1} = 7$ , $c_{2} = 4$ , $λ_{θ} = λ_{β} = λ_{g} = 0.3$ , and $σ (t) = 5 e^{- 0.1 t}$ .

To evaluate the control performance and the robustness of the proposed algorithm in the presence of variations in the unknown gain and system parameter, $M$ has been changed from $1 {kgm}^{2}$ to $3 {kgm}^{2}$ at $t = 10$ . The tracking performance has been shown in Figure 7. The simulation result shows that the desired trajectory has been tracked very fast although the system parameters have been changed, indicating the robustness of the proposed control scheme. The control input and its corresponding quantized values are demonstrated in Figure 8. Figure 9 shows variations of the adaptive parameters. Simulation results show that the control scheme is robust against the system uncertainty and variation in the control gain. As can be seen, all closed-loop signals remain bounded.

5. Conclusion

In this article, the design of an adaptive controller for a class of strict-feedback fractional-order nonlinear systems with unknown input quantization in the presence of unknown control direction, infinite number of actuator failures, and time-varying external disturbance has been addressed. With the help of the frequency-distributed model, a fractional-order adaptive backstepping controller has been designed which can be applied to both commensurate and noncommensurate order systems. To cope with input quantization and avoid chattering, the nonlinear decomposition and the hysteresis quantizer have been used. The Nussbaum gain functions have been used to tackle the problem of unknown control directions. Furthermore, it should be noted that the control gains can be functions of states. It has been shown that all closed-loop signals remain bounded and the output tracking error converges to zero asymptotically despite considered restrictions.

6. Future work

The controller design for the fractional-order systems is in its early stage. In most of published works in this area, the effectiveness of the proposed control scheme has been demonstrated via simulation and the experimental results are very rare. Evaluating the effectiveness of the proposed control scheme via experimental study can be considered as a future work.

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

Mohammad Shahrokhi

Appendix

Step 2

The fractional time derivative of $ε_{2}$ is expressed as (71)

D^{γ_{2}} ε_{2} = D^{γ_{2}} x_{2} - D^{γ_{2}} α_{1} = g_{2} ({\underline{x}}_{2}) (ε_{3} + α_{2}) + φ_{2}^{T} θ - D^{γ_{2}} α_{1}

With the help of Lemma 3, (71) can be transferred into the following frequency model (72)

{\begin{cases} \frac{\partial z_{2} (ω, t)}{\partial t} = - ω z_{2} (ω, t) + g_{2} ({\underline{x}}_{2}) (ε_{3} + α_{2}) + φ_{2}^{T} θ - D^{γ_{2}} α_{1} \\ ε_{2} = \int_{0}^{\infty} μ_{γ_{2}} (ω) z_{2} (ω, t) d ω \end{cases}

where

μ_{γ_{2}} (ω) = \sin (γ_{2} π) / ω^{γ_{2}} π

Choose the second Lyapunov function as (73)

V_{2} = V_{1} + \frac{1}{2} \int_{0}^{\infty} μ_{γ_{2}} (ω) z_{2}^{2} (ω, t) d ω

Using equation (36) and (72), it can be shown that the time derivative of $V_{2}$ satisfies the following inequality (74)

\begin{array}{l} {\dot{V}}_{2} \leq - \sum_{i = 1}^{2} \int_{0}^{\infty} ω μ_{γ_{i}} (ω) z_{i}^{2} (ω, t) d ω - c_{1} ε_{1}^{2} + (g_{1} (x_{1}) N (ξ_{1}) + 1) {\dot{ξ}}_{1} + {\hat{g}}_{u} ε_{2}^{2} \\ + ε_{2} (g_{2} ({\underline{x}}_{2}) ε_{3} + g_{2} ({\underline{x}}_{2}) α_{2} + φ_{2}^{T} θ - D^{γ_{2}} α_{1}) + \frac{1}{λ_{θ}} {\tilde{θ}}^{T} (τ_{1} - \dot{\hat{θ}}) \\ + \frac{1}{λ_{g}} {\tilde{g}}_{u} (ϱ_{1} - {\dot{\hat{g}}}_{u}) + σ (t) {\tilde{θ}}^{T} \hat{θ} + σ (t) {\tilde{g}}_{u} {\hat{g}}_{u} \end{array}

Using Young’s inequality and Assumption 2, one can obtain (75)

g_{2} ({\underline{x}}_{2}) ε_{2} ε_{3} \leq g_{2 u}^{2} ε_{3}^{2} + \frac{ε_{2}^{2}}{4} \leq g_{u} ε_{3}^{2} + \frac{ε_{2}^{2}}{4}

Substituting the above inequality into (74) and noting $θ = \tilde{θ} + \hat{θ}$ and $g_{u} = {\tilde{g}}_{u} + {\hat{g}}_{u}$ , we have (76)

\begin{array}{l} {\dot{V}}_{2} \leq - \sum_{i = 1}^{2} \int_{0}^{\infty} ω μ_{γ_{i}} (ω) z_{i}^{2} (ω, t) d ω - c_{1} ε_{1}^{2} + (g_{1} (x_{1}) N (ξ_{1}) + 1) {\dot{ξ}}_{1} \\ + ε_{2} (g_{2} ({\underline{x}}_{2}) α_{2} + {\hat{g}}_{u} ε_{2} + \frac{ε_{2}}{4} + φ_{2}^{T} \hat{θ} - D^{γ_{2}} α_{1}) + \frac{1}{λ_{θ}} {\tilde{θ}}^{T} (τ_{1} + λ_{θ} ε_{2} φ_{2}^{T} - \dot{\hat{θ}}) \\ + \frac{1}{λ_{g}} {\tilde{g}}_{u} (ϱ_{1} + λ_{g} ε_{3}^{2} - {\dot{\hat{g}}}_{u}) + {\hat{g}}_{u} ε_{3}^{2} + σ (t) {\tilde{θ}}^{T} \hat{θ} + σ (t) {\tilde{g}}_{u} {\hat{g}}_{u} \end{array}

The virtual control law $α_{2}$ , tuning functions $τ_{2}$ and $ϱ_{2}$ are chosen as given below (77)

α_{2} = N (ξ_{2}) (c_{2} ε_{2} + {\hat{g}}_{u} ε_{2} + \frac{ε_{2}}{4} + φ_{2}^{T} \hat{θ} - D^{γ_{2}} α_{1})

(78)

{\dot{ξ}}_{2} = ε_{2} (c_{2} ε_{2} + {\hat{g}}_{u} ε_{2} + \frac{ε_{2}}{4} + φ_{2}^{T} \hat{θ} - D^{γ_{2}} α_{1})

(79)

τ_{2} = τ_{1} + λ_{θ} ε_{2} φ_{2}^{T}

(80)

ϱ_{2} = ϱ_{1} + λ_{g} ε_{3}^{2}

where

c_{2} > 0

is a design constant.

Using (77)–(80), (76) can be written as (81)

\begin{array}{l} {\dot{V}}_{2} \leq - \sum_{j = 1}^{2} \int_{0}^{\infty} ω μ_{γ_{j}} (ω) z_{j}^{2} (ω, t) d ω - \sum_{j = 1}^{2} c_{j} ε_{j}^{2} + \sum_{j = 1}^{2} (g_{j} ({\underline{x}}_{j}) N (ξ_{j}) + 1) {\dot{ξ}}_{j} \\ + \frac{1}{λ_{θ}} {\tilde{θ}}^{T} (τ_{2} - \dot{\hat{θ}}) + \frac{1}{λ_{g}} {\tilde{g}}_{u} (ϱ_{2} - {\dot{\hat{g}}}_{u}) + {\hat{g}}_{u} ε_{3}^{2} + σ (t) {\tilde{θ}}^{T} \hat{θ} + σ (t) {\tilde{g}}_{u} {\hat{g}}_{u} \end{array}

Step i

( $i = 3, \dots, n - 1$ ). The fractional time derivative of $ε_{i}$ is given by (82)

\begin{array}{l} D^{γ_{i}} ε_{i} = D^{γ_{i}} x_{i} - D^{γ_{i}} α_{i - 1} \\ = g_{i} ({\underline{x}}_{i}) (ε_{i + 1} + α_{i}) + φ_{i}^{T} θ - D^{γ_{i}} α_{i - 1} \end{array}

The equivalent frequency model of (82) is as follows (83)

{\begin{cases} \frac{\partial z_{i} (ω, t)}{\partial t} = - ω z_{i} (ω, t) + g_{i} ({\underline{x}}_{i}) (ε_{i + 1} + α_{i}) + φ_{i}^{T} θ - D^{γ_{i}} α_{i - 1} \\ ε_{i} = \int_{0}^{\infty} μ_{γ_{i}} (ω) z_{i} (ω, t) d ω \end{cases}

where

μ_{γ_{i}} (ω) = \sin (γ_{i} π) / ω^{γ_{i}} π

3 \leq i \leq n - 1

Consider the following Lyapunov function candidate (84)

V_{i} = V_{i - 1} + \frac{1}{2} \int_{0}^{\infty} μ_{γ_{i}} (ω) z_{i}^{2} (ω, t) d ω

In view of (83) and the time derivative of $V_{i - 1}$ , it can be shown that the time derivative of $V_{i}$ satisfies the following inequality (85)

\begin{array}{l} {\dot{V}}_{i} \leq - \sum_{j = 1}^{i} \int_{0}^{\infty} ω μ_{γ_{j}} (ω) z_{j}^{2} (ω, t) d ω - \sum_{j = 1}^{i - 1} c_{j} ε_{j}^{2} + \sum_{j = 1}^{i - 1} (g_{j} ({\underline{x}}_{j}) N (ξ_{j}) + 1) {\dot{ξ}}_{j} + {\hat{g}}_{u} ε_{i}^{2} \\ + ε_{i} (g_{i} ({\underline{x}}_{i}) ε_{i + 1} + g_{i} ({\underline{x}}_{i}) α_{i} + φ_{i}^{T} θ - D^{γ_{i}} α_{i - 1}) + \frac{1}{λ_{θ}} {\tilde{θ}}^{T} (τ_{i - 1} - \dot{\hat{θ}}) \\ + \frac{1}{λ_{g}} {\tilde{g}}_{u} (ϱ_{i - 1} - {\dot{\hat{g}}}_{u}) + σ (t) {\tilde{θ}}^{T} \hat{θ} + σ (t) {\tilde{g}}_{u} {\hat{g}}_{u} \end{array}

Applying Young’s inequality and Assumption 2 to term $g_{i} ({\underline{x}}_{i}) ε_{i} ε_{i + 1}$ yields (86)

g_{i} ({\underline{x}}_{i}) ε_{i} ε_{i + 1} \leq g_{i u}^{2} ε_{i + 1}^{2} + \frac{ε_{i}^{2}}{4} \leq g_{u} ε_{i + 1}^{2} + \frac{ε_{i}^{2}}{4}

Utilizing (86) in (85) results in (87)

\begin{array}{l} {\dot{V}}_{i} \leq - \sum_{j = 1}^{i - 1} \int_{0}^{\infty} ω μ_{γ_{j}} (ω) z_{j}^{2} (ω, t) d ω - \sum_{j = 1}^{i - 1} c_{j} ε_{j}^{2} + \sum_{j = 1}^{i - 1} (g_{j} ({\underline{x}}_{j}) N (ξ_{j}) + 1) {\dot{ξ}}_{j} \\ - \int_{0}^{\infty} ω μ_{γ_{i}} (ω) z_{i}^{2} (ω, t) d ω + ε_{i} (g_{i} ({\underline{x}}_{i}) α_{i} + {\hat{g}}_{u} ε_{i} + \frac{ε_{i}}{4} + φ_{i}^{T} \hat{θ} - D^{γ_{i}} α_{i - 1}) \\ + \frac{1}{λ_{θ}} {\tilde{θ}}^{T} (τ_{i - 1} + λ_{θ} ε_{i} φ_{i}^{T} - \dot{\hat{θ}}) + \frac{1}{λ_{g}} {\tilde{g}}_{u} (ϱ_{i - 1} + λ_{g} ε_{i + 1}^{2} - {\dot{\hat{g}}}_{u}) \\ + {\hat{g}}_{u} ε_{i + 1}^{2} + σ (t) {\tilde{θ}}^{T} \hat{θ} + σ (t) {\tilde{g}}_{u} {\hat{g}}_{u} \end{array}

The virtual control law $α_{i}$ , tuning functions $τ_{i}$ and $ϱ_{i}$ are chosen as given below (88)

α_{i} = N (ξ_{i}) (c_{i} ε_{i} + {\hat{g}}_{u} ε_{i} + \frac{ε_{i}}{4} + φ_{i}^{T} \hat{θ} - D^{γ_{i}} α_{i - 1})

(89)

{\dot{ξ}}_{i} = ε_{i} (c_{i} ε_{i} + {\hat{g}}_{u} ε_{i} + \frac{ε_{i}}{4} + φ_{i}^{T} \hat{θ} - D^{γ_{i}} α_{i - 1})

(90)

τ_{i} = τ_{i - 1} + λ_{θ} ε_{i} φ_{i}^{T}

(91)

ϱ_{i} = ϱ_{i - 1} + λ_{g} ε_{i + 1}^{2}

where

c_{i} > 0

is a design constant.

Using (88)–(91) in (87), one can get (92)

\begin{array}{l} {\dot{V}}_{i} \leq - \sum_{j = 1}^{i} \int_{0}^{\infty} ω μ_{γ_{j}} (ω) z_{j}^{2} (ω, t) d ω - \sum_{j = 1}^{i} c_{j} ε_{j}^{2} + \sum_{j = 1}^{i} (g_{j} ({\underline{x}}_{j}) N (ξ_{j}) + 1) {\dot{ξ}}_{j} \\ + \frac{1}{λ_{θ}} {\tilde{θ}}^{T} (τ_{i} - \dot{\hat{θ}}) + \frac{1}{λ_{g}} {\tilde{g}}_{u} (ϱ_{i} - {\dot{\hat{g}}}_{u}) + {\hat{g}}_{u} ε_{i + 1}^{2} + σ (t) {\tilde{θ}}^{T} \hat{θ} + σ (t) {\tilde{g}}_{u} {\hat{g}}_{u} \end{array}

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