Adaptive actuator failure compensation for uncertain nonlinear fractional order strict feedback form systems

Abstract

In this paper, adaptive fractional control design is established for uncertain nonlinear fractional order strict feedback form systems with unknown actuator failures. The fractional actuator failure compensation problem is considered in the sense of two actuation shapes, and output matching conditions for these shapes are employed. By means of a fractional backstepping control method, two fractional adaptive state feedback control laws are designed to accomplish output tracking and to guarantee closed-loop stability in the presence of unknown actuator failures and unknown system parameters. A fractional order filter is proposed to avoid the problem of computational explosion of the backstepping design. The stability is proved via fractional order analysis method for the whole closed-loop system. Finally, simulation results for the control of fractional Chua’s circuit and fractional Genesio–Tesi systems demonstrate the effectiveness of the proposed actuator failure compensation method and output tracking property along with fast convergence of unknown parameters estimations.

Keywords

Fractional order strict feedback form systems actuator failure compensation adaptive fractional order backstepping output tracking fractional order filter

Introduction

In recent decades, fractional order (FO) systems have been extended and their applications have attracted a lot of attention of researchers. Fractional calculus that has been considered as the extension of the integer-order calculus to non-integer order calculus originated for the first time more than three centuries ago (Podlubny, 1999). With the development of physics, the application of differential equations has been extensively studied in the modeling of dynamical systems. This is mostly because of the fact that many physical processes are well characterized by FO differential equations (Jenson and Jeffreys, 1997; Si et al., 2017). For more facts on the applications of fractional calculus, one can refer to Bigdeli (2015), Kumar et al. (2016) and Aghababa (2017) and the references therein. In this regard, FO control systems have been well addressed in the literature and many advanced control methods have been developed, such as robust control (Hmed et al., 2017), sliding mode control (Onder, 2011), backstepping control (Ding et al., 2015; Wang et al., 2016) and so on. In Ding et al. (2015) and Wang et al. (2016), the adaptive stabilization of a class of FO nonlinear systems is investigated without considering the probable actuator failures. A novel adaptive sliding mode control disturbance-observer technique is suggested for the synchronization of fractional-order quadratic chaotic systems with time varying disturbances in Mofid and Mobayen (2017).

Adaptive control method adjusts controller parameters by using system response errors to achieve preferred performance. An adaptive fast terminal sliding mode combined with a global sliding mode control approach is designed to eliminate the reaching phase that improves the performance and robustness of the uncertain non-linear n-order systems in Golestani et al. (2016). Another adaptive global sliding mode control is suggested for performance improvement of disturbed nonlinear systems in Mobayen and Baleanu (2016). The study on adaptive control carries on developing with appearance of fresh interesting topics such as uncertain FO systems in recent years. For example, a sliding mode control strategy is designed for a second-order non-linear system using adaptive fuzzy compensator in Ullah (2016). Another fractional adaptive control scheme (Razmjou et al., 2017) with iterative learning sliding mode is investigated to control linear and nonlinear FO systems. A robust fractional-order adaptive intelligent controller is investigated in Bigdeli and Ziazi (2017) for stabilization of uncertain fractional-order chaotic systems. The intelligent neuro-fuzzy network is employed to estimate unknown dynamics of system.

For reliability and safety causes, actuator failure compensation (AFC) has long been an active topic in the control community (Tao et al., 2004; Wang et al., 2017). Actuator failures have a destructive outcome on the performance of control systems, resulting in unwanted consequences such as instability. Actuator failures are unknown in the sense of failure time moments, failure patterns and failure parameters. For system safety and reliability, the compensation of actuator failures is of both theoretical and practical importance.

The AFC problem has been studied via several different approaches such as multiple-model (Narendra and Balakrishnan, 1997). The main plan of multiple-model, switching and tuning designs is supposing that the controlled plant is one of the plant models in a predefined set. Regarding each plant model, a controller is designed to reach the control objectives. During plant operation, the models in the set run in parallel with the plant, and if one actuator failure occurs, the switching appliance will match the best equal model and switch to the proper controller. In Tan (2013), a novel multiple-model adaptive AFC control scheme is proposed for nonlinear systems inspired from a near-space vehicle control application.

The fault detection and diagnosis (FDD) techniques have also been employed for compensation purpose in systems with actuator failures. Using fault tolerant control methods such as fault identification based on parameter estimation, residual generation and other related techniques have been used (Blanke et al., 2016; Tao et al., 2011).

Robust control schemes (Mobayen and Tchier, 2017), which can handle parameter changes and model uncertainties, have also been utilized to accommodate certain assumed actuator failures by considering them as uncertainties. Consequently, system stability can be guaranteed and a desired closed-loop performance can be preserved in the presence of actuator failures. The robust control-based fault-tolerant approaches employ fixed parameter controllers, which are for the worst-case failures and do not adapt to variations of failure pattern and its values (Cai et al., 2013; Wu and Yang, 2015).

Another type of AFC designs are indirect or direct adaptive control based schemes. An adaptive approach, which is capable of controlling systems with uncertainties caused by failures and disturbances, is an effective scheme for AFC (Tao et al., 2014). Indirect adaptive control designs first estimate the system and failure parameters and then apply control law reconfiguration employing the healthy actuators. Direct adaptive control-based designs do not clearly engage with system and failure parameter estimation, and as an alternative they adaptively update control reconfiguration parameters online. In Yao et al. (2016), a feedback linearization-based adaptive control approach is designed for multivariable nonlinear systems in the presence of uncertain actuators failures. This adaptive controller includes a direct adaptive actuator failure compensator to neutralize the uncertain actuator failure, a nonlinear feedback to linearize the nonlinear dynamics, and a linear feedback to stabilize the linearized system.

The above-mentioned AFC methods and techniques have been used for systems described by integer order differential equations. According to our investigations in control literature, there is not any research and study regarding AFC for FO system. Therefore, we review some significant studies on FDD, and also fault tolerant control in FO systems. In Pisano et al. (2014), a discontinuous FDD observer based on second-order sliding mode techniques has been developed in order to solve certain fault detection and isolation problems for some classes of FO and integer-order uncertain switched dynamics. The aim of Aribi et al. (2014) is to propose diagnosis methods based on FO models and to validate their efficiency to detect faults occurring in thermal systems by using generalized dynamic parity space method and Luenberger diagnosis observer. The FO fault estimation observer based on the continuous frequency distributed equivalent model and indirect Lyapunov approach is proposed for a class of nonlinear FO systems in N’Doye and Laleg-Kirati (2015). The authors in Aribi et al. (2012) develop a fault detection and isolation scheme for FO systems. It is an extension to FO models of a scheme developed for integer order models to design generalized fractional observers scheme. This scheme allows to generate residuals perfectly robust to disturbances and to isolate faults.

In Shen et al. (2013), a method is developed to design FO dynamic output feedback controller for a class of FO systems with actuator faults and polytope-type uncertain parameters. The fault tolerant controllers were designed in Pettinari and Corradini (2014) and Talange and Joshi (2016) for FO systems. The problem of robust fault tolerant control for continuous-time FO systems with interval parameters and sensor faults has been studied in Song and Shen (2013). By constructing sensor fault model and state observer, an observer-based FO output feedback controller is designed.

In all previous researches on FO systems, fault has been considered as a partial loss of effectiveness of system components. These faulty components can be still operational with less efficiency, but the stability and performance of the control system reduces correspondingly. This problem has been widely studied in FO systems in last decade, and various solutions have been proposed to compensate it using different control strategies. However, failure that is a permanent fault or a total loss of effectiveness of system components requires to be studied in FO systems as a new important subject of research. Although, for AFC, actuator redundancy is the key solution, there are significant procedural concerns in the design of such control systems, such as cooperation of multiple actuators and their failure compensation. In addition, actuator failures bring together further and large system uncertainties which can be harmful for the control systems. Consequently, actuator failures not only lead to actuator gain variations, but also cause system uncertainties that need to be dealt with using new adaptive control approaches. Hence, the importance and challenge of failure compensation in control systems motivated us to fill this gap of literature in fractional fault tolerant control systems.

This paper investigates the problem of adaptive AFC for strict feedback form FO nonlinear systems. It is assumed that some parameters of system dynamics and actuator failures are unknown in advance. To deal with the unknown model, the appropriate adaptation laws are presented. Based on the update laws, an adaptive backstepping controller is proposed to stabilize the uncertain system and fulfill the output tracking based on matching designs for two different actuation models in the presence of actuator failures. By using a FO filter, the repeated differentiations of virtual control signal are avoided. The proposed compensation method does not need any information of actuator failures (time, value and pattern of failure). It only uses an adaptive feedback control strategy. The stability of the recommended schemes is mathematically proved. Numerical simulations for the control of fractional Chua’s circuit model and fractional Genesio–Tesi system with augmented actuators authenticate the desired performance of the proposed adaptive AFC approaches.

The main contributions of this research lie in that:

It is the first time that the failure is considered in FO strict feedback form systems.

Fractional adaptive laws are extracted to compensate the unknown actuator failures.

The backstepping controller is designed to ensure the stability and tracking in the presence of actuator failures and unknown system parameters.

A FO filter is proposed to avoid the problem of computational explosion of the backstepping design.

This paper is organized as follows: In Section 2, some definitions and Lemmas of FO systems and the plant model are presented. In Section 3, two different actuation models are introduced for designing procedure of the proposed adaptive AFC schemes. Two adaptive control laws for the uncertain fractional system based on the two actuation models are derived in section 4. Simulation results of designed methods are detailed in Section 5. Finally, conclusions are given in Section 6. Some directions to future works are mentioned in section 7.

Preliminaries and system description

In this section, some basic definitions of fractional calculus and necessary FO stability theorems are presented. Also, the plant model and a basic actuator failure characterization are introduced. It is worth mentioning that there are a different number of FO derivative definitions (Podlubny, 1999), but the definition of fractional integration is unique as follows:

Definition 1: (Podlubny, 1999) Riemann-Liouville fractional integration

ℐ^{α} f (t) ≜ \frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} f (τ) d τ

(1)

where $Γ (z) = \int_{0}^{\infty} t^{z - 1} e^{- t} d t$ , $t > 0, α \in R_{+}$ . Resolving Riemann–Liouville-type fractional differential equations needs the FO initial value conditions with no obvious practical denotations. Another definition has been declared by Caputo to overcome this limitation.

Definition 2: (Podlubny, 1999) Caputo fractional derivative

D^{α} f (t) = \frac{1}{Γ (m - α)} \int_{0}^{t} \frac{f^{m} (τ)}{{(t - τ)}^{α - m + 1}} f (τ) d τ

(2)

where

Γ (z) = \int_{0}^{\infty} t^{z - 1} e^{- t} d t, t > 0, α \in ℝ_{+}

(3)

Lemma 1: (Li and Sun, 2015) Consider the following FO system:

D^{α} x (t) \leq - λ_{0} x (t) + λ_{1}, α \in (0, 1)

(4)

then there exists a constant $t_{1} > 0$ such that for all $t \in (t_{1}, \infty)$ , we have

| x (t) | \leq \frac{2 λ_{1}}{λ_{0}}

(5)

where $x (t)$ is the state variable, $λ_{0}$ and $λ_{1}$ are two positive constants.

Lemma 2: (Aguila-Camacho, 2014) Let $f (t) \in R$ be a continuous and derivable function. Then

\frac{1}{2} D^{α} f^{2} (t) \leq f (t) D^{α} f (t), α \in (0, 1) .

(6)

Consider the uncertain nonlinear FO strict feedback system

D^{α} x_{i} = x_{i + 1} + θ^{T} ϕ_{i} (x_{1}, \dots, x_{i}), i = 1, 2, \dots, n - 1

(7)

D^{α} x_{n} = ϕ_{0} (x) + θ^{T} ϕ_{n} (x) + β^{T} (x) u

(8)

y = x_{1}

(9)

where $β (x) \in R^{m} - {0}$ for all $x \in R^{n}$ is a partly known vector with altered forms based on particular assumptions for special control purposes. $θ \in R^{q}$ is the unknown constant vector, $ϕ_{0} \in R$ , $ϕ_{i} \in R^{q}$ , $i = 1, 2, \dots, n$ are known smooth nonlinear functions, and $u (t) \in R^{m}$ is the input vector whose components can fail throughout system performance, and $y (t)$ is the plant output. The following assumptions are considered in this paper:

Assumption 1: All states of FO system (7)–(9) are measurable.

Assumption 2: All signals and their first n FO derivatives with FO $α$ are known and bounded.

The model of actuator failure pattern is considered as

u_{i} (t) = {\bar{u}}_{i}, t \geq t_{i}, i \in {1, 2, \dots, m}

(10)

where ${\bar{u}}_{i}$ is a constant and $t_{i}$ is the moment that the i^th actuator failure occurs. Equation (10) defines that from $t_{i}$ , the i^th actuator is stuck at some permanent value. Both ${\bar{u}}_{i}$ and $t_{i}$ are unknown.

Assumption 3: Up to m-1 actuators may suffer the actuator failures modeled as (10) and the remaining actuators can still accomplish a desired stabilization and output tracking control purpose, where applied with known parameters.

With actuator failures, the input vector $u (t)$ is

u (t) = v (t) + σ (\bar{u} - v (t))

(11)

where $v (t)$ is a vector of functional control inputs, and

\bar{u} = {[{\bar{u}}_{1}, {\bar{u}}_{2}, \dots, {\bar{u}}_{m}]}^{T}

(12)

σ = diag {σ_{1}, σ_{2}, \dots, σ_{m}}, σ_{i} = {\begin{matrix} 1 if the ith actuator fails, i . e ., u_{i} = {\bar{u}}_{i} \\ 0 otherwise . \end{matrix}

(13)

Note that numerous systems belong to the class characterized by (7)–(9). Table 1 shows the systems, including physical systems, which can be described by the model (7)–(9).

Table 1.

List of published systems, which can be characterized by the adopted model (7)–(9).

Name	Model
Magnetic bearings (Razmjou et al., 2017)	$D^{α} x_{1} = x_{2}$ $D^{α} x_{2} = - θ_{1} x_{1} - θ_{2} x_{2} + u$
Gyro system (Ding et al., 2015)	$D^{α} x_{1} = x_{2}$ $D^{α} x_{2} = - θ_{1} x_{2} - θ_{2} {x_{2}}^{3} - p (t) x_{1} + q (t) {x_{1}}^{3} + u$
Chaotic oscillator (Razmjou et al., 2017)	$D^{α} x_{1} = x_{2}$ $D^{α} x_{2} = x_{3}$ $D^{α} x_{3} = - θ (x_{1} + x_{2} + x_{3}) + sgn (x_{1}) + u$
Jerk model (Razmjou et al., 2017)	$D^{α} x_{1} = x_{2}$ $D^{α} x_{2} = x_{3}$ $D^{α} x_{3} = - θ x_{3} - x_{2} - 1.2 x_{1} + 2 sgn (x_{1}) + u$
Duffing system (Razmjou et al., 2017)	$D^{α} x_{1} = x_{2}$ $D^{α} x_{2} = - θ_{1} x_{1} - θ_{2} x_{2} - {x_{1}}^{3} + 0.3 \cos (t) + u$

For issues in fractional AFC such as controller structure, matching, unknown parameters, adaptive laws, stability and output tracking, new solutions are needed, which will be discussed in following sections.

Failure compensation schemes

Two different actuation schemes (Tao et al., 2004) are used for formulating the AFC problems.

First scheme

The first actuation shape which is employed for AFC purpose is

β^{T} (x) = [β_{1} (x), β_{2} (x), \dots, β_{m} (x)]

(14)

Consider the nominal plant as

D^{α} x_{i} = x_{i + 1} + θ^{T} ϕ_{i} (x_{1}, \dots, x_{i}), i = 1, 2, \dots, n - 1

(15)

D^{α} x_{n} = ϕ_{0} (x) + θ^{T} ϕ_{n} (x) + β_{0} (x) u_{0} (t)

(16)

where $u_{0} (x) \in R$ and $β_{0} (x) \in R - {0}$ is a known smooth nonlinear function.

Based on the Assumption 3 for AFC, when all but one actuator fail, equation (8) with (14) becomes (16) with

β_{i} (x) u_{i} (t) = β_{0} (x) u_{0} (t) .

(17)

To satisfy the output tracking purpose, we adopt the following assumption:

Assumption 4a: There exists constant scalar $k_{f 1 i}^{*}$ such that

k_{f 1 i}^{*} β_{i} (x) = β_{0} (x)

(18)

where $k_{f 1 i}^{*}$ is unknown, whereas the sign of $k_{f 1 i}^{*}$ is known and required in designing adaptive update laws.

A set of output matching conditions is designed for the system (7)–(9) with the actuation scheme (14) with actuator failures, on the basis of the controller structure

v (t) = k_{1} u_{0} (t) + k_{2}

(19)

where $k_{1}$ , $k_{2} \in R^{m}$ . Assuming that p actuators have failed, that is, $u_{j} (t) = {\bar{u}}_{j} (t)$ , $j = j_{1}, \dots, j_{p}$ , $1 \leq p \leq m - 1$ and using the actuation scheme (14), $u_{0} (t)$ is to be designed. Thus, to match $β^{T} (x) u$ to $β_{0} u_{0}$ , we need the following output matching conditions

\sum_{i \neq j_{1}, \dots, j_{p}} \frac{k_{1 i}}{k_{f 1 i}^{*}} = 1

(20)

\sum_{i \neq j_{1}, \dots, j_{p}} \frac{k_{2 i}}{k_{f 1 i}^{*}} + \sum_{i = j_{1}, \dots, j_{p}} \frac{{\bar{u}}_{i}}{k_{f 1 i}^{*}} = 0

(21)

Second scheme

The second actuation shape which is employed for AFC purpose is

β (x) = {[b_{1} β_{1} (x), b_{2} β_{2} (x), \dots, b_{m} β_{m} (x)]}^{T}

(22)

where $β_{i} (x) \in R, i = 1, 2, . . ., m,$ are known smooth nonlinear functions of x with $β_{i} (x) \neq 0,$ $\forall x \in R^{n},$ and $b_{i}, i = 1, 2, . . ., m,$ are unknown constants. To simplify the expression, we define

u_{a} (t) = {[u_{1} (t) β_{1} (x), u_{2} (t) β_{2} (x), \dots, u_{m} (t) β_{m} (x)]}^{T}

(23)

b = {[b_{1}, b_{2}, \dots, b_{m}]}^{T}

(24)

β_{a} (x) = [β_{1} (x), β_{2} (x), \dots, β_{m} (x)]

(25)

Then we rewrite (8) as

D^{α} x_{n} = ϕ_{0} (x) + θ^{T} ϕ_{n} (x) + b^{T} u_{a} .

(26)

In the presence of actuator failures, we have

u_{a} (t) = v_{a} (t) + σ ({\bar{u}}_{a} - v_{a} (t))

(27)

where $v_{a} (t) = {[β_{1} (x) v_{1} (t), β_{2} (x) v_{2} (t), \dots, β_{m} (x) v_{m} (t)]}^{T}$ with the control inputs $v_{i} (t),$ $i = 1, 2, . . ., m,$ to be designed

{\bar{u}}_{a} (t) = {[β_{1} (x) {\bar{u}}_{1} (t), β_{2} (x) {\bar{u}}_{2} (t), \dots, β_{m} (x) {\bar{u}}_{m} (t)]}^{T}

(28)

with ${\bar{u}}_{i},$ $i = 1, 2, . . ., m,$ some unknown constant actuator failure values, and the diagonal matrix $σ$ in (11) indicates the failure pattern. By using the solution to the nominal plant

D^{α} x_{i} = x_{i + 1} + θ^{T} ϕ_{i} (x_{1}, \dots, x_{i}), i = 1, 2, \dots, n - 1

(29)

D^{α} x_{n} = ϕ_{0} (x) + θ^{T} ϕ_{n} (x) + u_{0} (t)

(30)

Based on Assumption 3 for AFC, when all but one actuator fail, we obtain

b_{i} u_{i} β_{i} = b_{i} u_{ai} = u_{0} .

(31)

To satisfy the output tracking purpose, we adopt the following assumption:

Assumption 4b: There exists constant scalar $k_{f 1 i}^{*}$ such that

b_{i} k_{f 1 i}^{*} = 1, i = 1, 2, \dots, m

(32)

where $k_{f 1 i}^{*}$ is unknown, whereas the sign of $k_{f 1 i}^{*}$ is known and required in designing adaptive update laws.

A set of output matching conditions is designed for the system (7)–(9) with the actuation scheme (22) with actuator failures, on the basis of the controller structure

v_{a} (t) = k_{1} u_{0} (t) + K_{2} β_{a} (x)

(33)

where $k_{1} \in R^{m}$ and $K_{2} = {[K_{21}, K_{22}, \dots, K_{2 m}]}^{T} \in R^{m \times m}$ . Assuming that p actuators have failed, that is, $u_{aj} (t) = {\bar{u}}_{aj} (t)$ , $j = j_{1}, \dots, j_{p}$ , $1 \leq p \leq m - 1$ and using the actuation scheme (22), $u_{0} (t)$ is to be designed. Thus, to match $b^{T} u_{a}$ to $u_{0}$ , we need the following output matching conditions

\begin{matrix} \sum_{i \neq j_{1}, \dots, j_{p}} \frac{k_{1 i}}{k_{f 1 i}^{*}} = 1 \\ \sum_{i \neq j_{1}, \dots, j_{p}} \frac{K_{2 il}}{k_{f 1 i}^{*}} + \frac{{\bar{u}}_{l}}{k_{f 1 l}^{*}} = 0, l = j_{1}, \dots, j_{p} . \end{matrix}

(34)

Main results

Motivated by the developments in Ding et al. (2015) and Wang et al. (2016), the backstepping method is applied to extract the adaptive control laws for the fractional system (7)–(9) with actuator failures on the basis of aforementioned actuation schemes. Such a fractional adaptive AFC control system structure is demonstrated in Figure 1.

A FO filter is introduced as

D^{α} π_{i} = τ_{i} (μ_{i} - π_{i}), i = 1, \dots, n - 1

(35)

where $τ_{i}$ is a constant, $τ_{i} > 0, α \in (0, 1)$ and $μ_{i}$ is virtual control signal that will be given next. For bounded signal $μ_{i}$ , there is a positive constant $δ_{i}$ that satisfies $| μ_{i} - π_{i} | \leq δ_{i}$ . This FO filter avoids the complexity that arises owing to the problem of differentiating the virtual controllers repeatedly which leads to a complicated algorithm.

Theorem 1: For FO system (7)–(9) with actuation scheme (22) based on Assumption 4b and controller structure (33) and FO filter (35), the following adaptive controller guaranties system stability and the output tracking is achieved

u_{0} (t) = - C_{n} z_{n} - z_{n - 1} - ϕ_{0} - {\hat{θ}}^{T} ϕ_{n} + D^{α} π_{n - 1} (z_{1}, \dots, z_{n - 1}, \hat{θ})

(36)

μ_{i - 1} = - C_{i - 1} z_{i - 1} - z_{i - 2} - {\hat{θ}}^{T} ϕ_{i - 1} (x_{1}, \dots, x_{i - 1}) - D^{α} π_{i - 2}, i = 2, \dots, n

(37)

With adaptive laws

D^{α} \hat{θ} = Θ \sum_{l = 1}^{n} z_{l} ϕ_{l} (x_{1}, \dots, x_{l}) - \hat{θ}

(38)

D^{α} {\hat{k}}_{1 i} = - sign [k_{f 1 i}^{*}] Θ_{1 i}^{- 1} z_{n} u_{0} - {\hat{k}}_{1 i}, i = 1, \dots, m

(39)

D^{α} {\hat{k}}_{2 i} = - sign [k_{f 1 i}^{*}] Θ_{2 i}^{- 1} β_{a} z_{n} - {\hat{k}}_{2 i}, i = 1, \dots, m .

(40)

Where $C_{1}, \dots, C_{n} > \frac{1}{2}, Θ = diag [γ_{1}, \dots, γ_{q}] > 0$ , where $γ_{1}, \dots, γ_{q}$ are arbitrary positive constant numbers, $Θ_{1 i}, Θ_{2 i} \in R_{+}$ , all are free design parameters. ${\hat{k}}_{1 i}, {\hat{k}}_{2 i}$ and $\hat{θ}$ are the estimates of $k_{1 i}, k_{2 i}$ and $θ$ , respectively.

Proof: By use of adaptive backstepping technique, we have the following steps:

Step 1: Describing the tracking error $z_{i} = x_{i} - π_{i - 1}, i = 1, \dots, n$ , where $π_{0} = r (t)$ is the reference signal, hence $z_{1} = x_{1} - π_{0}$ and $z_{2} = x_{2} - π_{1}$ , we have

D^{α} z_{1} = z_{2} + π_{1} + θ^{T} ϕ_{1} - D^{α} π_{0}

(41)

By choosing the first Fractional Lyapunov Function (FLF) candidate as $V_{1} = \frac{1}{2} z_{1}^{2} + \frac{1}{2} {(\hat{θ} - θ)}^{T} Θ^{- 1} (\hat{θ} - θ)$ and using Lemma 2, we have

\begin{matrix} D^{α} V_{1} \leq z_{1} (z_{2} + (π_{1} - μ_{1}) + μ_{1} + {\hat{θ}}^{T} ϕ_{1} - D^{α} π_{0}) \\ + {(\hat{θ} - θ)}^{T} Θ^{- 1} (D^{α} \hat{θ} - Θ z_{1} ϕ_{1}) \\ \leq z_{1} (z_{2} + δ_{1} + μ_{1} + {\hat{θ}}^{T} ϕ_{1} - D^{α} π_{0}) \\ + {(\hat{θ} - θ)}^{T} Θ^{- 1} (D^{α} \hat{θ} - Θ z_{1} ϕ_{1}) . \end{matrix}

(42)

Selecting $μ_{1} = - C_{1} z_{1} - {\hat{θ}}^{T} ϕ_{1} + D^{α} π_{0}$ , we obtain

\begin{matrix} D^{α} V_{1} \leq z_{1} z_{2} - C_{1} z_{1}^{2} + δ_{1} z_{1} + {(\hat{θ} - θ)}^{T} Θ^{- 1} (D^{α} \hat{θ} - Θ z_{1} ϕ_{1}) \\ \leq z_{1} z_{2} - C_{1} z_{1}^{2} + \frac{1}{2} z_{1}^{2} + \frac{1}{2} {δ_{1}}^{2} + {(\hat{θ} - θ)}^{T} Θ^{- 1} (D^{α} \hat{θ} - Θ z_{1} ϕ_{1}) \\ \leq z_{1} z_{2} - (C_{1} - \frac{1}{2}) z_{1}^{2} + {(\hat{θ} - θ)}^{T} Θ^{- 1} (D^{α} \hat{θ} - Θ z_{1} ϕ_{1}) + \frac{1}{2} {δ_{1}}^{2} \end{matrix}

(43)

the first term $z_{1} z_{2}$ will be cancelled at the next step, and the adaptation law for $\hat{θ}$ as $D^{α} \hat{θ}$ in (43) is to be given later on.

Step 2: Let $z_{3} = x_{3} - π_{2}$ , we have

D^{α} z_{2} = z_{3} + π_{2} + θ^{T} ϕ_{2} - D^{α} π_{1}

(44)

By selecting the second FLF candidate as $V_{2} = V_{1} + \frac{1}{2} z_{2}^{2}$ , we have

D^{α} V_{2} \leq z_{1} z_{2} - C_{1} z_{1}^{2} + {(\hat{θ} - θ)}^{T} Θ^{- 1} (D^{α} \hat{θ} - \sum_{i = 1}^{2} Θ z_{i} ϕ_{i}) + z_{2} (z_{3} + δ_{2} + μ_{2} - D^{α} π_{1} + {\hat{θ}}^{T} ϕ_{2})

(45)

Selecting $μ_{2} = - z_{1} - C_{2} z_{2} - {\hat{θ}}^{T} ϕ_{1} + D^{α} π_{1}$ , we obtain

D^{α} V_{2} \leq z_{2} z_{3} - \sum_{i = 1}^{2} (C_{i} - \frac{1}{2}) z_{i}^{2} + {(\hat{θ} - θ)}^{T} Θ^{- 1} (D^{α} \hat{θ} - \sum_{i = 1}^{2} Θ z_{i} ϕ_{i}) + \frac{1}{2} {δ_{1}}^{2} + \frac{1}{2} {δ_{2}}^{2}

(46)

Step j: Let $z_{j + 1} = x_{j + 1} - π_{j}$ , we have

D^{α} z_{j} = z_{j + 1} + π_{j} + θ^{T} ϕ_{j} - D^{α} π_{j - 1} .

(47)

By selecting the j^th FLF candidate as $V_{j} = V_{j - 1} + \frac{1}{2} z_{j}^{2}$ , we have

D^{α} V_{j} \leq z_{j - 1} z_{j} - \sum_{i = 1}^{j - 1} C_{i} z_{i}^{2} + {(\hat{θ} - θ)}^{T} Θ^{- 1} (D^{α} \hat{θ} - \sum_{i = 1}^{j} Θ z_{i} ϕ_{i}) + z_{j} (z_{j + 1} + μ_{j} + {\hat{θ}}^{T} ϕ_{j} - D^{α} π_{j - 1})

(48)

Selecting $μ_{j} = - z_{j - 1} - C_{j} z_{j} - {\hat{θ}}^{T} ϕ_{j - 1} + D^{α} π_{j - 1}$ , we obtain

D^{α} V_{j} \leq z_{j} z_{j + 1} - \sum_{i = 1}^{j} (C_{i} - \frac{1}{2}) z_{i}^{2} + {(\hat{θ} - θ)}^{T} Θ^{- 1} (D^{α} \hat{θ} - \sum_{i = 1}^{j} Θ z_{i} ϕ_{i}) + \frac{1}{2} \sum_{i = 1}^{j} {δ_{j}}^{2}

(49)

Step n: First, we rewrite the n^th state equation for the actuation scheme (22) and utilizing the subsequent structure for the controller

v_{a} (t) = {\hat{k}}_{1} u_{0} (t) + {\hat{K}}_{2} β_{a} (x)

(50)

Presuming to have p actuator failures so $u_{j} (t) = {\bar{u}}_{j} (t)$ or $u_{aj} (t) = {\bar{u}}_{aj} (t)$ , $j = j_{1}, \dots, j_{p}, 1 \leq p \leq m - 1$ . From the second AFC scheme, Assumption 4b and the controller structure (50) we rewrite (8) as

\begin{matrix} D^{α} x_{n} = ϕ_{0} (x) + θ^{T} ϕ_{n} (x) + \sum_{i \neq j_{1}, \dots, j_{p}} b_{i} β_{i} v_{i} + \sum_{i = j_{1}, \dots, j_{p}} b_{i} β_{i} {\bar{u}}_{i} \\ = ϕ_{0} (x) + θ^{T} ϕ_{n} (x) + \sum_{i \neq j_{1}, \dots, j_{p}} b_{i} ({\hat{k}}_{1 i}) u_{0} + \sum_{i \neq j_{1}, \dots, j_{p}} b_{i} ({\hat{k}}_{2 i}^{T}) β_{a} \\ + \sum_{i = j_{1}, \dots, j_{p}} b_{i} β_{i} {\bar{u}}_{i} = ϕ_{0} (x) + θ^{T} ϕ_{n} (x) + u_{0} \\ + \sum_{i \neq j_{1}, \dots, j_{p}} \frac{({\hat{k}}_{1 i} - k_{1 i}) u_{0}}{k_{f 1 i}^{*}} + \sum_{i \neq j_{1}, \dots, j_{p}} \frac{{({\hat{k}}_{2 i} - k_{2 i})}^{T} β_{a}}{k_{f 1 i}^{*}} \end{matrix}

(51)

Now we apply the following last step of backstepping technique to the reshaped dynamics according to actuation scheme (22). Introducing

V_{n} = V_{n - 1} + \frac{1}{2} z_{n}^{2} + \sum_{i \neq j_{1}, \dots, j_{p}} Θ_{1 i} \frac{{({\hat{k}}_{1 i} - k_{1 i})}^{2}}{2 | k_{f 1 i}^{*} |} + \sum_{i \neq j_{1}, \dots, j_{p}} \frac{{({\hat{k}}_{2 i} - k_{2 i})}^{T} Θ_{2 i} ({\hat{k}}_{2 i} - k_{2 i})}{2 | k_{f 1 i}^{*} |}

(52)

\begin{array}{l} D^{α} V_{n} \leq - \sum_{i = 1}^{n - 1} C_{i} z_{i}^{2} + {(\hat{θ} - θ)}^{T} Θ^{- 1} (D^{α} \hat{θ} - \sum_{i = 1}^{n} Θ z_{i} ϕ_{i}) + z_{n} (z_{n - 1} + ϕ_{0} + u_{0} + {\hat{θ}}^{T} ϕ_{n} - D^{α} π_{n - 1}) \\ + z_{n} (\sum_{i \neq j_{1}, \dots, j_{p}} \frac{({\hat{k}}_{1 i} - k_{1 i}^{}) u_{0}}{k_{f 1 i}^{*}} + \sum_{i \neq j_{1}, \dots, j_{p}} \frac{{({\hat{k}}_{2 i} - k_{2 i})}^{T} β_{a}}{k_{f 1 i}^{*}}) + \sum_{i \neq j_{1}, \dots, j_{p}} Θ_{1 i} \frac{({\hat{k}}_{1 i} - k_{1 i})}{| k_{f 1 i}^{*} |} D^{α} {\hat{k}}_{1 i} \\ + \sum_{i \neq j_{1}, \dots, j_{p}} \frac{{({\hat{k}}_{2 i} - k_{2 i})}^{T} Θ_{2 i}}{| k_{f 1 i}^{*} |} D^{α} {\hat{k}}_{2 i} . \end{array}

(53)

Replacing one control law (36) and adaptive laws (38)–(40), we obtain

\begin{matrix} D^{α} V_{n} \leq - \sum_{i = 1}^{n - 1} (C_{i} - \frac{1}{2}) z_{i}^{2} - C_{n} z_{n}^{2} - {(\hat{θ} - θ)}^{T} Θ^{- 1} \hat{θ} \\ - \sum_{i \neq j_{1}, \dots, j_{p}} Θ_{1 i} \frac{({\hat{k}}_{1 i} - k_{1 i})}{| k_{f 1 i}^{*} |} {\hat{k}}_{1 i} - \sum_{i \neq j_{1}, \dots, j_{p}} \frac{{({\hat{k}}_{2 i} - k_{2 i})}^{T} Θ_{2 i}}{| k_{f 1 i}^{*} |} {\hat{k}}_{2 i} \\ + \frac{1}{2} \sum_{i = 1}^{n - 1} {δ_{i}}^{2} \end{matrix}

(54)

By completion of squares, we have

\begin{matrix} - {(\hat{θ} - θ)}^{T} Θ^{- 1} \hat{θ} = - {(\hat{θ} - θ)}^{T} Θ^{- 1} [(\hat{θ} - θ) + θ] \\ \leq - Θ^{- 1} ‖ (\hat{θ} - θ) ‖^{2} + Θ^{- 1} ‖ (\hat{θ} - θ) ‖ ‖ θ ‖ \\ \leq - \frac{1}{2} Θ^{- 1} ‖ (\hat{θ} - θ) ‖^{2} + \frac{1}{2} Θ^{- 1} ‖ θ ‖^{2} \end{matrix}

(55)

and similarly, one can obtain

- \frac{Θ_{1 i}}{| k_{f 1 i}^{*} |} ({\hat{k}}_{1 i} - k_{1 i}) {\hat{k}}_{1 i} \leq - \frac{Θ_{1 i}}{2 | k_{f 1 i}^{*} |} ({\hat{k}}_{1 i} - k_{1 i})^{2} + \frac{Θ_{1 i}}{2 | k_{f 1 i}^{*} |} k_{1 i}^{2}

(56)

{({\hat{k}}_{2 i} - k_{2 i})}^{T} \frac{Θ_{2 i}}{| k_{f 1 i}^{*} |} {\hat{k}}_{2 i} \leq - \frac{Θ_{2 i}}{2 | k_{f 1 i}^{*} |} ‖ ({\hat{k}}_{2 i} - k_{2 i}) ‖^{2} + \frac{Θ_{2 i}}{2 | k_{f 1 i}^{*} |} ‖ (k_{2 i}) ‖^{2}

(57)

Replacing (55)–(57) into (54), we have

\begin{array}{l} D^{α} V_{n} \leq - \sum_{i = 1}^{n - 1} (C_{i} - \frac{1}{2}) z_{i}^{2} - C_{n} z_{n}^{2} - \frac{1}{2} Θ^{- 1} {‖ (\hat{θ} - θ) ‖}^{2} - \sum_{i \neq j_{1}, \dots, j_{p}} \frac{Θ_{1 i}}{2 | k_{f 1 i}^{*} |} {({\hat{k}}_{1 i} - k_{1 i})}^{2} \\ - \sum_{i \neq j_{1}, \dots, j_{p}} \frac{Θ_{2 i}}{2 | k_{f 1 i}^{*} |} {‖ ({\hat{k}}_{2 i} - k_{2 i}) ‖}^{2} + \sum_{i \neq j_{1}, \dots, j_{p}} \frac{Θ_{2 i}}{2 | k_{f 1 i}^{*} |} {‖ (k_{2 i}) ‖}^{2} \sum_{i \neq j_{1}, \dots, j_{p}} \frac{Θ_{1 i}}{2 | k_{f 1 i}^{*} |} {k_{1 i}}^{2} + \frac{1}{2} Θ^{- 1} {‖ θ ‖}^{2} \\ + \frac{1}{2} \sum_{i = 1}^{n - 1} {δ_{i}}^{2} \leq - λ_{n 0} V_{n} + λ_{n 1} \end{array}

(58)

where

\begin{matrix} λ_{n 0} : = \\ \min (2 C_{1} - 1, \dots, 2 C_{n - 1} - 1, 2 C_{n}, \frac{1}{γ_{1}}, \dots, \frac{1}{γ_{q}}, \frac{Θ_{1 i}}{| k_{f 1 i}^{*} |}, \frac{Θ_{2 i}}{| k_{f 1 i}^{*} |}), \\ i = 1, \dots, m \\ λ_{n 1} : = \sum_{i \neq j_{1}, \dots, j_{p}} \frac{Θ_{2 i}}{2 | k_{f 1 i}^{*} |} ‖ (k_{2 i}) ‖^{2} + \sum_{i \neq j_{1}, \dots, j_{p}} \frac{Θ_{1 i}}{2 | k_{f 1 i}^{*} |} k_{1 i}^{2} \\ + \frac{1}{2} Θ^{- 1} ‖ θ ‖^{2} + \frac{1}{2} \sum_{i = 1}^{n - 1} {δ_{i}}^{2} \end{matrix}

(59)

To ensure that $λ_{n 0} > 0$ , the design parameters are positive and $2 C_{i} - 1 > 0, i = 1, \dots, n - 1$ . By using Lemma 1, there exists a constant $t_{1} > 0$ , such that for all $t \in (t_{1}, \infty)$ , we obtain $| V_{n} | \leq \frac{2 λ_{n 1}}{λ_{n 0}}$ . Combining with (52), we have $| z_{1} | \leq \sqrt{\frac{4 λ_{n 1}}{λ_{n 0}}}$ . Regarding Lemma 1, the error signals $z_{i} (t)$ , $\hat{θ} - θ$ , ${\hat{k}}_{1 j} - k_{1 j}$ , ${\hat{k}}_{2 j} - k_{2 j}$ converge to a compact set, where $j = 1, \dots, m$ and $i = 1, \dots, n$ . Thus, the proof is completed.

By considering $β_{1} (x) = β_{2} (x) = \dots = β_{m} (x) = β_{0} (x)$ in (22), one can find that the actuation scheme I is the special case of the actuation scheme II. Therefore, the results for scheme I are represented briefly in Remark 1.

Remark 1: Using the results of Theorem 1, the following simplified adaptive controller is obtained for system (7)–(9) with actuation scheme I (14), which is the special case of the actuation scheme II

u_{0} (t) = \frac{1}{β_{0} (x)} [- C_{n} z_{n} - z_{n - 1} - ϕ_{0} - {\hat{θ}}^{T} ϕ_{n} + D^{α} π_{n - 1} (z_{1}, \dots, z_{n - 1}, \hat{θ})]

(60)

μ_{i - 1} = - C_{i - 1} z_{i - 1} - z_{i - 2} - {\hat{θ}}^{T} ϕ_{i - 1} (x_{1}, \dots, x_{i - 1}) - D^{α} π_{i - 2}, i = 2, \dots, n

(61)

with adaptive laws

D^{α} \hat{θ} = Θ \sum_{l = 1}^{n} z_{l} ϕ_{l} (x_{1}, \dots, x_{l}) - \hat{θ}

(62)

D^{α} {\hat{k}}_{1 i} = - sign [k_{f 1 i}^{*}] Θ_{1 i}^{- 1} β_{0} z_{n} u_{0} - {\hat{k}}_{1 i}, i = 1, \dots, m

(63)

D^{α} {\hat{k}}_{2 i} = - sign [k_{f 1 i}^{*}] Θ_{2 i}^{- 1} β_{0} z_{n} - {\hat{k}}_{2 i}, i = 1, \dots, m .

(64)

Remark 2: It is clear that the parameters of the proposed control laws are not having any relationship with the unknown parameters of the system and those of actuator failures.

Numerical simulations

In this section, numerical simulations are presented to investigate the effectiveness of the proposed adaptive AFC design for two examples of uncertain nonlinear FO strict feedback form systems. The predictor-corrector method defined in Aguila-Camacho et al. (2014) with a step time of 0.005 is employed to solve the FO equations in MATLAB.

Example 1

A typical Chua’s circuit and its characteristics may be found in Kebriaei and Yazdanpanah (2010), whose nonlinear equations are given by

{\overset{\cdot}{χ}}_{1} (t) = \overset{\cdot}{a} (χ_{2} (t) - χ_{1} (t) - f (χ_{1} (t))

(65)

{\overset{\cdot}{χ}}_{2} (t) = χ_{1} (t) - χ_{2} (t) + χ_{3} (t)

(66)

{\overset{\cdot}{χ}}_{3} (t) = - \overset{\cdot}{b} χ_{2} (t) - \overset{\cdot}{c} χ_{3} (t)

(67)

where $f (χ_{1} (t)) = m_{1} χ_{1} (t) + \frac{1}{2} (m_{0} - m_{1}) (| χ_{1} (t) + 1 | - | χ_{1} (t) - 1 |)$ and the parameters $\overset{\cdot}{a} = 10, \overset{\cdot}{b} = 15, \overset{\cdot}{c} = 0.0385, m_{0} = 1.31$ and $m_{1} = 0.75$ are set accordingly.

To obtain the system dynamics form described by (7)–(9), first the normal form of Chua’s circuit equations described by (65)–(67) should be derived. By using the following diffeomorphism transformation

Ψ (χ (t)) = [\begin{matrix} x_{1} (t) \\ x_{2} (t) \\ x_{3} (t) \end{matrix}] = [\begin{matrix} χ_{3} (t) \\ - \overset{\cdot}{b} χ_{2} (t) - \overset{\cdot}{c} χ_{3} (t) \\ - \overset{\cdot}{b} χ_{1} (t) + \overset{\cdot}{b} (1 + \overset{\cdot}{c}) χ_{2} (t) + ({\overset{\cdot}{c}}^{2} - \overset{\cdot}{b}) χ_{3} (t) \end{matrix}]

(68)

With some mathematical calculations, the system (65)–(67) can be converted to

{\overset{\cdot}{x}}_{1} (t) = x_{2} (t)

(69)

{\overset{\cdot}{x}}_{2} (t) = x_{3} (t)

(70)

\begin{matrix} {\overset{\cdot}{x}}_{3} (t) = - \overset{\cdot}{a} \overset{\cdot}{b} x_{1} (t) - ((\overset{\cdot}{a} + 1) \overset{\cdot}{c} + \overset{\cdot}{b}) x_{2} (t) - (1 + \overset{\cdot}{a} + \overset{\cdot}{c}) x_{3} (t) \\ + \overset{\cdot}{a} \overset{\cdot}{b} f (\frac{- (\overset{\cdot}{b} + \overset{\cdot}{c}) x_{1} (t) - (1 + \overset{\cdot}{c}) x_{2} (t) - x_{3} (t)}{b}) + b_{u 1} u_{1} (t) \\ + b_{u 2} u_{2} (t) + b_{u 3} u_{3} (t) \end{matrix}

(71)

Based on equations (69)–(71), the following normalized FO model with three augmented actuation parameters $(b_{u 1}, b_{u 2}, b_{u 3})$ for the Chua’s circuit is presented

D^{α} x_{1} (t) = x_{2} (t)

(72)

D^{α} x_{2} (t) = x_{3} (t)

(73)

D^{α} x_{3} (t) = - θ_{1} x_{1} (t) - θ_{2} x_{2} (t) - θ_{3} x_{3} (t) + θ_{4} f (x) + b_{u 1} u_{1} (t) + b_{u 2} u_{2} (t) + b_{u 3} u_{3} (t)

(74)

y = x_{1}

(75)

where $b_{ui}$ , $i = 1, 2, 3$ , and, $θ_{j}, j = 1, 2, 3, 4$ are unknown constant parameters, which may be caused by modeling uncertainties.

Considering the plant (72)–(75) as the first actuation shape given in (14), the adaptive AFC scheme with the controller (60), FO filter (35) and update laws (62)–(64) is applied to ensure the stability and to achieve the output tracking objective, in the presence of the actuator failures

u_{1} (t) = {\begin{matrix} v_{1} (t) for t \in [02] \\ {\bar{u}}_{1} for t \in [2 \infty), \end{matrix}

(76)

u_{2} (t) = {\begin{matrix} v_{2} (t) for t \in [05] \\ {\bar{u}}_{2} for t \in [5 \infty), \end{matrix}

(77)

While $u_{3} (t) = v_{3} (t)$ , $u_{3} (t) = v_{3} (t), t \in [0 \infty)$ .

Step 1: Describing the tracking error $z_{1} = x_{1} - π_{0}, π_{0} = r (t)$ and $z_{2} = x_{2} - π_{1}$ , we have $D^{α} z_{1} = z_{2} + π_{1} - D^{α} π_{0}$ . By choosing the first FLF candidate as $V_{1} = \frac{1}{2} z_{1}^{2}$ and $μ_{1} = - C_{1} z_{1} + D^{α} π_{0}$ , we obtain $D^{α} V_{1} \leq z_{1} z_{2} - (C_{1} - \frac{1}{2}) z_{1}^{2} + {(\hat{θ} - θ)}^{T} Θ^{- 1} (D^{α} \hat{θ} - Θ z_{1} ϕ_{1}) + \frac{1}{2} {δ_{1}}^{2}$

Step 2: Let $z_{3} = x_{3} - π_{2}$ , we have $D^{α} z_{2} = z_{3} + π_{2} - D^{α} π_{1}$ . By selecting the second FLF candidate as $V_{2} = V_{1} + \frac{1}{2} z_{2}^{2}$ and $μ_{2} = - z_{1} - C_{2} z_{2} + D^{α} π_{1}$ , we obtain $D^{α} V_{1} \leq z_{1} z_{2} - (C_{1} - \frac{1}{2}) z_{1}^{2} + {(\hat{θ} - θ)}^{T} Θ^{- 1} (D^{α} \hat{θ} - Θ z_{1} ϕ_{1}) + \frac{1}{2} {δ_{1}}^{2}$

Step 3: Letting

$V = V_{2} + \frac{1}{2} z_{3}^{2} + \frac{1}{2} {(\hat{θ} - θ)}^{T} Θ^{- 1} (\hat{θ} - θ) + \sum_{i \neq j_{1}, \dots, j_{p}} Θ_{1 i} \frac{{({\hat{k}}_{1 i} - k_{1 i})}^{2}}{2 | k_{f 1 i}^{*} |} + \sum_{i \neq j_{1}, \dots, j_{p}} Θ_{2 i} \frac{{({\hat{k}}_{2 i} - k_{2 i})}^{2}}{2 | k_{f 1 i}^{*} |}$ be the last FLF candidate. The adaptive controller with the actuation scheme (14) based on Assumption 4a and controller structure (19), is

u_{0} (t) = \frac{1}{β_{0} (x)} [- C_{3} z_{3} - z_{2} - {\hat{θ}}^{T} ϕ_{3} + D^{α} π_{2}]

(78)

with adaptive laws

D^{α} \hat{θ} = Θ z_{3} ϕ_{3} - \hat{θ}

(79)

D^{α} {\hat{k}}_{1 i} = - sign [k_{f 1 i}^{*}] Θ_{1 i}^{- 1} z_{3} β_{0} u_{0} - {\hat{k}}_{1 i}, i = 1, 2, 3

(80)

D^{α} {\hat{k}}_{2 i} = - sign [k_{f 1 i}^{*}] Θ_{2 i}^{- 1} β_{0} z_{3} - {\hat{k}}_{2 i}, i = 1, 2, 3

(81)

For simulation purposes, the parameters are chosen as $α = 0.93,$ $τ_{1} = τ_{2} = 110$ , $γ_{1} = γ_{2} = γ_{3} = γ_{4} = 0.1$ , $C_{i} = 6$ , $Θ_{1 i}$ = 0.1, $Θ_{2 i}$ = 0.1, ${\hat{k}}_{1 i} (0) = 1$ , ${\hat{k}}_{2 i} (0) = 1$ , $i = 1, 2, 3$ , and ${\hat{θ}}_{j} (0) = 0$ , $j = 1, 2, 3, 4$ , with the true value: $θ_{1} = 150$ , $θ_{2} = 15.4235$ , $θ_{3} = 11.0385$ , $θ_{4} = 150$ . The initial conditions for the system are: $x_{1} (0) = 0.1, x_{2} (0) = 0.7$ and $x_{3} (0) = 1.2$ . The true value for other unknown parameters of the system and the actuator failures are: $b_{ui} = 1 / 3$ , $i = 1, 2, 3,$ and ${\bar{u}}_{1} = 20$ , ${\bar{u}}_{2} = - 20$ . The simulation results are shown in Figures 2 –5 with $r (t) = 1$ and $r (t) = \sin 2 t$ , respectively. As shown in these figures, the closed-loop system can be stabilized. Then, from Example 1 one can conclude that the adaptive AFC controller works efficiently. Besides, from Figure 2(d) and 4(d), it can be realized that the output tracking is accomplished while $u_{1}$ and $u_{2}$ fail at the 2nd and the 5th seconds, respectively, whereas $u_{3}$ does not. Figure 2(c) and 4(c) show the boundedness of control inputs although they are large in the initial time. The values of the parameter estimates ${\hat{θ}}_{1}, {\hat{θ}}_{2}, {\hat{θ}}_{3}$ and ${\hat{θ}}_{4}$ , which are shown in Figures 2(b) and 4(b), have small deviations when actuator failures occur, and then converge to constant values. In addition, the boundedness of control parameters estimations is shown in Figures 3 and 5.

Example 2

In this subsection, the compensation scheme is applied to the FO Genesio–Tesi system (Faieghi and Delavari, 2012). According to the searches of the authors, no study has been done before on failure compensation in FO strict feedback form systems. However, to show the efficiency of the proposed AFC method, a study by Aghababa published in 2016 on canonical FO systems using sliding mode controller (Aghababa, 2016) is considered, in which the stability of a healthy Genesio–Tesi system has been obtained successfully. In order to compare our proposed method with Aghababa’s work, it is supposed that a failure detection unit will act in his control system, which will switch to healthy auxiliary actuators in the case of actuator failures occurrence. The performance of sliding mode control approach in the presence of actuator failure is simulated and investigated.

Considering the plant (72)–(75) as the fractional Genesio–Tesi system dynamics, these parameters are changed, $α = 0.94$ , $f (x) = {x_{1}}^{2}$ , and the true values of $θ_{j}$ , $j = 1, 2, 3, 4$ are $θ_{1} = 6$ , $θ_{2} = 2.92$ , $θ_{3} = 1.2$ , $θ_{4} = 1$ .

Now, the adaptive fractional AFC scheme (22) with the controller (36), FO filter (35) and update laws (38)–(40) is applied to ensure the stability and to achieve the output tracking objective, in the presence of the actuator failure

u_{1} (t) = {\begin{matrix} v_{1} (t) for t \in [05] \\ {\bar{u}}_{1} for t \in [5 \infty), \end{matrix}

(82)

while $u_{2} (t) = v_{2} (t)$ and $u_{3} (t) = v_{3} (t)$ , $t \in [0 \infty)$ .

The first and second steps of controller design are exactly the same as subsection 5.1, so we proceed to step 3.

Step 3: Letting $V = V_{2} + \frac{1}{2} z_{3}^{2} + \sum_{i \neq j_{1}, \dots, j_{p}} Θ_{1 i} \frac{{({\hat{k}}_{1 i} - k_{1 i})}^{2}}{2 | k_{f 1 i}^{*} |} + \sum_{i \neq j_{1}, \dots, j_{p}} Θ_{2 i} \frac{{({\hat{k}}_{2 i} - k_{2 i})}^{2}}{2 | k_{f 1 i}^{*} |}$ be the last FLF candidate. The adaptive controller with the actuation scheme (22) based on Assumption 4b and controller structure (50), is

u_{0} (t) = - C_{3} z_{3} - z_{2} - {\hat{θ}}^{T} ϕ_{3} + D^{α} π_{2}

(83)

with adaptive laws

D^{α} \hat{θ} = Θ z_{3} ϕ_{3} - \hat{θ}

(84)

D^{α} {\hat{k}}_{1 i} = - sign [k_{f 1 i}^{*}] Θ_{1 i}^{- 1} z_{3} u_{0} - {\hat{k}}_{1 i}, i = 1, 2, 3

(85)

D^{α} {\hat{k}}_{2 i} = - sign [k_{f 1 i}^{*}] Θ_{2 i}^{- 1} β_{a} z_{3} - {\hat{k}}_{2 i}, i = 1, 2, 3

(86)

where $C_{1}, C_{3}, C_{3} > \frac{1}{2}$ are constants, $ϕ_{3} = {[- x_{1} (t) - x_{2} (t) - x_{3} (t) f (x)]}^{T}$ , $Θ = diag [γ_{1}, \dots, γ_{4}] > 0$ , $Θ_{1 i} > 0 \in R$ and $Θ_{2 i} > 0 \in R$ . ${\hat{k}}_{1 i}$ , ${\hat{k}}_{2 i}$ and $\hat{θ} = {[{\hat{θ}}_{1} {\hat{θ}}_{2} {\hat{θ}}_{3} {\hat{θ}}_{4}]}^{T}$ are the estimates of $k_{1 i}$ , $k_{2 i}$ and $θ$ , respectively. For simulation purposes, we choose the parameters as $γ_{1} = γ_{2} = γ_{3} = γ_{4} = 0.1$ , $C_{i} = 6$ , $Θ_{1 i}$ = 0.1, $Θ_{2 i}$ = 0.1, $τ_{1} = τ_{2} = 90$ , ${\hat{k}}_{1 i} (0) = 1$ , ${\hat{k}}_{2 i} (0) = 1$ , $i = 1, 2, 3$ , and ${\hat{θ}}_{j} (0) = 0$ , $j = 1, 2, 3, 4$ . The initial conditions for the system are: $x_{1} (0) = - 0.3, x_{2} (0) = 1.3$ and $x_{3} (0) = 1$ . The true value for other unknown parameters of the system and the actuator failures are: $b_{ui} = 1 / 3$ , $i = 1, 2, 3,$ and ${\bar{u}}_{1} = - 15$ . The reference signal is $r (t) = 0$ . As shown in Figure 4(b), failure occurs at the 5th second, and immediately, the proposed AFC technique starts to compensate the first actuator failure by using healthy actuators. It can be seen that the fault detection unit of the sliding mode approach switches to the healthy actuator at the 6th second for replacing the failed one. Comparison of Figures 6(a) and 6(c) shows the efficiency of the proposed AFC method. The post-failure state trajectories converge to zero faster and have less oscillation amplitudes.

Conclusions

The problem of adaptive AFC for strict feedback form FO nonlinear systems is studied in this paper. It is assumed that some parameters of system dynamics and actuator failures are unknown in advance. To tackle the unknown parameters, the appropriate adaptation laws are derived. Based on the update laws, an adaptive backstepping controller is proposed to stabilize the uncertain system and fulfill the output tracking based on matching designs for two different actuation models in the presence of unknown actuator failures. The stability of the suggested schemes is mathematically proved. Numerical simulations for the control of fractional Chua circuit and fractional Genesio–Tesi system with augmented actuators, have verified the effectiveness of the proposed adaptive AFC approaches.

Future recommendations

This paper provides opportunities for future studies as following: improvement of adaptive control system performance can be studied, particularly at the moment when actuator fails; the relaxation of the assumption on known signs of control coefficients can be investigated; and the other adaptive or intelligent feedback control strategies such as neural networks-based methods can be employed for handling uncertain actuator or sensor failures.

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.

References

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(2016) Control of non-integer-order dynamical systems using sliding mode scheme. Complexity. 21(6): 224–233.

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(2017) Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology. Nonlinear Dynamics 89(2): 1357–1370.

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