1 Introduction

With the rapid advancement of technology, the demand for a system's reliability and safety under fault scenarios is an evident and critical issue [1], especially for large-scale systems (LSS), which are complicated high-dimensional systems containing coupling, nonlinearity, and uncertainty. External disturbance and interconnection during operation are additional difficulties that increase the probability of system failure. Therefore, developing and maintaining interconnected systems with reliable control are not straightforward [2]. There are three basic control mechanisms for LSS: centralized, decentralized, and distributed control. Researchers are particularly interested in decentralized control since each subsystem relies on local information without sharing it with other subsystems. Numerous studies are concerned with decentralizing LSS [3,4,5]. In most practical applications for interconnected systems, components or subsystems can suddenly fail or break down, making the whole system unstable or malfunctioning [6,7,8]. Fault-tolerant control (FTC) is a system which automatically compensates for potential faults to ensure the entire system remains stable while providing the desired performance [9, 10]. Due to the complexity of nonlinear systems, recent research has focused on the methods that tackle the nonlinearity of the FTC system, especially for interconnected nonlinear systems; for example, the issue of decentralized adaptive tracking with actuator faults and strong interconnections is presented in [11]. In [12], a decentralized predictor control is studied, which suffers from input delay and weakly interconnections between subsystems. In [13], a fuzzy decentralized FTC with unmodeled dynamics affected by actuator faults and disturbances is discussed with unknown time delay. Fault diagnosis is the most traditional method for acquiring fault information that relies on residual signals.

In contrast, fault estimation (FE) provides essential information directly involving faults' size, time occurrence, and location without complex procedures. Research has yielded significant advances in the field of FE design based on state observers like the adaptive observer (AO) [14], the extended state observer (ESO) [15], and the sliding mode observer (SMO) [16]. Although the methods mentioned above are effective for fault estimation, the unknown input observer (UIO) ability to entirely decouple disturbances makes it an ideal architecture for FE [9]. UIO is created in this article to simultaneously estimate the fault and state, where the estimated signals are used to compensate for faults, hence achieving overall system stability. The robustness problem of the interactions between the FE and FTC was recently described in detail by Lan and Patton [17]. This strategy has been extended to encompass a variety of applications [18,19,20].

Faults increase the effort of the actuator to counteract undesirable consequences, increasing the probability of actuator saturation [21]. However, this critical topic has paid less emphasis on the FE/FTC approach, particularly when the system suffers from disturbances and uncertainty [14]. Saturation of actuators is inevitable in LSS, impairing the closed-loop system's performance and stability. So, it is essential to study the control of LSS with actuator saturation to avoid this issue. Numerous noteworthy findings concerning actuator saturation in control systems were published [22, 23]. The author in [23] proposed a decentralized non-fragile control against actuator faults and saturation. An optimization method that uses convexity based on linear matrix inequalities (LMI) formulation is employed to study bounded disturbances and actuator saturation [24]. A model predictive control (MPC) is presented in [25], but none of the previous studies mention using the FTC system.

On the other hand, a strategy of an adaptive FTC with a nonlinear system subjected to actuator fault with external disturbance and actuator saturation based on sliding mode control (SMC) with radial neural network technique is discussed in [26]. A surface with a fixed time sliding mode is created to compensate for a malfunctioning spaceship attitude system to guarantee closed-loop stability [27]. In [28], the FTC is used to handle the problem of actuator saturation, assuming that the effectiveness of the actuator loss should be constant. However, none of the mentioned research used these techniques with LSS. This fact motivates the paper to propose a new strategy based on the integration design of FE/FTC for interconnected LSS with actuator faults, saturation, external disturbance, and nonlinear interactions. The combination of the SMC with the control allocation (CA) is applied to design the FTC scheme, where the virtual control signals are converted into physical actuator requirements via the CA scheme. At the same time, virtual controllers rely on the UIO, which is developed to estimate the fault. The main contributions of this paper are:

  1. 1.

    Compared with previous studies, this work addresses the problem of saturation level within the framework of FE and compensation-based FTC for nonlinear interconnected LSS.

  2. 2.

    A new strategy combining SMC, CA, and UIO is presented based on the integration technique of FE/FTC for nonlinear interconnected LSS under actuator saturation, actuator faults, external disturbance, and nonlinear interactions. Both the controller and observer gains can be obtained concurrently through a technique of LMI.

The paper is structured as follows: The FTC scheme includes the UIO-based FE with sliding mode control allocation is presented in Sect. 2. The integrated strategy of FE/FTC is proposed in Sect. 3. The simulation results are illustrated by applying a 3-DOF helicopter model in Sect. 4. Finally, Sect. 5 discusses the conclusion of the suggested approach.

2 The proposed design of FTC based on FE

The nonlinear interconnected system for the ith subsystem is presented as follows:

$$ \begin{aligned} \dot{x}_{i} &= A_{i} x_{i} + B_{i} u_{i} + f_{i} \left( {x_{i} ,t} \right) + h_{i} \left( {x_{i} ,t} \right) + D_{i} d_{i} \hfill \\ y_{i} &= C_{i} x_{i} \hfill \\ \end{aligned} $$
(1)

where \(x_{i} \in R^{in} , y_{i} \in R^{ir}\), and \(d_{i}\) are the state, output, and external disturbance, respectively. The \(A_{i} \in R^{in \times in} ,\) \(B_{i} \in R^{in \times im}\), \(C_{i} \in R^{ip \times in} , \) and \(D_{i} \in R^{ip \times iq}\) are constant matrices. \(f_{i} \left( {x_{i} ,t} \right) \in R^{in} \) and \({\text{h}}_{i} \left( {x_{i} ,t} \right) \in R^{in} \) represent the nonlinear term and nonlinear interaction, respectively.

As shown in Fig. 1, the control input applied to the system can be described as:

$$ u_{i}^{{{\text{sat}}}} = {\text{sat}}\left( {u_{i}^{*} + f_{ai} } \right)\quad ,i = 1,2, \ldots ,n $$
(2)

where \( u_{i}^{*}\) represents the design control input, \(f_{ai} \) is the actuator fault, and \({\text{sat}}:R^{im} \to R^{im}\) represent the standard saturation function, where sat(.) can be described as:

$$ {\text{sat}}\left( {\Lambda_{i} } \right) = \left\{ {\begin{array}{*{20}c} {{\text{sign}}\left( {\Lambda_{i} } \right)u_{i\max } } & {\left| {\Lambda_{i} } \right| \ge u_{i\max } } \\ {\Lambda_{i} } & {\left| {\Lambda_{i} } \right| < u_{i\max } } \\ \end{array} } \right. $$
(3)

where \(\Lambda_{i} = u_{i}^{*} + f_{ai}\) and \(u_{\max }\) is the maximum voltage of the actuator, respectively.

Fig. 1
figure 1

The model of the actuator with fault and saturation

Full size image

Substitute the control input in (1) as:

$$ u_{i} = u_{i}^{*} + f_{ai}^{*} $$
(4)

where \( f_{ai}^{*} \) represents the actuator fault influenced by the saturation, which is described as:

$$ f_{ai}^{*} = {\text{sat}}\left( {u_{i}^{*} + f_{ai} } \right) - u_{i}^{*} $$
(5)

Due to the intervention of the saturation function, which is well known as non-differential, the composite fault \(f_{ai}^{*}\) represents the actuator fault influenced by the saturation and is non-differentiable, leading to theoretical challenges for subsequent design work of fault estimation. So, to tackle this problem, the saturation function is approximated as in (6):

$$ \overline{{{\text{sat}}}} \left( {\Lambda_{i} } \right) = \left\{ {\begin{array}{*{20}l} {\Lambda_{i} } \hfill & {0 \le \left| {\Lambda_{i} } \right| \le u_{i\max } } \hfill \\ {\Lambda_{i} - \frac{{\left[ {\Lambda_{i} - u_{i\max } {\text{sign}} \left( {\Lambda_{i} } \right)} \right]^{2} {\text{sign}}\left( {\Lambda_{i} } \right)}}{{2\delta_{i} }}} \hfill & {u_{i\max } \le \left| {\Lambda_{i} } \right| \le u_{i\max } + \delta_{i} } \hfill \\ {\left( {u_{i\max } + \frac{{\delta_{i} }}{2}} \right){\text{sign}} \left( {\Lambda_{i} } \right)} \hfill & {\left| {\Lambda_{i} } \right| \ge u_{i\max } + \delta_{i} } \hfill \\ \end{array} } \right. $$
(6)

where \(\delta_{i}\) is a constant, and \(\overline{{{\text{sat}}}} \left( {\Lambda_{i} } \right) \) is a new function that is differentiable with constrained estimation error, which can be restricted further by choosing a suitably small \(\delta_{i} .\)

Remark 1

Different approaches have been utilized to approximate the saturation function in the literature [29, 30]. However, the approximation in [17] is considered in this work.

The new control input described in (4) can be written as:

$$ \begin{aligned} u_{i} & = u_{i}^{*} + f_{ai}^{*} + \Delta u_{i} \\ f_{ai}^{*} & = \overline{{{\text{sat}}}} \left( {u_{i}^{*} + f_{ai} } \right) - u_{i}^{*} \\ \Delta u_{i} & = {\text{sat}}\left( {u_{i}^{*} + f_{ai} } \right) - \overline{{{\text{sat}}}} \left( {u_{i}^{*} + f_{ai} } \right) \\ \end{aligned} $$
(7)

This study includes the following assumptions:

Assumption 1

Assuming that the pair \((A_{i} ,C_{i} )\) is observable, and the pair \((A_{i} ,B_{i} )\) is controllable for (i = 1,2,…,n).

Assumption 2

Assume that fi (xi,t) achieves the Lipschitz constant as follows:

$$ f_{i} \left( {\hat{x}_{i} ,t} \right) - f_{i} \left( {x_{i} ,t} \right) \le L_{f} \hat{x}_{i} - x_{i} , $$
(8)

where Lf denotes the Lipschitz constant.

Assumption 3

The interaction \(h_{i} \left( {x_{i} ,t} \right)\) achieves:

$$ h_{i}^{T} \left( {x_{i} ,t} \right)h_{i} \left( {x_{i} ,t} \right) \le \sigma_{i} x_{i}^{T} H_{oi}^{T} H_{oi} x_{i} , $$
(9)

where the \(\sigma_{i}\) indicates the positive scalar, which denotes the uncertain interaction bound; Hoi is a constant matrix. For the overall system, (9) can be rewritten as follows:

$$ h^{T} \left( {x,t} \right)h\left( {x,t} \right) \le x^{T} H_{o}^{T} H_{o} x, $$
(10)

where \( h\left( {x,t} \right) = \left[ {h_{1} \left( {x_{i} ,t} \right), \ldots ,h_{n} \left( {x_{n} ,t} \right)} \right]\) aggregates the interaction for the overall interconnected system where \(H_{o} = \left[ {\sqrt {\sigma_{i} } H_{o1}^{T} , \ldots ,\sqrt {\sigma_{n} } H_{0n}^{T} } \right]\).

Assumption 4

It is presumed that the actuator's usable control is restricted by:

$$ \left| {u_{i} } \right| \le u_{\max } \quad {\text{for}}\;\left( {i = 1,2, \ldots ,n} \right) $$
(11)

in which all actuators have the same saturation value of \(u_{\max }\).

From Fig. 2, SMC is an attractive choice for designing FTC due to its inherent advantages against uncertainties. Since actuator faults can be modeled as matched uncertainty, the SMC approach "naturally" handles actuator faults from an FTC perspective. If the actuator fails to operate as required, it cannot respond to control signals; therefore, redundant actuators are essential [31]. With CA, redundant control effectors can be used efficiently to create fault tolerance without changing the underlying control rule, where the control effect is equally distributed among the actuators.

Fig. 2
figure 2

The block diagram of the suggested strategy, including the design of the observer and FTC system

Full size image

As a result of CA's flexibility, the underlying "virtual" effort can be established using any applicable control approach; therefore, CA is considered a robust candidate to be integrated with SMC to address actuator fault and saturation.

The following subsections discuss the method for developing decentralized SMC allocation for nonlinear LSS to accomplish the goal.

2.1 Control allocation (CA)

FTC researchers are interested in CA because it can tolerate actuator faults or failures without modifying the law of baseline control [32]. When the fault occurs, the control allocation unit redistributes control signals to the healthy one depending on their effectiveness level without changing the controller. Consider the system in (1) with redundant actuators to gain a better understanding of how the CA approach works by assuming that the distributed matrix Bi can be analyzed into two components:

$$ B_{i} = {\text{B}}_{iv} {\text{B}}_{iu} $$
(12)
$$ {\text{Let}}\;v_{i} = B_{iu} u_{i} $$
(13)

where \(v_{i}\) is the virtual control effort. According to (13), the control input can rewrite as:

$$ u_{i} = B_{iu}^{\dag , W} v_{i} $$
(14)

where \(B_{iu}^{\dag , W}\) represents the weighted of the right pseudo-inverse of \(B_{iu}\), that provides some freedom to the design, which can be defined as (15), and \(W = {\text{diag}}\left[ {w_{1} , \ldots , w_{n} } \right]\) represent the efficiency level of the actuators.

$$ B_{iu}^{\dag , W} = WB_{iu}^{T} \left( {B_{iu} WB_{iu}^{T} } \right)^{ - 1} $$
(15)

In case of \(w_{i} = 1\) indicates that the ith actuator is operating normally with free faults. In contrast, if \(1 > w_{i} > 0\) means that a problem is present, and the actuator is operating at a lower efficiency level, and if \(w_{i} = 0;\) this signifies that the ith actuator has failed and no longer responds to the control signal.

Substituting Eqs. (12)–(15) into (1) to obtain the following description of the new system:

$$ \begin{aligned} \dot{x}_{i} & = A_{i} x_{i} + B_{iv} B_{iu}^{\dag , W} v_{i} + f_{i} \left( {x_{i} ,t} \right) + h_{i} \left( {x_{i} ,t} \right) + D_{i} d_{i} \\ & = A_{i} x_{i} + B_{iv} \hat{v}_{i} + f_{i} \left( {x_{i} ,t} \right) + h_{i} \left( {x_{i} ,t} \right) + D_{i} d_{i} \\ \end{aligned} $$
(16)

where \(\hat{v}_{i} = B_{iu}^{\dag , W} v_{i}\).

UIO is advantageous for fault estimation in over-actuated systems. With input redundancy, control can be reallocated to restrict the use of malfunctioning actuators once estimated. It is possible to integrate the CA with the FE technique to expand the number of faulty events that can be tackled, as illustrated in Sect. 2.3.

2.2 FE based on unknown input observer (UIO)

The design of UIO has gained much attention in recent decades. Unknown input decoupling is one of the most important characteristics of a UIO for state estimation in systems exposed to unknown inputs. In most practical applications, control systems are affected by unknown inputs, including external disturbances, different types of faults, etc., resulting in system performance deficiencies. This paper is motivated by the challenge of making the error signal zero despite the effect of unknown inputs, where the estimated states are asymptotically converging toward the actual states.

An augmented UIO with substituting the virtual control can illustrate as follows:

$$ \begin{aligned} \dot{x}_{i} & = \overline{A}_{i} \overline{x}_{i} + \overline{B}_{i} \hat{v}_{i} + \overline{f}_{i} \left( {x_{i} ,t} \right) + \overline{h}_{i} \left( {x_{i} ,t} \right) + \overline{D}_{i} \overline{d}_{i} \\ y_{i} & = \overline{C}_{i} \overline{x}_{i} \\ \end{aligned} $$
(17)

where \( \overline{x}_{i} = \left[ {\begin{array}{*{20}c} {x_{i} } \\ {f_{i} } \\ \end{array} } \right]\), \(\overline{A}_{i} = \left[ {\begin{array}{*{20}c} {A_{i} } & {B_{i} } \\ 0 & 0 \\ \end{array} } \right]\), \(\overline{f}_{i} \left( {x_{i} ,t} \right) = \left[ {\begin{array}{*{20}c} {f_{i} (x_{i} , t)} \\ 0 \\ \end{array} } \right]\),

\(\overline{h}_{i} \left( {x_{i} ,t} \right) = \left[ {\begin{array}{*{20}c} {h_{i} (x_{i} , t)} \\ 0 \\ \end{array} } \right]\), \(\overline{B}_{i} = \left[ {\begin{array}{*{20}c} {B_{i} } \\ 0 \\ \end{array} } \right]\), \(\overline{D}_{i} = \left[ {\begin{array}{*{20}c} {D_{i} } & 0 \\ 0 & {I_{q} } \\ \end{array} } \right],\)

\(\overline{d}_{i} = \left[ {\begin{array}{*{20}c} {d_{i} } \\ {f_{i} } \\ \end{array} } \right]\), \(\overline{C}_{i} = \left[ {\begin{array}{*{20}c} {C_{i} } & 0 \\ \end{array} } \right]\).

The UIO can estimate the augmented state (\(\overline{x}_{i}\)) as follows:

$$ \begin{aligned} \dot{z}_{i} & = F_{i} z_{i} + G_{i} \hat{v}_{i} + K_{i} y_{i} \\ \dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\overline{x}} }_{{\varvec{i}}} & = z_{i} + H_{i} y_{i} \\ \end{aligned} $$
(18)

where \(z_{i}\) is the observer state system \(\in R^{in + iq}\), and \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\overline{x}} \in R^{in + iq }\) is the estimation of \(\overline{x}\). The following matrices \(\left[ {F_{i} , G_{i} , T_{i} , K_{i} , {\text{and}} H_{i} } \right] \) must be held to synthesize the UIO:

$$ \left( {I_{in + iq} - H_{i} \overline{{C_{i} }} } \right)\overline{D}_{i} $$
(19)
$$ T_{i} = I_{in + iq} - H_{i} \overline{C}_{i} $$
(20)
$$ F_{i} = I_{i} \overline{A}_{i} - K_{i1} \overline{C}_{i} $$
(21)
$$ G_{i} = T_{i} \overline{B}_{i} $$
(22)
$$ K_{i2} = F_{i} H_{i} $$
(23)
$$ K_{i} = K_{i1} + K_{i2} $$
(24)
$$ K_{i} = K_{i1} + F_{i} H_{i} $$
(25)

The state estimation error can be determined as: \(\dot{e}_{i} = \overline{x}_{i} - \dot{\hat{\overline{x}}}_{i} ,\) and then, the dynamic error is:

$$ \begin{aligned} \dot{e}_{i} & = \left( {T_{i} \overline{A}_{i} - K_{i1} \overline{C}_{i} } \right)e_{i} + \left( {T_{i} \overline{A}_{i} - K_{i1} \overline{C}_{i} - F_{i} } \right)z_{i} \\ & \quad + \left( {T_{i} \overline{B}_{i} - G_{i} } \right)\hat{v}_{i} + \left[ {\left( {T_{i} \overline{A}_{i} - K_{i1} \overline{C}_{i} } \right)H_{i} - K_{i2} } \right]y_{i} \\ & \quad + T_{i} \overline{f}_{i} \left( {x_{i} ,t} \right)+ T_{i} \overline{h}_{i} \left( {x_{i} ,t} \right) + T_{i} \overline{D}_{i} \overline{d}_{i} \\ \end{aligned} $$
(26)

By substituting the matrices (19)–(25), the dynamic error becomes:

$$ \dot{e}_{i} = \left( {T_{i} \overline{A}_{i} - K_{i1} \overline{C}_{i} } \right)e_{i} + T_{i} \overline{f}_{i} \left( {x_{i} ,t} \right) + T_{i} \overline{{h_{i} }} \left( {x_{i} ,t} \right) + T_{i} \overline{D}_{i} \overline{d}_{i} $$
(27)

The matrices \([F_{i} ,T_{i} ,\) and \( K_{i2} ]\) can be derived, while the matrices [\(H_{i} , K_{i1}\)] must be determined.

Because the system under consideration is over-actuated, there remains a set of control inputs that can be used to tolerate faults. This is achieved via CA system which re-allocate control efforts from failing actuators to healthy ones if any of the actuators are prone to fault. The most frequent solution is to alter the weighting matrix (W), which relies on fault information from the fault estimation module. The greater the weighting matrix gain, the lower the control input to the associated actuator. The weighting matrix is modified based on the FE module's implying that there is no control effort transferred to a specific actuator.

The signal of the virtual control will be designed and synthesized using the SMC technique, as discussed in Sect. 2.4.

2.3 Design of sliding mode control (SMC)

The SMC is among the most potent control schemes widely used in FTC applications due to its robustness against uncertainty and disturbances [33]. To construct a sliding mode controller, there are two steps. Firstly, a sliding surface is created, upon which sliding motion will happen. The next stage is to develop a control rule that relies on the switching function and compels the system state routes to slide optimally upon the sliding surface [34, 35]. Also, the reachability requirement, which ensures the sliding mode exists on the sliding surface, is an essential criterion in the sliding mode study. In SMC theory, the selected control law is composed of two components, linear and nonlinear, designed as follows:

$$ \hat{v}_{i} = \hat{v}_{il} + \hat{v}_{in} $$
(28)

where \(\hat{v}_{il}\), \(\hat{v}_{in}\) are linear and nonlinear components, respectively, described as:

$$ \hat{v}_{il} = - K_{xi} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\overline{x}}_{i} $$
(29)

and

$$ \hat{v}_{in} = - \eta_{i} {\text{sign}}\left( {s_{i} } \right), $$
(30)

with \(\eta_{i} = \hat{\omega }_{si} + \varepsilon_{si}\), where \( \varepsilon_{si}\) represents the positive scaler, and \(\hat{\omega }_{si}\) is an unknown scalar used to estimate of \(\omega_{si}\) for each subsystem, this scalar can be updated by:

$$ \dot{\hat{\omega }}_{si} = \varsigma_{i} s_{i} ,\hat{\omega }_{si} \left( 0 \right) \ge 0 $$

where \(\varsigma_{i} > 0\) represents the learning rate, and \(s_{i}\) represents the sliding surface which is defined as:

$$ s_{i} = N_{i} \hat{x}_{i} $$
(31)

where \(N_{i} = B_{i}^{ + }\) and \(B_{i}^{ + } = (B_{i}^{T} B_{i} )^{ - 1} B_{i}^{T}\). The virtual control \(\hat{v}_{i}\) must be determined to achieve the reachability condition for designing the SMC.

Remark 2

The rationale for using the adaptive term in Eq. (30) is to tackle the chattering and high activity control action associated with SMC. Further, adaptive approach enables designing SMC for systems with disturbance of unknown upper bound. Hence, by using Eq. 30 (\(\eta_{i} = \hat{\omega }_{si} + \varepsilon_{si}\), \(\dot{\hat{\omega }}_{si} = \varsigma_{i} s_{i} ,\hat{\omega }_{si} \left( 0 \right) \ge 0\), \(\varsigma_{i} > 0)\), the magnitude of the discontinuous control action becomes adaptive and can be minimized to the lowest acceptable sliding condition. Extensive explanations on adaptive SMC approaches can be found in [36, 37]

The differentiating \(s_{i}\) respective with time is:

$$\begin{aligned} \dot{s}_{i} &= N_{i} A_{i} x_{i} + \hat{v}_{i} + N_{i} f_{i} \left( {x_{i} ,t} \right) + N_{i} {\text{h}}_{i} \left( {x_{i} ,t} \right) \\ & \qquad + N_{i} D_{i} d_{i} - N_{i} \dot{e}_{i} \end{aligned}$$
(32)

Both components work together to drive the system states toward the \(s_{i}\), where the appropriate selection of the \(s_{i} \) assists in stabilizing the system after reaching it. Assuming the Lyapunov function is:

$$ V_{si} = \frac{1}{2}s_{i}^{T} s_{i} $$
(33)

and the derivative of \(V_{si}\) is:

$$ \begin{aligned} \dot{V}_{si} & = s_{i}^{T} \Big[ N_{i} A_{i} x_{i} + \hat{v}_{i} + N_{i} f_{i} \left( {x_{i} ,t} \right) + N_{i} h_{i} \left( {x_{i} ,t} \right) \\ & \qquad + N_{i} D_{i} d_{i} - N_{i} \dot{e}_{i} \Big] \\ & = s_{i}^{T} \Big[ N_{i} A_{i} x_{i} + \hat{v}_{il} + N_{i} f_{i} \left( {x_{i} ,t} \right) + N_{i} h_{i} \left( {x_{i} ,t} \right) \\ & \qquad + N_{i} D_{i} d_{i} - N_{i} \dot{e}_{i} \Big] - \hat{v}_{in} ] \\ \end{aligned} $$
(34)

Substituting (29) and (30) in (34) gives (35):

$$ \begin{aligned} \dot{V}_{si} & = s_{i}^{T} [N_{i} A_{i} x_{i} - K_{xi} \hat{x}_{i} + N_{i} f_{i} \left( {x_{i} ,t} \right) + N_{i} h_{i} \left( {x_{i} ,t} \right) \\ & \quad + N_{i} D_{i} d_{i} - N_{i} \dot{e}_{i} \left] { - \eta_{i} {\text{sign}}\left( {s_{i} } \right)} \right] \\ & \le \left( {\Psi_{si} + \omega_{si} - \eta_{i} } \right)s_{i} \\ \end{aligned} $$
(35)

where \(\Psi_{si} = N_{i} A_{i} - K_{xi}\), \( \omega_{si} = N_{i} D_{i} d_{i} + N_{i} \dot{e}_{i}\).

After the system reaches the proposed \( s_{i}\) with satisfying the sliding requirement \(s_{1} = \dot{s}_{1} = 0\), the equivalent equation is stated as:

$$ \hat{v}_{{{\text{equ}}}} = - \left[ {N_{i} A_{i} x_{i} + N_{i} f_{i} \left( {x_{i} ,t} \right) + N_{i} h_{i} \left( {x_{i} ,t} \right)N_{i} D_{i} d_{i} } \right] + \hat{v}_{il} $$
(36)

So, the system in (1) is preserved in the sliding mode with the equivalent control (36). The nonlinearity and the interactions must be reduced to ensure closed-loop robustness, which can be accomplished using the H∞ optimization described in the following section.

3 The integrated design of FE/FTC

For the overall LSS, the augmented closed-loop system can be represented as:

$$ \begin{aligned} \dot{x} & = \tilde{A}x + \tilde{F}e + f\left( {x,t} \right) + h\left( {x,t} \right) \\ \dot{e} & = \tilde{A}_{e} e + T\overline{f}\left( {x,t} \right) + T\overline{h}\left( {x,t} \right) + T\overline{D}\overline{d} \\ z & = {\text{diag}} [C_{x} x,C_{e} e] \\ \end{aligned} $$
(37)

where

$$ e \, = [e_{1} , \ldots ,e_{n} ],\overline{f}\left( {x,t} \right) = \left[ {\overline{f}_{1} \left( {x_{i} ,t} \right), \ldots ,\overline{f}_{n} \left( {x_{i} ,t} \right)} \right], $$
$$ \overline{h}\left( {x,t} \right) = \left[ {\overline{h}_{1} \left( {x_{i} ,t} \right), \ldots ,\overline{h}_{n} \left( {x_{i} ,t} \right)} \right],\overline{D} = {\text{diag}} \left[ {\overline{D}_{1} , \ldots ,\overline{D}_{n} } \right], $$
$$ T = {\text{diag}} \left[ {T_{1} , \ldots .,T_{n} } \right],\tilde{F} = {\text{diag}} \left[ {B_{1} K_{x1} \ldots .,B_{n} K_{xn} } \right], $$

\(\tilde{A} = {\text{diag}} \left( {A_{1} - B_{1} K_{x1} , \ldots ., A_{n} - B_{n} K_{xn} } \right)\),\( \tilde{A}_{e} = {\text{diag}} \left( {T_{1} \overline{A}_{1} - K_{11} \overline{C}_{1} , \ldots ,T_{n} \overline{A}_{n} - K_{n1} \overline{C}_{n} } \right)\),\( z \in R^{{2\overline{n} + \overline{q}}}\), represent the output measured that used to confirm the closed-loop system by the matrices \(C_{x} \in R^{{\overline{n} \times \overline{n}}} ,\) and \(C_{e} \in R^{{\left( {\overline{n} + \overline{q}} \right) \times \left( {\overline{n} + \overline{q}} \right)}} ,\) where, \(\overline{n} = \left[ {n_{1} , \ldots ,n_{in} } \right]\), \(\overline{q} = \left[ {q_{1} , \ldots q_{in} } \right], \) respectively.

The design problem of FE and FTC can be handled through the H∞ optimization and the LMI approach, as illustrated in Theorem 1.

Theorem 1

According to Assumptions (1–3), the system is stable with the performance of H∞ \(G_{{z\overline{d}\infty }} < \beta ,\) if the following optimization is possible:

Min \(\mathop \sum \limits_{i = 1}^{n} (\mu_{i} + \beta_{i} ),\) with subject to:

$$ \left[ {\begin{array}{*{20}c} {\Pi_{11} } & {\Pi_{12} } & {\Pi_{13} } & 0 & {\Pi_{15} } & 0 & {\Pi_{17} } \\ * & {\Pi_{22} } & 0 & {\Pi_{24} } & 0 & {\Pi_{26 } } & 0 \\ * & * & { - \delta_{3}^{ - 1} Z} & 0 & 0 & 0 & 0 \\ * & * & * & { - \delta_{2} Z} & 0 & 0 & 0 \\ * & * & * & * & { - \epsilon I} & 0 & 0 \\ * & * & * & * & * & { - \delta_{1} I} & 0 \\ * & * & * & * & * & * & { - I} \\ * & * & * & * & * & * & * \\ * & * & * & * & * & * & * \\ \end{array} } \right. $$
$$ \qquad \qquad \left. {\begin{array}{*{20}c} 0 & 0 \\ {\Pi_{28} } & {\Pi_{29} } \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ { - I} & 0 \\ * & { - \overline{\beta }} \\ \end{array} } \right] < \, 0 $$
(38)

where

\(\in = {\text{diag}} \left( { \in_{1} , \ldots , \in_{n} } \right), \mu = {\text{diag}} \left( {\left[ {\mu_{1} , \ldots ,\mu_{n} } \right]} \right),\) \(\Pi_{11} = {\text{diag}} \left( {\Pi_{111} , \ldots ,\Pi_{11n} } \right), \Pi_{11i} = {\text{He}}\left( {A_{i} Z_{i} } \right) + \delta_{i2}^{ - 1} I\), \(\Pi_{12} = {\text{diag}}\left( {\left[ {0,F_{1} } \right],..,\left[ {0,F_{n} } \right]} \right),\Pi_{13} = {\text{diag}}\left( {B_{1} M_{1i} , \ldots , B_{n} M_{1n} } \right)\).

\(\Pi_{15} = \left( {\begin{array}{*{20}c} {Z_{1} M_{011}^{T} } & \cdots & {Z_{1} M_{0n1}^{T} } \\ \vdots & \ddots & \vdots \\ {Z_{n} M_{01n}^{T} } & \cdots & {Z_{n} M_{0nn}^{T} } \\ \end{array} } \right)\), \(\Pi_{17} = \left[ {\begin{array}{*{20}c} {Z_{1} C_{x1}^{T} } \\ \vdots \\ {Z_{n} C_{xn}^{T} } \\ \end{array} } \right]\), \(\Pi_{22} = {\text{diag}} \left( {\Pi_{221} , \ldots ,\Pi_{22n} } \right) ,\)\(\Pi_{22i} = \left[ {\begin{array}{*{20}c} {T_{22i} } & {T_{23i} } \\ * & {T_{33i} } \\ \end{array} } \right],\)

\(T_{22i} = {\text{He}}\left( {Q_{i1} A_{i} - M_{4i} C_{i} A_{i} - M_{2i} C_{i} } \right)\),

\(T_{23i} = Q_{i1} F_{i} - M_{4i} C_{i} F_{i} - (M_{5i} C_{i} A_{i} + M_{3i} C_{i} )^{T}\),

\(T_{33i} = {\text{He}}\left( { - M_{5i} C_{i} F_{i} } \right)\), \(\Pi_{24} = \left[ {0,I} \right]\),

\(\Pi_{26} = {\text{diag}}\left( {Q_{1} T_{1} , \ldots , Q_{n} T_{n} } \right)\), \(Q_{i} T_{i} = \left[ {\begin{array}{*{20}c} {Q_{i1} - M_{4i} C_{i} } & 0 \\ { - M_{5i} C_{i} } & {Q_{i2} } \\ \end{array} } \right],\)

\(\Pi_{28} = \left[ {\begin{array}{*{20}c} {C_{e1}^{T} } \\ \vdots \\ {C_{en}^{T} } \\ \end{array} } \right]\), \(\Pi_{29} = {\text{diag}}\left( {\Pi_{291} , \ldots , \Pi_{29n} } \right)\), \(\Pi_{29i} = \left[ {\begin{array}{*{20}c} 0 \\ {Q_{i2} } \\ \end{array} } \right]\).

The gains of the state feedback control and the observer are:

\( K_{xi} = M_{1i} Z_{i}^{ - 1} ,\) \(K_{i11} = Q_{i1}^{ - 1} M_{2i}\), \(K_{i12} = Q_{i1}^{ - 1} M_{3i}\), \(H_{i1} = Q_{i1}^{ - 1} M_{4i}\), \(H_{i2} = Q_{i2}^{ - 1} M_{5i} \)

Proof

Consider the Lyapunov functions with s.p.d P and Q matrices:

\(V_{1} = e^{T} P\) e and \(V_{2} = x^{T} Q{\text{x}}\).

The time derivative of \(V_{1}\) concerning (37) is:

$$ \begin{aligned} \dot{V}_{1} & = e^{T} P\dot{e} + \dot{e}^{T} Pe \\ & = e^{T} P\left[ {\tilde{A}_{e} + T\overline{f}\left( {x,t} \right) + T\overline{h}\left( {x,t} \right) + T\overline{D}\overline{d}} \right] \\ & \quad + \left[ {\tilde{A}_{e} + T\overline{f}\left( {x,t} \right) + T\overline{h}\left( {x,t} \right) + T\overline{D}\overline{d}} \right]^{T} Pe \\ & = e^{T} He[P\left( {\tilde{A}_{e} } \right) + \delta_{1}^{ - 1} PTT^{T} P + \delta_{1} L_{f}^{2} I + \alpha_{1}^{ - 1} PT \\ & T^{T} P]e + \alpha_{1} x^{T} H_{0}^{T} H_{0} x + He\left( {e^{T} PT\overline{D}\overline{d}} \right) \\ & \le x^{T} [He\left( {P\tilde{A}_{e} } \right) + \delta_{1}^{ - 1} PTT^{T} P + \delta_{1} L_{f}^{2} I + \alpha_{1}^{ - 1} PT \\ & T^{T} P]e + He\left( {e^{T} PT\overline{D}\overline{d}} \right) + \alpha_{1} x^{T} H_{0} H_{0}^{T} x \\ \end{aligned} $$
(39)

where δ1 and α1 are positive scalars, with s.p.d matrice P, the system is stable \(\dot{V}_{1} < 0\).□

The time derivative of \(V_{2}\) is:

$$ \dot{V}_{2} = x^{T} Q\dot{x} + \dot{x}^{T} Qx $$
$$ \begin{aligned} \dot{V}_{2} & = x^{T} Q[\tilde{A}x + \tilde{F}e + f\left( {x,t} \right) + h\left( {x,t} \right) + D\overline{d}] \\ & \quad + \left[ {\tilde{A}x + \tilde{F}e + f\left( {x,t} \right) + h\left( {x,t} \right) + D\overline{d}} \right]^{T} Qx \\ & = x^{T} Q\tilde{A}x + x^{T} Q\tilde{F}e + x^{T} Qf\left( {x,t} \right) + x^{T} Qh\left( {x,t} \right) \\ & \quad + x^{T} QD\overline{d} + x^{T} \tilde{A}^{T} Qx + e^{T} \tilde{F}^{T} + f(x,t)^{T} \\ & \quad + h(x,t)^{T} + x^{T} QHe\left( {Q\tilde{A}} \right)x + He\left( {x^{T} Q\tilde{F}} \right) \\ & \quad + \delta_{2}^{ - 1} x^{T} Q\varphi \varphi^{T} Qx + \delta_{2} L_{f}^{2} I + x^{T} \alpha_{2}^{ - 1} QQx \\ & \quad + \alpha_{2} x^{T} H_{0} H_{0}^{T} x, \\ & \le x^{T} [He\left( {Q\tilde{A}} \right) + \delta_{2}^{ - 1} Q\varphi \varphi^{T} Q + \delta_{2} L_{f}^{2} \\ & \quad + \alpha_{2}^{ - 1} QQ]x + He\left( {x^{T} Q\tilde{F}e} \right) + He\left( {x^{T} QD\overline{d}} \right) \\ & \quad + \alpha_{2} x^{T} H_{0} H_{0}^{T} x, \\ \end{aligned} $$
(40)

where \(\varphi = I_{n} - BN,\) and δ2 and α2 are some positive scalars.

Clearly, the inequalities in (39) and (40) are obtained by substituting the upper bound of Assumptions 2 and 3 in equations of \(\dot{V}_{1}\) and \(\dot{V}_{2}\). The closed-loop system in (37) is stable if:

$$ \dot{V}_{1} + \dot{V}_{2} < 0, $$
(41)

Assuming that \(\Im = \left[ {x^{T} , e^{T} } \right]\), the \(H_{\infty }\) optimization can

be obtained as:

$$ \Omega = \mathop \int \limits_{0}^{\infty } \left( {\Im^{T} \Im - \mu^{2} \overline{d}^{T} \overline{d}} \right){\text{d}}t < 0, $$
(42)

with (zero) initial conditions:

$$ \begin{aligned} \Omega & = \mathop \smallint \limits_{0}^{\infty } \left( {\Im^{T} \Im - \mu^{2} \overline{d}^{T} \overline{d} + \dot{V}_{1} + \dot{V}_{2} } \right){\text{d}}t - \mathop \smallint \limits_{0}^{\infty } \left( {\dot{V}_{1} + \dot{V}_{2} } \right){\text{d}}t \\ & = \mathop \smallint \limits_{0}^{\infty } \left( {\Im^{T} \Im - \mu^{2} \overline{d}^{T} \overline{d} + \dot{V}_{1} + \dot{V}_{2} } \right){\text{d}}t - (V_{1} \left( \infty \right) \\ & \quad + V_{2} \left( \infty \right) + V_{1} \left( 0 \right) + V_{2} \left( 0 \right)) \\ & \le \mathop \smallint \limits_{0}^{\infty } \left( {\Im^{T} \Im - \mu^{2} \overline{d}^{T} \overline{d} + \dot{V}_{1} + \dot{V}_{2} } \right){\text{d}}t, \\ &{\text{Then}},\Im^{T} \Im - \mu^{2} \overline{d}^{T} \overline{d} + \dot{V}_{1} + \dot{V}_{2} < 0 \\ \end{aligned} $$
(43)

Substituting (39) and (40) then:

$$ \Omega = \left[ {\begin{array}{*{20}c} x \\ e \\ {\overline{d}} \\ \end{array} } \right]^{T} \left[ {\begin{array}{*{20}c} {\Omega_{11} } & {\Omega_{12} } & {\Omega_{13} } \\ * & {\Omega_{22} } & {\Omega_{23} } \\ * & * & {\Omega_{33} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} x \\ e \\ {\overline{d}} \\ \end{array} } \right] < 0 $$
(44)

Inequality (44) must equal to:

$$ \left[ {\begin{array}{*{20}c} {\Omega_{11} } & {\Omega_{12} } & {\Omega_{13} } \\ * & {\Omega_{22} } & {\Omega_{23} } \\ * & * & {\Omega_{33} } \\ \end{array} } \right] < 0 $$
(45)

where

$$\begin{aligned} \Omega_{11} &= He\left( {Q\tilde{A}} \right) + \delta_{2}^{ - 1} Q\varphi \varphi^{T} Q + \delta_{2} L_{f}^{2} I + \alpha_{2}^{ - 1} QQ^{T} \\ & \quad + \alpha_{2} H_{0} H_{0}^{T} + C_{x}^{T} C_{x} \Omega_{12} = \Omega_{12} = Q\tilde{F},\Omega_{13} = QD,\end{aligned} $$
$$\begin{aligned} \Omega_{22} &= He\left( {P\tilde{A}_{e} } \right) + \delta_{1}^{ - 1} PTT^{T} P + \delta_{1} L_{f}^{2} I + \alpha_{1}^{ - 1} PTT^{T} P \\ & \quad + C_{e}^{T} C_{e} ,\Omega_{23} = PTD, \Omega_{33} = - \mu^{2} I. \end{aligned}$$

Let Z = P-1, by multiplying both sides of (45) by the diagonal (Z, I, I), then (45) becomes:

$$ \left[ {\begin{array}{*{20}c} {\Omega_{11} } & {\Omega_{12} } & {\Omega_{13} } \\ * & {\Omega_{22} } & {\Omega_{23} } \\ * & * & {\Omega_{33} } \\ \end{array} } \right] < 0 $$
(46)
$$ \begin{aligned} \Omega_{11} & = He\left( {\tilde{A}Z} \right) + \delta_{2}^{ - 1} \varphi \varphi^{T} + \delta_{2} L_{f}^{2} ZZ + \alpha_{2}^{ - 1} I \\ & \quad + \alpha_{2} ZH_{0} H_{0}^{T} Z + ZC_{x}^{T} C_{x} Z, \Omega_{12} = \tilde{F},\Omega_{13} = D, \\ \end{aligned} $$
$$ \begin{aligned} \Omega_{22} & = He\left( {P\tilde{A}_{e} } \right) + \delta_{1}^{ - 1} PTT^{T} P + \delta_{1} L_{f}^{2} I+ \alpha_{1}^{ - 1} PTT^{T} P \\ & \quad + C_{e}^{T} C_{e} , \Omega_{23} = PTD, \Omega_{33} = - \mu^{2} I. \\ \end{aligned} $$

Remark 3

By using Schur complement theorem and after simple manipulation, the matrix inequality (38) can be easily obtained [38]. LMI toolbox can be used for obtaining the design variable that achieve negativity of (38). Hence, by satisfying Assumptions (1–3), solution of the optimization problem of Theorem 1 guarantees stability of the system with H∞ performance of \(G_{{z\overline{d}\infty }} < \beta\).

Remark 4

It is worth mentioning that references [5, 9, 15] neither deal with the actuator saturation nor the integrated design between the fault estimation and fault-tolerant control. For instance, [5] has dealt with the class of uncertain linear interconnected systems based on the backstepping method, considering the disturbance and interconnection consist of matched and mismatched parts, using the cyclic-small-gain to handle the problem of mismatched interconnections. On the other hand, the authors in [9] have dealt with uncertain nonlinear large-scale systems, using the radial basis function neural network to approximate the unknown nonlinear interconnections. Similarly, the nonlinear system with actuator fault has been considered in [15]. Compared with the research above, the proposed control strategy has employed the sliding mode control allocation fault-tolerant control to deal with nonlinear interconnected systems affect by actuator fault, saturation, and interconnections.

4 Simulation results

A three degree of freedom (3-DOF) helicopter [39] is described in this paper in order to demonstrate the efficacy of the proposed approach (see Fig. 3), which can be briefly described as follows:

Fig. 3
figure 3

The model of a 3-DOF helicopter [39]

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where ϵ(t), p(t), and γ(t) represent the elevation angle, pitch angle, and travel angle, respectively, and ϵo is the initial arm-base angle. Ff, Fb are the control voltages for the front and back motors. J1, J2, and J3 are the moments of inertia that pertain to the axis of elevation, pitch, and travel; Kf is the coefficient of force-thrust between propellers; and m is the helicopter's effective mass. The gravity constant denotes by g equals 9.8 m/s2; La is the distance of the travel axis to the helicopter's body; Lh is the distance of the pitch axis to each motor. The mathematical model of the proposed model is [32]:

$$ \begin{aligned} J_{1} \ddot{ \in } & = K_{f} L_{a} \cos p\left( {F_{f} + F_{b} } \right) - mgL_{a} \sin \left( { \in + \in_{0} } \right) + d_{1} \\ J_{2} \ddot{p} & = K_{f} L_{h} \left( {F_{f} - F_{b} } \right) + d_{2} \\ J_{3} \ddot{\gamma } & = K_{f} L_{a} \sin p\left( {F_{f} + F_{b} } \right) + d_{3} \\ \end{aligned} $$
(47)

The control system has two inputs and two outputs. Only two available control inputs are ready for control in this model: the helicopter's elevation and pitch angles. The state-space can be described as:

$$ \dot{x}_{11} = x_{12} $$
$$ \dot{x}_{12} = b_{1} u_{1} $$
$$ \dot{x}_{21} = x_{22} $$
$$ \dot{x}_{22} = b_{2} u_{2} $$
$$ y_{1} = x_{11} $$
$$ y_{2} = x_{21} $$

where \(x^{T} = \left[ {x_{11} x_{12} x_{21} x_{22} } \right] = \left[ { \in \dot{ \in }p \dot{p}} \right]\) represent the state vector, \(u^{T} = \left[ {u_{1} u_{2} } \right] = \left[ {u_{f} u_{b} } \right]\) is the vector of control inputs for front and back motors, and \(y^{T} = \left[ {y_{1} y_{2} } \right] = \left[ { \in p} \right]\) is the output vector. The values of bi for (i = 1, 2) are given by:

\(b_{1} = \frac{{K_{f} L_{a} }}{{J_{1} }} \cos x_{21}\) and \(b_{2} = \frac{{K_{f} L_{h} }}{{J_{2} }} \)

Remark 5

The pitch angle is limited during the interval.

\(\left( {\frac{{ - {\varvec{\pi}}}}{{\user2{ }2}},\user2{ }\frac{{\varvec{\pi}}}{2}} \right)\) assuming that \(x_{21} \in \left[ {\frac{ - \pi }{{ 2}} + \theta_{0} , \frac{\pi }{2} - \theta_{0} } \right]\), where \(\theta_{0}\) is a positive constant.

The helicopter's significant parameters are mentioned in Table 1.

Table 1 The parameters of a 3-DOF helicopter [39]
Full size table

Due to mechanical constraints, the pitch angle is limited to ± 31 degrees, the elevation angle is to ± 30.5 degrees, and the restriction of the maximum voltage is ± 0.2. According to Assumption 1, all subsystems are verified locally and globally controllable and observable. The required parameters are given as \(\beta = 0.2,\delta_{11} = \delta_{12} = 0.01,\delta_{21} = \delta_{22} = 10,\) \( \alpha_{1} = \alpha_{2} = 10.\) The integrated design is simulated based on the parameters solved by Theorem 1, which gives: \(\overline{\mu }_{1} = 242.031,\overline{\mu }_{2} = 139.553.\) The external disturbance \(d_{1} = 0.01,{ }\) and \(d_{2} = 0.02,{ }\) and \(h_{i} \left( {x_{i} ,t} \right)\) \(= \mathop \sum \limits_{j = 1,j \ne i}^{n} p_{ij} G_{ij} g_{ij} (x_{i} , x_{j} )\), where:

\(g_{ij} \left( {x_{i} , x_{j} } \right) = \sin \left( {\sigma_{i} - \sigma_{j} } \right) - \sin \left( {\sigma_{i0} - \sigma_{j0} } \right)\).

For the first subsystem

$$ K_{x1} = \left[ {\begin{array}{*{20}c} {9.136 } \\ {1.724} \\ \end{array} } \right], G_{1} = \left[ {\begin{array}{*{20}c} 0 \\ { - 29.102} \\ 0 \\ \end{array} } \right], $$
$$ H_{1} = \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ {0.002} & {0.015} \\ \end{array} } \right],K_{1} = \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & 0 \\ { - 0.031} & { - 0.020} \\ \end{array} } \right], $$
$$ F_{1} = \left[ {\begin{array}{*{20}c} {0.220} & { - 0.089} & { - 1.302} \\ { - 0.035} & {0.220} & { - 0.247} \\ 0 & {0.021} & { - 21.042} \\ \end{array} } \right], $$

For the second subsystem

$$ K_{x2} = \left[ {\begin{array}{*{20}c} {7.028 } \\ {2.301} \\ \end{array} } \right], G_{2} = \left[ {\begin{array}{*{20}c} 0 \\ { - 18.089} \\ 0 \\ \end{array} } \right],H_{2} = \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ {0.001} & 0 \\ \end{array} } \right], $$
$$ K_{2} = \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & 0 \\ { - 0.011} & { - 0.032} \\ \end{array} } \right], $$
$$ F_{2} = \left[ {\begin{array}{*{20}c} {0.782} & { - 0.105} & { - 1.408} \\ { - 0.301} & {0.102} & { - 0.235} \\ 0 & {0.042} & { - 18.056} \\ \end{array} } \right] $$

For the design of the FTC, the SMC is constructed when the sliding surface is obtained as:\( s_{i} = \left( { - 0.0246 - 0.8} \right)\hat{x}\),\( \partial_{i} = \hat{\omega }_{si} + 0.8\),\( \hat{\omega }_{si} \left( 0 \right) = 1\). Keeping the typical controller magnitude below 0.2 Nm to meet the actuator saturation constraint is necessary.

When adding the actuator fault to the proposed system, the performance of the control inputs, the elevation and pitch angles response for each subsystem is shown in Figs. 4, 5 and 6, respectively.

Fig. 4
figure 4

The control input for each subsystem when exceeding the saturation level

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Fig. 5
figure 5

The system state for the first subsystem without FTC

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Fig. 6
figure 6

The system state for the second subsystem without FTC

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Figure 4 illustrates the control input when the permissible value of saturation level (uimax = 0.2) is exceeded.

Figures 5 and 6 illustrate the performance of the system states for each subsystem under nominal control; it can be noticed that the states are still stable without actuator malfunction despite some overshoot at the starting due to the disturbance and nonlinear interactions. However, they exhibit unacceptable performance once the fault occurs at 50 s.

Figure 7 shows the actuator faults simulation results and their estimation for each subsystem. It is clear that the actuator faults completely match their estimation in the first and second subsystems; in contrast, some overshoot occurs at starting. Such overshoots can be limited by modifying the performance index and the parameters of pole placement.

Fig. 7
figure 7

Actuator faults and their estimation for: a the first subsystem, b the second subsystem

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According to Fig. 8, the fault estimation error will eventually converge to a value around zero. We notice a small increase occurs after 50 s due to the effect of the additive fault.

Fig. 8
figure 8

Fault estimation error

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The FTC system can be used to compensate for the influence of fault and maintains system stability. The control input responses, elevation and pitch angles are shown in Figs. 9 and 10, from which it may be noted that good control performance is attained. Moreover, the proposed controller has limited the control input to less than 0.2 Nm, mainly after the fault occurs.

Fig. 9
figure 9

The control inputs under FTC

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Fig. 10
figure 10

The system state under FTC for: a the first subsystem, b the second subsystem

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By summarizing the simulation results, compared to the nominal approach, the integrated FE/FTC design can stabilize the proposed system with optimum performance, while avoiding the actuator saturation once a fault occurs.

5 Conclusions

This paper has presented a FTC for nonlinear interconnected systems under actuator saturation. The proposal combines the SMC, CA, and fault estimation/compensation methodologies to ensure robust closed-loop performance. The integrated approach of decentralized FE/FTC is illustrated using the 3-DOF helicopter suffering from saturation, actuator fault, nonlinear interaction, and external disturbance. In the presented framework, the problem of actuator saturation within FE/FTC has been highlighted and tackled efficiently. This has been achieved via employing the decoupling capability of SMC and UIO within integrated FE/FTC loop. Simulation results have shown the sliding mode FTC allocation technique efficacy in compensating for the hard effect of saturation and actuator fault, thereby achieving the stability of the whole system.