A novel robust fixed‐time fault‐tolerant tracking control of uncertain robot manipulators

Funding information Key Research and Development Program of Shaanxi Province of China, Grant/Award Number: 2019GY086; Natural Science Foundation of Shaanxi Province, Grant/Award Number: 2020JQ-847; Scientific Research Program Funded by Shaanxi Provincial Education Department, Grant/Award Number: 20JK0914 Abstract This paper presents a novel robust fixed-time fault-tolerant control for global fixed-time tracking of uncertain robot manipulators with actuator effectiveness faults. With the sufficient consideration of the effects on uncertain dynamics, external disturbances and actuator effectiveness faults to the trajectory tracking performance, a singularity-free robust fault-tolerant control with an auxiliary vector is constructed for the fixed-time tracking control of uncertain robot manipulators. Lyapunov stability theory is employed to prove the global fixed-time stability ensuring that both the position and velocity tracking errors converge globally to the origin within a fixed time. The appealing advantages of the proposed control are as follows: (i) it is easy to implement with the global robust fixed-time fault-tolerant tracking control for uncertain robot manipulators featuring with faster transient convergence rate and higher steady-state tracking precision; (ii) the settling time is independent of the initial states of closed-loop system and can be calculated in advance for robot manipulators with uncertain dynamics, external disturbances and actuator faults. Extensive simulations on a two-DOFs robot are presented to demonstrate the effectiveness and improved performances of the proposed approach.


INTRODUCTION
Tracking control of robot manipulators with a high reliability requirement on accuracy, stability, and safety has been a critical issue in both academic and industrial applications [1]. With the sufficient consideration of uncertain dynamics, external disturbances and actuator effectiveness faults, it is still a challenge to develop a simple robust fixed-time tracking control with an improved tracking performance and transient respond for uncertain robot manipulators in the research community [2]. Since robot manipulators are a typical mechanical interconnected system, several actuator faults such as low input voltage and larger load, which causes the actuators to lose its partial effectiveness of actuators, have been affecting the tracking precision of robot system. To improve the tracking performance, transient respond and reliability of robot, thus fault-tolerant control (FTC) schemes [3] have been developed in these robot manipulators. Generally, they can be divided into two categories, that is, passive FTC (PFTC) and active FTC (AFTC).
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. For AFTC, it is worth noting that the control input depends on the feedback of fault estimation obtained from a fault diagnosis (FD) observer [4,5]. Since the requirement of an additional observer, nevertheless, these approaches featuring with high computational complexity and fault feedback time delay may decrease the tracking performance and increase the computational complexity even become unstable. In contrast, for PFTC, no requirement of any feedback from fault observer has been encountered in the design of robust control for both normal and fault operation [6]. As a result, the PFTC has faster speed to compensate the actuator fault than AFTC. However, since the fault effects imposed on the PFTC system are heavier than that of the active approaches. Consequently, it is necessary to develop a robust fault-tolerant control with higher robustness for robot manipulators in the presence of uncertain dynamics, external disturbances and partial loss of actuator effectiveness faults.
With this purpose, several approaches have been developed to improve the tracking performance of robot manipulators in the presence of uncertain dynamics, external disturbances, and partial loss of actuator effectiveness faults. In the initial approaches, PID control [7], intelligent and learning controls [8,9], optimal controls [10], robust controls [11,12] etc., have been developed. Among them, robust controls show a higher robustness and disturbance and/or fault rejection capability. Furthermore, sliding mode controls (SMC) are a well-known robust control technique since its strong robustness against uncertainties and disturbances [13][14][15]. This great feature of SMCs have been applied in the formulation of FTC systems [16,17]. A novel nonsingular fast terminal sliding mode control based on adaptive backstepping technique is developed in [18] for the fault-tolerant control of robot manipulators. By utilising the backstepping control technique, furthermore, the fault-tolerant controls [19,20] have been proposed for a class of nonlinear systems, which can obtain the improved tracking performance with finite-time convergence. Nevertheless, the settling time of these mentioned approaches is related to the initial states of the closed-loop systems, which cannot be used in a non-linear system with time-constraints tracking control.
One minor drawback of finite-time controls is that it has a slower convergence than exponential stable systems if the system states are far away from the equilibrium point. The reason behind this is that the settling time of these finite-time controls depends on the initial states of closed-loop systems. As a result, the closed-loop system under the different initial states has different convergence performance. In light of above analysis, a strong finite-time stable system named as fixed-time control has been developed in [21,22]. In comparison with conventional finite-time control systems, the fixed-time control guarantees that the settling time independently of initial states is uniformly bounded by a fixed time and calculated in advance [23]. Recognising these advantages, Tian et al. [24] proposes a continuous output feedback control scheme for the fixed-time stabilization of the double integrator system; while Zhang et al. [25] developed a prescribed fixed-time tracking control for robot manipulators by utilising a disturbance observer. Both of these approaches [24,25] use the bi-limit homogeneous technique to obtain fixed-time stable controller and observer designs. Nevertheless, the settling time of these fixed-time controls [24,25] may not be given as an exact time constant in advance since it uses the bi-limit homogeneous technique to obtain the fixed-time stable control. Based on seminal work, Zuo [26] proposes a TSMC for fixed-time stabilization of double integrators and applies for consensus tracking of second-order multi-agent systems. This fixed-time control is later extended to a class of non-linear second-order systems in the form of double integrators with matched uncertainties and perturbation [27]. By utilising a non-linear function, an approximate fixed-time sliding mode control is proposed in [28], which guarantees the system states converge to an arbitrarily small region centred at the origin. Upon analysing the existing robust fault tolerant controls, there still has one or more of following drawbacks that limit its performance in real applications [24]: (i) it does not provide a fixed time convergence; (ii) although it possesses a good transient response in normal operation, it is worse at tackling the fast variation effects of the disturbance and/or faults; and (iii) its control input has high computational complexity. Consequently, it is more desirable to develop a simple robust fixed-time fault-tolerant tracking control featuring with simplicity and robustness subject to uncertain dynamics, external disturbances and actuator effectiveness faults.
Motivated by the above analysis, this paper visits the robust fixed-time fault-tolerant tracking of robot manipulators with uncertain dynamics, external disturbances and partial loss of actuator effectiveness faults. Inspired by the works in [18,27], a robust fixed-time fault-tolerant tracking control (RFTC) is proposed. The contributions of our paper are as follows: (i) the proposed RFTC is constructed without using the acceleration of joints or the assumption that the lumped uncertainty involving the acceleration of joints are bounded by a constant, which not only overcomes the algebraic loop problem [29] but also obtains the fixed-time fault-tolerant tracking of uncertain robot manipulators; (ii) in comparison with the existing finitetime tracking controls [13,14], the convergence time of the proposed RFTC is independent of the initial states of robotic system and can be calculated in advance; (iii) compared with the fixed-time tracking controls [24,26], the proposed RFTC considers the effects of actuator faults on tracking performance, and also has more higher steady-state tracking precision and simpler control structure in both position and velocity trajectory tracking for robot manipulators with uncertainties, external disturbances and actuator faults; (iv) in comparison with the fixed-time fault-tolerant control [25], the proposed RFTC did not use the bi-limit homogeneous technique to prove the fixed-time stability, thus its settling time can be calculated from controller parameters in advance. Lyapunov stability theory is employed to prove the global fixed-time stability ensuring that both the position and velocity tracking errors converge globally to the origin within a fixed time. Simulation comparisons have been performed for uncertain robot manipulators in the presence of uncertain dynamics, external disturbances and actuator effectiveness faults. The simulation results demonstrate that the proposed controller gains the improved tracking performance including faster transient and higher steady-state tracking precision in both position and velocity trajectory tracking.
The reminder of this paper is organised as follows. In Section 2, Some preliminaries including the model and properties of robot manipulators and fixed-time stability of dynamical systems are introduced. The controller design and stability analysis are presented in Section 3. In Section 4, numerical comparisons are performed. Finally, a conclusion is included in Section 5.

Robot manipulator model and properties
The n-joints rigid manipulators are described as [30] M (q)q + C (q,̇q)̇q + g(q) = Γ + d, where q,̇q,q ∈ ℜ n denote the vectors of position, velocity and acceleration, respectively, M (q) ∈ ℜ n×n is the symmetric positive definite inertia matrix, C (q,̇q) ∈ ℜ n×n stands for the centrifugal-Coriolis matrix, g(q) ∈ ℜ n denotes the vector of gravitational torque, d ∈ ℜ n denotes the bounded external disturbances and is upper bounded by ‖d ‖ ≤ d m with a known constant d m , ∈ ℜ n is the control input, Γ= diag{ i (t )}, i = 1, … , n denotes the actuator health condition with 0 ≤ i (t ) ≤ 1, and 0 ∈ (0, 1] stands for a known positive constant.

Remark 1.
Such an actuator fault formulation described by system (1) can be found in many previous results, such as [31][32][33]. Since the faults of control circuit or servo system was often happened in the application of robot manipulators, partial loss of actuator effectiveness is a kind of common fault of robot systems, which always affects the tracking precision and even stability of closed-loop system. Thus, the assumption of actuator faults for system (1) is reasonable and a question worth studying.

Remark 2.
It will be observed from system (1) that i (t ) = 1 indicates that the robot manipulators is fault-free; while 0 ≤ i (t ) ≤ 1 represents the ith actuator partial loses its effort.
In order to facilitate the following analysis, the proposed robust fixed-time fault-tolerant tracking control for uncertain robot manipulators will be accomplished on the following fundamental facts [13,30].
The subsequent development is based on the assumption that q anḋq are available, and the desired trajectory q d ∈ ℜ n be C 2 for the robotic system. Additionally, the following assumption will be exploited [13].
Without loss of generality, it is assumed that the norms of desired vectors are upper bounded by the following positive constants where q d ,̇q d ,q d ∈ ℜ n are the vectors of desired position, velocity and acceleration, respectively, and P p , P v and P a are some known positive constants.
To facilitate the following design and analysis, we define the vector Sgn( ) ∈ ℜ n and Sig ( ) ∈ ℜ n as follows: where > 0, sig ( i ) = | i | sign( i ), i = 1, ⋅ ⋅ ⋅, n with i denotes the ith components of ∈ ℜ n , and sign(⋅) denotes the standard signum function.
The objective of this paper is to design a novel robust fixedtime fault-tolerant tracking control (RFTC) for robot manipulators subject to the uncertain dynamics, external disturbances and actuator effectiveness faults such that both the position and velocity tracking errors converge globally to the origin within a fixed time. To quantify this objective, the definition of the position and velocity tracking errors are defined as follows:

Fundamental facts
To accomplish the subsequent design and analysis, the following fundamental facts are introduced.

x) is said to be globally fixed-time stable if the settling time function T is globally bounded, that is, there exists a fixed constant T
Lemma 1. Consider a scalar system [26] where v 1 and v 2 are two known positive constants satisfied v 1 > 1 and 0 < v 2 < 1, and > 0 and > 0 denote some positive constants which depend on controller parameters. Then, the equilibrium point of system (7) is fixed-time stable, and the settling time T is derived as Lemma 2. For ∈ ℜ + , x ∈ ℜ, the following inequalities hold [14] d dt |x| where sign(⋅) denotes the standard signum function.

CONTROL DESIGN AND STABILITY ANALYSIS
To facilitate the subsequent control design and stability analysis, we begin with the open-loop error system development aimed to obtain an upper bound of the lumped

Open-loop system development
Based on Assumption 1, the system (1) can be rewritten as where the lumped uncertainty ∈ ℜ n is defined as In light of (1) and (2), it follows that where E ∈ ℜ n×n defined by [29] with I n denoting the n × n identity matrix.
Observed by the work [29], once M 0 (q) is chosen as where 1 and 2 are two known positive constants given by then E is upper bounded by [29] ‖E‖ ≤ with standing for a known positive constant given by Note that M 0 (q) is written as M 0 in the subsequent development owing to the M 0 (q) defined by (14) is defined as a constant matrix. By virtue of Assumption 1, the lumped uncertainty given by (11) is upper bounded by [15] where a i , i = 0, 1 denote two positive constants that depend on the robotic system, and is defined by (17). The proof of (18) can be found in Appendix.

Control formulation
Upon substituting (6) into (10), the error closed-dynamic equation forë takes where M 0 is a constant matrix given by (14), the lumped uncertainty ∈ ℜ n is defined by (11) and the nominal part ∈ ℜ n is described as follows: To facilitate the subsequent control design, we defined an auxiliary vector as follows: where ∈ ℜ + denotes a given constant, anḋe and e are given by (6).
For system (19) and the auxiliary vector (21), then, a robust fixed-time fault-tolerant tracking control (RFTC) is defined as where ∈ ℜ n is defined by (20) and with is given by (21) and where > 0, > 0, p > 1 and r > 1 are some positive constants, 1 , 2 , , a 0 and a 1 are defined by (15), (17) and (18), respectively, and 0 is defined by (1) satisfied 0 > . (22)- (26), the formulation of control components 1 and 2 does not include its upper bounds ‖ 1 ‖ and ‖ 2 ‖, respectively. As a result, the proposed RFTC can overcomes the algebraic loop problem [29] completely. Compared with the existing robust controls, the uncertain dynamics, external disturbances and partial loss of actuator effectiveness faults will be considered adequately in the formulation of the proposed RFTC given by (22)- (26), which also preserve a simple control structure and appropriate controller gains to implement the trajectory tracking of robot manipulators.

Remark 4.
Observed by (15) and (17), for most robot manipulators systems, we also can obtain a small enough because of the derivation of 1 and 2 comes from the fact 1 On the other hand, the parameters can be modified as = 1 − 2 1 ∕( 1 + 2 ) from (17), which can also prove that is small enough since the fact 2 ≫ 1 can always be established on robot manipulators. Accordingly, the assumption 0 > is realistic and reasonable observed from (15). (22)- (26), the nominal parts of robot manipulators have been used in the formulation of the proposed approach. Moreover, the control term given by (20) can also be modified as = − M 0qd , which implies that no prior knowledge of robot manipulators have been used in the formulation of the proposed approach observed from (22)- (26). As a result, the proposed RFTC can be further converted to a fixed-time robust tracking controller in which no prior knowledges of robot manipulators are used in the formulation of the proposed approach.

Stability analysis
For system (27), we are in a position to state the following result.
This completes the proof. □

Remark 6. The proposed RFTC does not refer to model parameters in the control law formulation and would gain global robust fixed-time fault-tolerant tracking of robot manipulators in the presence of uncertain dynamics, bounded external disturbances and and partial loss of actuator effectiveness faults. Compared with the existing robust finite-time stable controls, the settling time of the proposed approach is independent of the initial states and can be calculated in advance. As a result, observed from the section "Stability analysis", the proposed RFTC has higher steady-state tracking precision due to the position and velocity tracking errors e anḋe converge globally to the origin within a fixed time T r simultaneously.
Remark 7. Different from the work [25], the proposed approach removes the assumption that the acceleration has an upper bound. Since the work [25] uses bi-limit homogeneous technique to prove the fixed-time stability in sliding phase of sliding mode control (see (9) of [25]); moreover, it cannot provide an exact convergence time in advance. In other words, the fixed settling time of the work [25] cannot be calculated from the controller parameters. In contrast, the settling time of the proposed approach has been obtained from (28) in advance. Accordingly, the proposed RFTC offers an alternative approach for improving the tracking performance of robot manipulators subject to uncertain dynamics, external disturbances and actuator faults. (26) proposed in this paper has some components that contains a signum function, which may result in chattering. The chattering situation not only decreases the tracking performance but also even damages the actuator of robot manipulators. The chattering will be eliminated by replacing the signum function to the following function [35] sign( i )

Remark 8. The robust fixed-time fault-tolerant control (22)-
where denotes a known constant.

SIMULATION COMPARISONS
Consider the dynamics of two-DOFs robot are given by [13] M (q) = [ with The sampling period is 1 ms. The initial conditions are In this section, we verify the effectiveness of the proposed RFTC in the following two aspects: (i) upon the sufficient consideration of uncertain dynamics, external disturbances and actuator effectiveness faults, we have focused on the improved convergence property of the proposed RFTC in both position and velocity trajectory tracking; (ii) it is emphasised on the advantage of the proposed RFTC in convergence time, which means the settling time is independent of the initial conditions and only related to the controller parameters.

Tracking performance with actuator effectiveness faults
In order to show the improved tracking performance, the uncertain dynamics (1), external disturbances (1) and the following actuator effectiveness faults will be considered in the following simulation comparisons. The actuator health condition (1) will be given as where the first and second actuators are assumed to be lost up to its 31% and 35% effectiveness from the time 8 s, respectively. By considering the uncertain dynamics (2) and partial loss of actuator effectiveness faults (43), accordingly, we have involved the tracking performance of the proposed RFTC in fault operations compared with finite-time integral backstepping control (FIBC) [20], computed torque controller (CTC) [36], PID controller [37], PID-based SMC (PID-SMC) [38], and non-singular terminal SMC (NTSMC) [39,40]. Upon these seminal works, they are designed as follows [18].
For FIBC controller [20], it is given as wherė= z, Sig (⋅) is defined by (5) with 0 < < 1, K i ∈ ℜ n×n , i = 1, ⋅ ⋅ ⋅, 6 stand for the diagonal positive matrixes, k 1i and k 4i denote the ith components of the matrixes K 1 and K 4 , respectively, and i , , and Δ i stand for some small positive constants.
For the CTC [36] and PID [37] controllers, they can be designed as where K p , K d and K i denote the proportional, differential and integral gain matrix, respectively. Then, the PID-SMC sliding surface and controller [38] are given as with eq = − M 0 where K p , K d and K i are given by (50), Sgn(⋅) is defined by (4),Δ denotes the upper bound of the lumped uncertainty and actuator faults with ‖Δ‖ ≤Δ and denotes arbitrarily small positive constant. While the NFTSM controller and sliding surface [39,40] are designed as where k i = diag{k i1 , … , k in } ∈ ℜ n×n , i = 1, 2 stand for the diagonal constant matrixes, p and q are two odd constants satisfying 1 < p∕q < 2, > 1 denotes a positive constant, Sgn(⋅) and Sig r (⋅) are defined by (4) and (5), respectively, andΔ and are defined by (54). For above controllers, by using (36) and the above given system parameters, the lower and upper bounds of the inverse inertial matrix M (q) defined by (15) are given by 1 = 0.09 and 2 = 0.2, and hence M 0 = 6.89I n and = 0.38 given by (14) and (17) in [29]. Furthermore, C 0 (q,̇q) and g 0 (q) are chosen by replacing m 1 and m 2 of (37)-(39) with the nominal onesm 1 and m 2 . The selected parameters of these controllers are reported in Table 1.
First, we complete the simulation comparisons with the typical robust controllers (FIBC (44)-(48), CTC (49) and PID (50)). Figure 2 depicts the position tracking of the proposed RFTC. Figures 3 and 4 show the position and velocity tracking   Figures 3 and  4 at the range 0 to 8 s represents a better tracking performance including faster transient rate and higher steady-state precision than the FIBC, PID and CTC. Obviously, the CTC shows worse robustness than the FIBC, PID and the proposed RFTC especially in the presence of uncertain dynamics and external disturbance. Furthermore, the uncertain robot manipulators will be affected by the actuator faults after the time 8 s from Figures 3 and 4, while the FIBC, CTC and PID provide worse tracking performance. No matter whether the system is affected by the uncertain dynamics, external disturbance and/or actuator effectiveness faults, consequently, we can conclude that the proposed RFTC always provides faster transient rate and higher steady-state precision in both position and velocity trajectory tracking. Moreover, these superior tracking performances of the proposed RFTC have been obtained without using the excessive control input observed by Figure 5.
Secondly, we have accomplished the simulation comparisons with the typical robust fault-tolerant controllers (PID-SMC and NFTSMC given by (51)-(58)). Figures 6 and 7 show the position and velocity tracking errors of the PID-SMC, NFTSMC and the proposed RFTC with its zoomed plots, respectively; while Figure 8 depicts their control inputs. Similarly, the proposed RFTC without considering the actuator faults has a better tracking performance including faster transient rate and higher steady-state precision than the PID-SMC and NFTSMC from Figures 6 and 7 before the time 8 s. Since the integral component of PID are crucially significant, the PID-SMC shows a better robustness than the NFTSMC especially in the presence of uncertain dynamics and external disturbance. Furthermore, the uncertain robot manipulators have been affected by the actuator faults after the time 8 s from Figures 6 and 7, then the NFTSMC provides worse tracking performance in both position and velocity trajectory tracking; especially the velocity tracking performance of the NFTSMC tends to deteriorate immediately when the faults occur. Consequently, we have obtained the conclusion from After that, in order to show the effects of the controller parameters on the tracking performance, the following simulation comparisons have been accomplished with different six groups parameters as shown in Figures 9 and 10. These different controller parameters are as follows:  Table 1. Figures 9 and 10 show the position tracking errors and control torque input of the RFTC with six different cases, respectively. Observed by the cases P1 and P2 of Figure 9, the transient convergence performance depends mainly on the parameter . As a result, the convergence time of the proposed RFTC can be decreased with the increased , but the control inputs will be increased with the larger . Moreover, the objective of the cases P3, P4, P5 and P6 is to verify how does , , r and p affect the system tracking performance. Obviously, the increased , , r and p cannot significantly improve the tracking performance as shown in Figures 9 and 10. However, these increased parameters may enlarge the control torque input from Figure 10. Based on the above discussion and analysis, accordingly, we can conclude that the proposed RFTC selects the parameters on a trade-off between the tracking performance and control inputs.
Finally, to further quantize the improved performance of the proposed RFTC and also for an easier comparisons, the position and velocity tracking precisions, and its efforts on the control torque are compared after 2 s at the beginning of simulation where N denotes the total number of samples, e(k),̇e(k) and (k) stand for the position and velocity tracking errors and the control input at the k − th sampling instant, respectively. The comparisons of three performance indexes are summarised in Table 2.
As shown in Table 2, the proposed RFTC without using an excessive control torque input can obtain the minimal position and velocity tracking errors than the CTC, PID, PID-SMC and NFTSMC. The comparison results of Table 2 are to further verify the improved transient and steady-state tracking performance of the proposed RFTC in both position and velocity trajectory tracking. In particular, the proposed RFTC gains more smaller velocity tracking errors than others (See E v of Table 2), which provides more higher tracking precision for uncertain robot manipulators.

Tracking performance with different initial conditions
Another advantage of the proposed RFTC is that its settling time is independent of the initial conditions of closed-loop sys- By utilising different initial states to the proposed RFTC, Figures 11 and 12 depict the position and velocity tracking errors with their zoomed plots, respectively, where the same controller parameters and different initial states are adopted in Table 1 and (62), respectively. According to Theorem 1 and (28), the position and velocity tracking errors always arrive at the origin within a fixed time T r = 3.28 s. Based on this seminal analysis, we have concluded from these simulation FIGURE 11 Position tracking errors of RFTC in Type I

FIGURE 12
Velocity tracking errors of RFTC in Type I results from Figures 11 and 12 that the position and velocity tracking errors can arrive at the range (−2, 2) × 10 −4 (rad) and (−5, 5) × 10 −4 (rad∕s) within a fixed time T r = 3.28 s, respectively. The simulation results of Figures 11 and 12 are to further verify Theorem 1 in which the tracking performance of the proposed RFTC depends mainly on the controller parameters instead of the initial conditions. The simulation comparisons of Figures 11 and 12 will be further verified the effectiveness of the proposed RFTC for uncertain robot manipulators in both position and velocity trajectory tracking. Figures 11 and 12, the periodic steady-state tracking error comes mainly from the discrete nature of simulations. The sampling frequency can be added to further improve the tracking precision. However, the computation complexity will be increased with the increased sampling frequency. Accordingly, the selection of sampling period is a tradeoff between the tracking precision and the computation complexity.

Remark 9. Observed by
Upon the basis of the above simulation comparisons A and B, no matter the systemic states start from any positions in state space, the proposed RFTC always achieves the improved tracking performance such as faster transient and higher steady-state precision in both position and velocity trajectory tracking with a fixed time T r .

CONCLUSION
In this paper, a novel robust fixed-time fault-tolerant control has been developed for global fixed-time tracking for robot manipulators subject to uncertainties, disturbances and actuator faults. Fixed-time stability analysis of the tracking system has been accomplished on Lyapunov theory. Numerical simulations demonstrate the enhanced tracking performance of the proposed approach in comparison with the traditional robust controls and the fault-tolerant controls with sliding mode in both position and velocity tracking. As a result, the proposed approach gets higher tracking precision with a fixed time than other robust controllers. Meanwhile, the developed approach provides higher robustness subject to uncertain dynamics, external disturbances and actuator faults. Future efforts will focus on finding a robust fixed-time tracking control with actuator constraints and continuity for robot manipulators with uncertain dynamics, external disturbances and actuator faults.