An Observer-Based Composite Nonlinear Feedback Controller for Robust Tracking of Uncertain Nonlinear Singular Systems With Input Saturation

This study proposes an observer-based Composite Nonlinear Feedback (CNF) controller for the robust tracking of uncertain singular systems with input saturation, nonlinear function, time-delay, and disturbances. The suggested control law is designed based states reconstructed using a singular observer so as to increase steady-state accuracy and improve robustness. The CNF controller is developed based on Generalized Riccati Equations (GRE) and Lyapunov–Krasovskii functional. Additionally, the proposed theorem verifies the stability conditions of the system in the presence of uncertainties and disturbances. Among the advantages of this method, are its fewer restrictive assumptions, transient and high-speed performance improvement and steady-state precision. The uniform boundedness of the tracking error in the presence of the input saturation and external disturbancesis also a prominent feature of this method. The performance of the proposed approach is assessed using a simulation study.

challenging problem [8]- [11]. For example, in addition to stability, other issues such as regularity and impulsefree are also addressed in the control of singular systems. In the last two decades, many researches have been done on singular control systems. Most articles in this field are based on the generalization and extension of the theory of non-singular systems to singular systems. One of the important factors that cause instability and weak performance in control systems is the delay. Therefore, time-delay control systems are one of the issues that researchers have addressed recently. Due to the nature of these systems, the problem of controlling singular systems with timedelay is highly complex. On the other hand, there are often uncertainties in systems due to errors in modeling and changes in environmental and operating conditions. Since uncertainties affect systems' optimal performance it is important to design control laws that are robust against them. The problem of robust stability analysis, as well as robust stabilization of indefinite singular systems and indefinite delayed descriptive systems, has been studied in [12]- [14]. Albeit the existence of several approaches in the literature for descriptive systems, model uncertainties and time delays were overlooked in those designs. Stability is another important feature of the dynamical systems that express the system's responsive behavior to disturbances and primary conditions. Lyapunov stability theory is appropriate for the stability analysis of singular systems. Unlike conventional systems, descriptive system response may include impact expressions that can cause saturation in the plant input and even damage the system. So eliminating impulsive behavior through specified feedback control is a very basic issue in singular systems theory. In system design, not all variables can be directly available from measurements. In these cases, the estimation of the immeasurable variables is required for state feedback control implementation. Designing observerbased controllers is very desirable for stabilizing systems [15]- [17]. Observer designs for descriptive systems have attracted significant attention in the past decades [18]- [22]. Input saturation constraint is a common constraint in practical systems that does not allow the control inputs to exceed one specified limit. Ignoring the input saturation limitation in the process of controller design for systems in many cases can lead to undesirable system behavior and even instability. Therefore, designing control laws that take into consideration input saturation is essential. Robustness and optimum performance are two desired features in control systems.
In [23], the stability analysis and design of robust dynamic output feedback controller and controller based on observer for uncertain continuous singular systems with time-delay have been investigated but the nonlinearity and disturbances are not considered. In [24], a new controller and a fault-tolerant observer for a class of nonlinear continuous singular control systems are discussed regardless of uncertainty and time delay. In [25], a H ∞ control based on observer for uncertain descriptive systems with time-delay and actuator saturation is investigated but system nonlinearities were overlooked. In [26], the observer design problem for one-sided Lipschitz nonlinear continuous-time singular systems with unknown input is considered but the nonlinearity and time delay are not investigated. In [27], observer design for a class of nonlinear singular systems with multi-outputs are considered without the nonlinearity, disturbance, and time delay. In [28], the problem of robust passive control based on observer is applied for uncertain singular time-delay systems with actuator saturation without considering the nonlinearities. A new functional observers design method is applied for descriptor systems via LMI in [29]. In [30], a method of designing full order observers is studied for time-delay descriptive systems with Lipschitz nonlinearities, then the Lyapunov-Krasovskii functional and the convexity principle are applied to investigate the stability of the singular systems. In [31], a robust adaptive observer is offered for a class of singular nonlinear non-autonomous uncertain systems with unstructured unknown system and derivative matrices, and unknown bounded nonlinearities and no strong assumption such as Lipschitz condition is applied on the recommended system. In [32], the linear multivariable feedback control is used for multi-input multioutput (MIMO), linear time-invariant (LTI) singular systems to improve the transient response to descriptor systems and follow a step reference with zero over-shoot. In [33], the modified composite nonlinear feedback method is considered for output tracking of non-step signals in singular systems with actuator saturation and external disturbances. In this article, the composite nonlinear feedback control law is not applied for the tracking of reference signals in singular systems. In [34], a robust composite nonlinear feedback controller is considered for descriptor systems with input saturation, this method can ensure general reference tracking for the singular systems with input saturation. In [35], a composite nonlinear feedback control method for tracking control problems is developed for the output regulation problem of singular linear systems with input saturation. In [36], an output-feedback sliding mode control is designed for a class of nonlinear singular systems with time delay and uncertainties, but input saturation was not considered. In [37], the problem of stability and stabilization is examined for singular networked control systems with short timevarying delay. In [38], the non-step tracking control problem is investigated for multi-input multi-output (MIMO) linear discrete-time descriptive systems with input saturation. In [39], the robust stability of uncertain fractional-order singular systems with neutral and time-varying delays is studied. In [40], an observer-based controller is considered for a class of singular nonlinear systems with state and exogenous disturbance-dependent noise. In [41], a finitetime observer-based controller is proposed for time-delay descriptor systems with time-varying disturbances, model uncertainties, and one-sided Lipschitz nonlinearities. In [42], the problem of adaptive output-feedback neural tracking control is investigated for a class of uncertain switched multiple-input multiple-output (MIMO) nonstrict-feedback nonlinear systems with time delays. The adaptive intelligent asymptotic tracking control is studied for a class of stochastic nonlinear systems with unknown control gains and full state constraints in [43]. A low-conservative composite nonlinear feedback controller is studied for singular time-delay systems with time-varying delay in [44]. For intermittent control, its control signal is updated in a continuous manner on control time intervals. To overcome the limitation, time-triggered intermittent control (TIC) is considered in [45]. In [46], the time-triggered intermittent control (TIC) is offered to examine the exponential synchronization issue of chaotic Lur'e systems.
Designing control approaches that ensure fast response with reasonable transient dynamics is essential in many applications. Composite nonlinear feedback (CNF) has recently emerged as a good solution for improving the transient performance of tracking control problems [47]- [48]. CNF combines a tracking control law that ensures a quick tracking performance with a nonlinear feedback law designed to smoothly change the damping ratio of the closed-loop system as this latter approaches the reference input so as to reduce the overshoots caused by the tracking law. The CNF method requires fewer limiting assumptions and as a result, more optimal design conditions are achieved. However, we cannot use the CNF control technique directly for singular systems. To overcome the problems of impulse terms and input derivatives in singular systems, we adopt state feedback to make the singular system free of impulses. For this purpose, a singular observer will be designed, based on the state variables, to estimate the states of the singular system. Since the problem is time-delays output tracking, the reference signal is generated by a reference generator and the tracking problem becomes a stabilization problem. The error vector and stability analysis will be carried over based on the Lyapunov's approach. Obviously, in the CNF combination, the CNF control law leads to a linear controller when the nonlinear phrase tends to zero. As a result, the added nonlinear expression allows modifying the linear control law to recover system transient performance and then the error converges to zero. In all the mentioned works, nonlinearity, disturbance, uncertainty, and even time delay are not considered together. To the best of our knowledge, no research has been performed to improve system performance for nonlinear and uncertain singular systems with input saturation, time delay, and external disturbances using the observer-based CNF control method.
This paper proposes an observer-based CNF control law for nonlinear singular systems with time delay, disturbances and input saturation. Its main contributions are as follows: • An observer-based CNF control design for singular systems that takes into consideration time-delays, nonlinear dynamics and control input saturation.
• A design that yields improved transient performance and steady-state precision in the presence of time-delays, nonlinearities and disturbances.
• A control scheme that guarantees the robustness and stability of uncertain nonlinear singular systems.
The remainder of this paper is organized as follows. The problem formulation and required assumptions are provided in section II. The proposed observer-based composite nonlinear feedback controller is derived in section III. Simulation results are provided in section IV. Finally, some conclusions are presented in section V.

II. PROBLEM FORMULATION AND PRELIMINARIES
Consider the uncertain nonlinear singular system with input saturation and time delay defined by: , C, and constant singular matrix E are matrices of proper dimensions, where rank(E) =r < n and τ i ∈ R + is the time-delay. In addition, d (t) ∈ R n is the unknown external disturbance vector. The saturation function is described by: where u i (t) is the maximum value of the i th control input.
A. PRELIMINARIES • Singular system (E,A,B,C) is calledregular, if there exists a scalar s ∈ C, so that det(sE − A) = 0.
solution does not have impulse terms and • Singular system (E, A, B, C) is called admissible if it is stable and impulse-free.
= n for all s ∈ C, s finite.
Adequate conditions for a singular system to be stabilizable and detectable are C-controllability and C-observability. Lemma 1: Let V : [0, ∞) × D → R be a continuously differentiable Lyapunov functionfor the singular system (1) so that: where η 1 (·) and η 2 (·) are class K functions, is a continuous positive definite function on the open set D. Then the solution x(t) of the system (1) for θ ∈ [−τ, 0] is uniformly ultimately bounded. Lemma 2: Suppose the singular system is regular and impulse-free, then the response of such a system will be on [0, ∞) unique and the impulse-free.
Assumption 1:The external disturbance vector is bounded with d(t) ≤ d max , where d max is a known positive real constant and d max < u imax for i = 1, 2, . . . , m.
Assumption 2: The following inequality holds for the nonlinear function f (ζ ) : where M = diag{M 0 , M 1 , . . . , M N } and M i ∈ R n×n , i = 0, 1, . . . , N are some known matrices. Assumption 3: Let F ∈ R n1×n2 , G ∈ R n2×n1 , for any ρ > 0 the following inequality holds : Remark 1: Assumption 1 is a reasonable condition considered in practical cases and indicates that the norm of the disturbance vector does not exceed the actuator saturation level in each input channel.

III. MAIN RESULTS
In this section, the observer-based control law is first designed then its stability and accuracy are proven using the Lyapunov stability analysis for three different cases of input saturation.

A. TIME VARYING REFERENCE GENERATION
The purpose of this study is to design an observed-based CNF lawfor the uncertain singular system so that the output y(t) can follow the reference y m (t). The reference signal may be created by a source generator system. The reference signal could be described based on the basic system matrices (E, A, B, C) as follows: where x m (t) ∈ R n , u m (t) ∈ R m , y m (t) ∈ R l are the state, the control input, and the output vectors of the reference signal, respectively. Consider the control input defined by where F m is a static feedback gain that is chosen so that the pair (E, A + BF m ) is stable and impulse-free. Moreover, r s (t) is an adjustable signal elected by the designer. As a result, the reference model is described as follows The auxiliary state vector of the output tracking error is specified as x e (t) = t 0 e(t)dt and the output tracking error as e(t) = y(t) − y m (t). The dynamic equation of the auxiliary state vector is determined as beloẇ where x e (t) ∈ R l . Thus, the augmented system is achieved as follows: where , , VOLUME 10, 2022 The reference model can also be written as follows: ∈ R l+n , and tracking error vector , therefore the dynamical equations of tracking error are obtained as the following equation

B. SINGULAR OBSERVER DESIGN
Since full state availability is not always possible in control system design, state estimation is often required to implement state feedback control. That's why the singular system must be C-observable or detectable. The singular system must also be impulse-observable to assure the lack of impulse terms in the observer response. The singular observer can be constructed for the system (10) as follows: where x v (t) ∈ R n+l is the state observer vector, and L is the observer gain and should be constructed so that the pair (E * , A * + LC * ) is stable and impulse-free. The observer error is determined asx v (t) = x v (t) − x * (t) and the dynamical equations of observer error are created as the following equation C. OBSERVER-BASED CNF CONTROL LAW The CNF includes two linear and nonlinear components. The linear feedback part is constructed in such a way that it can cause a slight damping ratio in the system in order to obtain a faster response. The nonlinear feedback control is applied to increment the damping ratio of the closed-loop system as the system output follows the reference model to lower the overshoot created by the linear portion. The linear part of the observer-based CNF control law is described as: where F is determined so that the pair (E, A * + B * F) is stable and impulse-free. Also, the nonlinear part of the CNF law is displayed as where P is determinedfrom the following Generalized Riccati Equations (GRE) for any Q> 0: where µ 1 , µ 2 , µ 4 , µ 5 , µ 7 , µ 9 , ρ 1i , ρ 2i , ρ 6i , ρ 7i are positive constants, and φ(y m (t),y(t)) is defined as φ (y m (t) , y (t))=diag[φ 1 (y m1 (t),y 1 (t)),φ 2 (y m2 (t),y 2 (t)) · · · , φ m (y ml (t)(y ml (t), y l (t))] where φ i are negative constant functions locally Lipschitz in y(t). The CNF law based on observer can be expressed as follows: The non-unique function φ(.) can be effective in the transient performance improvement. Some examples and criteria for selecting appropriate φ(.) are presented in [47].
Theorem 1: Consider the system (12) and assume that assumptions 1-3 hold. let > 0 be a solution of the GRE as follows for any H > 0 where µ 2 , µ 3 , µ 6 , µ 8 , µ 9 , ρ 3i , ρ 4i , ρ 5i are positive constants, and for any δ ∈ (0, 1), let d δ be the largest positive constant so that: where matrices P and satisfy the GRE (17) and (20), f i is the i th raw of matrix F, and u m i is the i th component of the control input u m . Thus, there exist scalars φ * i , for i= 1, 2, . . . , m such that for any φ(y m (t),C * x v (t)) locally Lipschitz in C * x v (t), and |φ(y * m (t),C * x v (t))| ≤ φ * i , the CNF lawbasedon observer (19) ensures the following: • By applying the control law, the controlled output y(t) can track asymptotically the reference input y m (t) in the presence as well as the absence of the external disturbances and uncertainties with a tracking error that is bounded and limited.
• θ δ is a positively invariant set for the closed-loop singular system. Proof: Due to conditions (21) and (22), the linear part of the control law in all input channels does not exceed the saturation bound. The closed-loop singular system can be attained using the control law (19) and the system's equations that are deduced from Eqs. (12) and (14) as where To examine the performance of the suggested control law in the presence of external disturbances, uncertainties, and nonlinear functions, we present the following Lyapunov function as follows: Deriving the Lyapunov function along with the directions of the closed-loop system in (24), yields: Then, we obtaiṅ . . .
In what follows, we investigate three different states of the saturation function.
Case 1: If all input factors are not saturated, i.e. |u i | ≤ū i , it can be easily shown that: where b * i is the i th column of B * and u N i is the i th component of u N .
Case 2: In this case, all input channels are higher than the upper saturation bound, i.e. u i ≥ū i so we have where u L i is the i th component of u L . According to the (25), it is obvious that ω i (t) =ū i − u L i , and thus ω i (t) ≥ū i − |u L i | ≥ 0. Moreover, according to (38) as u N i ≥ u i − u L i = ω i (t), the following inequality is achieved Case 3: In this case, all input channels are smaller than their lower saturation bound, so we have Considering According to (37), (39) and (41), ω i (t) can be considered as Hence, ω(t) can be achieved as follows: whereφ = ξ φ and ξ = diag[ξ 1 , ξ 2 , . . . , ξ m ]. Because φ is negative and according to Razumikhin Theorem . . . , N), the relation (36) is as follows: As a result, we will have the following forṁ it can be concluded that > 0 and for convenience one can writė where υ = χ(t) χ v (t) , and finally, it can be written aṡ By introducing a new positive invariant set as θ µ := E * υ (t) | ||E * υ|| 2 2 ≤ λ min ( ) ⊂ θ δ and := θ δ − θ µ . According to Eqs. (45) and (46) and the Lemma 1, it geṫ where is a positive-definite function soV is bounded and negative in . Because of the structure of the Lyapunov function, decrease V leads to a decrease in the norm of the closed-loop singular system's states. So, it can be concluded that x x v ≤ γ ; thus, ultimately bounded tracking error and observer error is obtained. Considering Lemma 1, Eq. (47) ensures that the tracking error will be bounded (even with saturation and uncertainty). If the system is not affected by the disturbance, that is d(t) = 0, the region θ µ is removed. Then, the observer-based control lawwill cause the states of the closed-loop singular system (12) to be bounded, and the output of the system (1) will follow thereference signal y m (t) with a tracking error that is bounded and limited in the presence of disturbance and input saturation.

IV. SIMULATION RESULTS
The effectiveness and performance of the proposed approach is examined in this section using two different examples. The first considers a singular system with time delay, perturbation, input saturation, and uncertainties. The second example, considers a DC motor modeled using a singular state-space representation with nonlinearity. Example 1: Consider the nonlinear singular system with time delays (1), whose parameters are given as: and A di , i= 1, 2 are fixed parameters, r 1 (t),r 2 (t) and A di (ν(t)),i= 1, 2 are the uncertain parameters and the external disturbance is as Considering Q = I 4 and using (17) and (20 ), the P and the descriptor observer L can be determined as The nonlinear function φ(y m (t),y(t)) is selected as follows: φ(y m (t),y(t)) = − βe −α 0 α|y(t)−y m (t)| , where    For comparison purposes, we consider the approach proposed in [44]. Fig. 1-4 depicts the dynamics of the states estimated by the designed singular observer. The trajectories of the tracking error are illustrated in Fig. 5. Also Fig. 6 displays the output responses of y m (t) and y(t). The dynamics of the nonlinear state-feedback controller are depicted in Fig. 7. The obtained results show that the system is robust to time delays and disturbances compared to those of [44], and also the proposed controller has good convergence rate. The purpose here is not to get zero tracking error, but rather ensure the is error bounded. The simulation results show that the system output tracks the reference signal with a tracking error that is bounded and limited by using the proposed observer-based controller. Example 2: Consider a DC motor with a singular statespace representation in the form of system (1) with the following information [50]: where f (x) = 0.5cos(x 1 ) − 0.5 and A di , i= 1, 2 are fixed parameters, and A di (ν(t)),i= 1, 2 are the uncertain   parameters and q(t) is the disturbance, and Using (17) and (20 ), the P and the descriptor observer L can be determined as For simulation use, take q (t) = 3 sin (0.01t) ,      Simulation results are represented in Figs. 8-13. Fig. 8-10 shows the vectors of the observer error. Fig. 11 demonstrates the tracking error at two different time intervals. Fig. 12 illustrates the output tracking by the suggested CNF control law based on observer. Fig.13 displays the response of the suggested control law.

V. CONCLUSION
This paper proposed an observer-based CNF controller for the robust tracking of uncertain singular systems subject to input saturation, nonlinear functions, time-delay, and disturbances. The control law was designed based on states reconstructed using a singular observer. A theorem was proposed to prove the uniform boundedness of the tracking error albeit the presence of external disturbances and nonlinear dynamics. Implementation of the proposed approach to two case studies confirmed the accuracy and effectiveness of the proposed approach in controlling uncertain nonlinear singular systems with input saturation. He is currently an Associate Professor with the Department of Electronic and Telecommunication Engineering, King Mongkut's University of Technology Thonburi, Bangkok. He has published a research monograph and more than 20 research articles in international refereed journals indexed by SCI/SCIE. His research interests include fuzzy control, robust control, optimal control, system and control theory, and computational intelligence. He also serves as an Associate Editor for International Journal of Innovative Computing, Information and Control.
AFEF FEKIH (Senior Member, IEEE) received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from the National Engineering School of Tunis, Tunisia, in 1995, 1998, and 2002, respectively. She is currently a Full Professor with the Department of Electrical and Computer Engineering and the Chevron/BORSF Professor of engineering at the University of Louisiana at Lafayette. She has authored or coauthored more than 200 publications in international journals, chapters, and conference proceedings. Her main research interests include control theory and applications, including nonlinear and robust control, optimal control, fault tolerant control with applications to power systems, wind turbines, unmanned vehicles, and automotive engines. She is a member of the Editorial Board of the IEEE Conference on Control Technology and Applications, the IEEE TC on Education, and IFAC TC on Power and Energy Systems. VOLUME 10, 2022