Fuzzy Reset-Based H∞ Unknown Input Observer Design for Uncertain Nonlinear Systems with Unmeasurable Premise Variables

This paper proposes an H∞ reset unknown input observer (R-UIO) based on the Takagi-Sugeno (T-S) fuzzy model for the state estimation of nonlinear uncertain systems. Firstly, H∞ unknown input observer (UIO) is designed for TSFM-based nonlinear systems with measurable and unmeasurable premise variables. Then, according to the importance of observers based on unmeasurable premise variables, the results on UIO is modified to propose R-UIO. The sufficient conditions for the stabilization of the estimation error are derived in terms of linear matrix inequalities (LMIs). The proposed R-UIO benefits from less computation complexity to find the feasible parameters, improvement of the estimation process in viewpoints of convergence speed and overshoot. To verify the effectiveness of the recommended approaches, the methods are applied to a practical system.


I. INTRODUCTION
Takagi-Sugeno (T-S) fuzzy model (TSFM) has attracted serious consideration in recent decades to deal with complicated nonlinear systems [1], [2]. The interest for consideration TSFM is principally that it represent the complex nonlinear systems by blending of local sub models [3]. However, for many practical systems, the states are not easily available or the sensors to measure them are costly or hard to protect. Accordingly, the observer design for state estimation is necessary. Great efforts have been put into observer design for T-S fuzzy systems and excellent results have been reported (see, e.g. [4], [5]).
Moreover, in practical applications, the systems are influenced by unknown inputs (UIs), such as parameter uncertainties [6], [7], external disturbance [8]. As a result, the unknown input observer (UIO) design is a basic concern which attracted special attentions in both theory and application [9]. In [10], a filter is recommended for a class of stochastic nonlinear system to estimate the system states with the presence of external disturbances and no consideration of the uncertainties. In [5], UIO is designed for TSFM subject to UIs and external disturbance affecting both outputs and states of the system and sufficient conditions are derived in terms of linear matrix inequalities (LMIs). Similar results are investigated for switched systems [11], descriptor systems [12] and so on.
Though acceptable researches have been published in designing UIO for TSFM-based nonlinear systems, little research has been focused on UIO for TSFM with nonlinear output equations. The reason is that designing UIOs for TSFM-based nonlinear systems with multiple output matrices is led to a non-convex problem that cannot be easily solved (see Remark 3 in [13]). However, TSFM of many practical systems such as vehicle lateral systems [14], inverted pendulum controlled by DC motor [15] and etc. share different multiple output matrices . Accordingly, recent studies are focused on designing UIO for TSFMbased nonlinear systems with nonlinear output equations [13], [15]- [17].
In the same way, if the premise variables of the observer depend on the unavailable states of the system, then the conventional observer with the measurable-based premise variables cannot estimate the system states. A fuzzy observer is designed for a class of T-S fuzzy neural-network systems in [16] with nonlinear output equations based on the measured premise variables. Similar results are developed for continuous-time T-S fuzzy systems with unmeasurable premise variables of the observer in [17], [18]. In [13], a new structure for UIO is established that can be used for discretetime TSFM with a nonlinear output equation. A modified UIO for discrete-time TSFM with multiple output matrices is presented in [15]. In this approach, the fuzzy system is reformulated to overcome the non-convex problem raised by designing UIO for TSFM with a nonlinear output equation. However, in [13], [15]- [18], the uncertainties are not considered in the problem formulation.
In the presence of uncertainties, it is difficult to design a fuzzy observer such that the state estimation error converges asymptotically to zero. Meanwhile, some approaches in designing UIO for uncertain TSFM are developed to ensure the asymptotic stability of the state estimation error [19]- [21]. However, the uncertainties in these developed strategies must satisfy a particular condition and be bounded. To solve this issue, in [22], a new method is investigated to design UIO for an uncertain TSFM-based nonlinear system. But, in these studies, the system outputs must be linear.
Moreover, reset theory is a particular type of control strategy focusing to reduce the performance drawbacks such as transient response oscillatory [23]. Based on the fruitful properties of reset control approach, this line of research attracted the researchers' attentions. Furthermore, the reset method can be implemented to conventional observers to enhance their performance [23]. More specifically, synthesizing the reset method with the linear observers led to reducing the overshoot and the settling-time of the state estimation error, simultaneously [23]. Therefore, a reset observer is safer and more reliable in the industry.
In [24], for the first time, the reset theory was applied to the adaptive observer. In this study, it is shown that synthesizing the reset method to the observer decreases the settling-time and overshooting of the estimation process. The authors in [25] investigate a reset observer for linear timevarying Wing-In-Ground craft. A similar study was extended for linear time-varying delay systems in [26].
Recently, given advantage applications of reset observer for a linear system, the research is continued for nonlinear systems [23], [27]- [29]. A fuzzy reset observer with discrete/continuous measurement is developed in [27] for a class of TSFM-based nonlinear systems. A fuzzy reset observer is designed for a biodiesel production process to provide online estimation in [28]. Compared with the classical fuzzy observer and extended Kalman filter, the performance of the state estimation error by the fuzzy reset observer has been improved. The problem of fault estimation using reset observer for a class of TSFM-based time-delay system is studied in [23]. In [30], reset observer-based eventtriggered control is considered for the multiagent system in the presence of disturbance. In comparison with the conventional observer-based event-triggered control, it was demonstrated that the performance of the system significantly improved. However, in [23], [27], [28], [30], the authors were not concerned with unknown input problems.
Recently, according to the importance of UIO, reset strategy is synthesized with UIO and R-UIO is developed to performance enhancement of the conventional UIO (C-UIO). R-UIO is designed for a class of switched Lipschitz nonlinear systems with UIs in [31]. The existence of the recommended R-UIO is given via the Riccati equation in which led to solving complicated equations. An R-UIO for state estimation of linear time-invariant systems with unknown input is presented in [32] to decrease the 2 norm and settling-time of the state estimation error. Extended results are presented in [33] for a class of nonlinear uncertain systems with no external disturbances.
To the best of the author's knowledge, little research has been focused on designing R-UIO for nonlinear systems, and in view of fruitful applications of TS fuzzy methodologies for complicated nonlinear systems, the problem of R-UIO for TSFM-based nonlinear system has not been addressed in two modes: with measurable premise variables and with unmeasurable premise variables. Also, in the formulation of the previous studies, external disturbance and uncertainties do not exist, together.
Motivated by the above discussions, in this article, a comprehensive fuzzy R-UIO is proposed for a continuoustime uncertain nonlinear system in the presence of external disturbance. At first, we propose a UIO for an uncertain TSFM-based nonlinear system in the presence of external disturbance. The proposed UIO is applicable for both single output matrix and multiple output matrices. To overcome the non-convex conditions in designing UIO for an uncertain TSFM-based nonlinear system with nonlinear output equations, a mathematical transformation is adopted. Besides, to improve the performance of the proposed UIO, the reset mechanism is incorporated to design R-UIO. According to the importance of designing observers with unmeasurable premise variables in practice, the proposed UIO and R-UIO are also modified with unmeasurable premise variables. The main contributions of this paper can be summarized as follows:  Compared with [5], [34], [35], the proposed UIO is designed with ∞ performance for uncertain TSFM-based nonlinear systems in the presence of external disturbance. The uncertainties are transformed to unknown inputs for the TSFM which removes the drawbacks of particular constrained conditions of the uncertainties [19]- [21].  The proposed ∞ UIO is applicable for any TSFM with both the linear and the nonlinear output equations in comparison with [5], [11], [12], [34].  Compared to [15]- [22], for the first time, the reset theory is synthesized to UIO for the uncertain TSFM-based nonlinear system. A more practical UIO/ R-UIO are designed in two modes: with measurable and unmeasurable premise variables which is more reasonable compared to [5], [11]- [13], [16], [34]. Lastly, to verify the effectiveness and superiority of the proposed results obtaining in this article, a practical system is simulated.
The organization of this article is as: Section II concerns with preliminaries and problem formulation. In section III, the main results of UIO/RUIO design procedure are given for uncertain TSFM. Simulation results are brought in Section IV to verify the performance of the estimation process. Finally, section V draws the conclusions along with some recommended future works.
Notation: In this paper, the ( * ) indicates the transpose element of the symmetric matrix.
Remark 3: The existence of the disturbance ( ( )) in the practical systems in real applications is unavoidable which leads to degrade the performance of the control systems and observers. Accordingly, designing a proper observer such that minimizes the effect of the disturbance is important. One of the most popular and efficient tools to minimize the effect of the undesired signal is to use ∞ optimization. The ∞ performance index, which is considered in this paper, is expressed as follows: where ( ) ≜̂( ) − ( ) and ( ∞ ) must be minimized.

III. MAIN RESULTS
In this section, we firstly propose an ∞ UIO for an uncertain TSFM-based nonlinear system. Secondly, we incorporate reset method with the designed UIO to improve the performance of the state estimation error process. We discuss the existence of UIO/R-UIO which is led to the LMI-based optimization problem.

A. UIO DESIGN WITH MEASURABLE PREMISE VARIABLES
Consider the following measured-states based-UIO: where ( ) is the state variable of the observer, Ψ ( ) is as defined in (6.b), Γ ( ) is the estimated of Γ ( ), the matrices , , , are the observer matrices with appropriate dimensions and will be designed later.
In the following theorem, the sufficient conditions for the existence of the UIO with ∞ performance assuming the measurable premise variables are given.
where ℧ = [ So, the inequality (10.d) is sufficient to hold (16) and this completes the proof. ∎ Remark 4: The sufficient condition (10.d) is not a convex optimization problem. To transform it into LMI, a systematic approach is proposed in the following sub-section.

B. PARAMETRIZATION OF THE UIO WITH MEASURABLE PREMISE VARIABLES
In this sub-section, an efficient procedure is generated to get the matrices gains of (8). To solve the bilinear matrix inequality (BMI) (10.d), the following Lemma is needed.
Lemma 1 [15]: For any matrices ∈ ℝ × and ∈ ℝ × where + = , the solution of ℒ = is ℒ = + + ( − + ), where ∈ ℝ × is an arbitrary matrix. The matrices and can be generated as the following to satisfy the condition (10.a): Besides, based on Lemma 1, the general solution of from the condition (10.c) is obtained as: Defining ≜ − ( , (17) can be reformulated as follows: Accordingly, by replacing (20) in (10.d) and rewrittening − , the BMI condition (10.d) is reformed and the following modified condition is obtained: where Π 1 ≜ ( . Thus, the problem of obtaining and is transformed to find , , , and .

C. UIO DESIGN WITH UNMEASURABLE PREMISE VARIABLES
In practice, the states of the system are not always measurable. In this case, if the premise variables of the observer (8) depend on the unmeasurable states, it is impossible to follow the procedure in subsection A. Therefore, the results in subsection A are modified in this sub-section to design a more practical observer with the premise variables based on the estimated states.
Consider the following UIO whose premise variables depend on the estimated states: where ( ) is state variable of the observer, Ψ ( ) as defined in (6.b), Γ ( ) is the estimated of Γ ( ), the matrices , , , are the observer matrices with appropriate dimensions and will be designed later. The lemma 2 is necessary to obtain the essential results in Theorems 2 and 3.
Applying Schur-complement [38], (30) is equivalent to the following conditions: where Ξ 1 and Ξ 2 were defined in Theorem 2. Pre-and postmultiplying (31) by matrix { , , , }, the inequality (23.d) is obtained and the proof is completed. ∎ Remark 5: Note that the condition (23.d) is not an LMI. To change the BMI condition (23.d) to LMI one, the following procedure is presented:

D. PARAMETRIZATION OF THE UIO WITH UNMEASURABLE PREMISE VARIABLES
Referring to sub-section III.B, to satisfy the conditions (23.a)-(23.c), the same as the conditions (10.a)-(10.c), the matrices , , should be obtained from (18)- (20). By substituting (18)- (20) in (23.d) and according to the definition ≜ + , (23.d) is reformed as the following: where Λ 1 ≜ + + , Λ 2 ≜ ( ≜ and ≜ . Therefore, generating and is transformed to obtain , , , and . In the following sub-section, reset strategy is incorporated with the proposed C-UIO to develop the R-UIO. R-UIO improves the transient performance of the observer. The formulations are proposed and derived for more general unmeasurable premise variables case. The results can be derived for the measurable premise variables case by substituting ℎ by ĥ .

E. RESET UNKNOWN INPUT OBSERVER (R-UIO) DESIGN WITH UNMEASURABLE PREMISE VARIABLES
In this sub-section, we deal with the situation that the premise variables are based on the unmeasurable states. In this case, the R-UIO is presented as follows: If ( ) ∈ 1 , Then, If ( ) ∈ 2 , Then, where is the after reset matrix, Θ 1 = { ( ) ∈ ℜ | ( ) ( ) ≥ 0} is the flow set and Θ 2 = { ( ) ∈ ℜ | ( ) ( ) ≤ 0} is the jump set and as ( ) ∈ Θ 2 , jump will occur. The matrices and will be obtained in the following.
The discrete-error is provided as follows based on (33.b): The following theorem is concerned with the stability analysis of the state estimation error (36).
Theorem 3: For the system (5), the R-UIO (33) that stabilized the state estimation error (36) and minimized in the ∞ performance of Remark 3 can be generated if there exists symmetric matrices = > 0, , and positive constants 1 , 2 , 3 , 4 and 0 < 5 ≤ 1 such that: Proof: Consider the following Lyapunov function: The state estimation error dynamics (35) is stabilized and satisfies the ∞ performance in Remark 3, if the following conditions Substituting (36.a) in (39) and follow the procedure in (16) for the continuous error dynamic along with some simplifications, yields to: where ≜ + + + + 2 + 1 and ϱ = − . Applying S-procedure [38] and taking ( ) ( ) ≥ 0 in (40), one gets: where ℴ = + 2 . Consequently, from (41), (37.a) is obtained. Besides, for the discrete error dynamic (36.b), combining (36.b) and (39) and follow the procedure in (16), results: Again with the aid of S-procedure [38] and taking ( ) ( ) < 0 in (42), the following results can be obtained: Applying Schur complement [38] for (43), results in: Multiplying (44) by { , , } from left and right, the following inequality is achieved: Replacing in (45) and using the variable change ≜ , the inequality (45) can be rewritten as follows: In addition, it is needed after the jump, the error trajectories drop out of the jump set, in other words: Using S-procedure [38] in (47), results in: Consequently, the state estimation error is stabilized and satisfy ∞ performance in Remark 3, if the conditions (37.a)-(37.c) are satisfied and this completes the proof.
∎ Remark 6: It should be noted that if the output of the system be nonlinear, the reset unknown input observer design is led to a non-convex problem that cannot be solved easily [15]. In our paper, a new mathematical definition in (6.a) and (6.b) was proposed to overcome the corresponding non-convex problem. Furthermore, in the presence of uncertainties it is hard to design an observer such that the state estimation error converges to zero. In our paper, a transformation is adopted to transform the uncertainties to unknown inputs with no bounded conditions. Besides, a more reasonable reset unknown input observer based on the unmeasurable premise variables.
The pseudo code of the proposed R-UIO is illustrated in algorithm 1.

V. SIMULATION RESULTS
To verify the effectiveness of the proposed observer, a twodegree freedom helicopter system is simulated by using MATLAB software. We revisit the following model of a two degree freedom helicopter [41]: The performance evaluation of the proposed controller is presented in two scenarios. In the first scenario, the performance analysis of the proposed C-UIO and R-UIO is evaluated with the measurable premise variables. In this case, based on Theorems 1 and 3 and taking 1 = 1.2, 2 = 0.01, 3 = 0.2, 4 = 0.6, 5 = 0.5, the parameters of the observer with measurable premise variables are obtained as follows:  ]. Moreover, the ∞ index is obtained as = 0.001. The states estimation by the proposed UIO and R-UIO based on the measurable premise variables are illustrated in Figure 1. Compared our approach to [36], the proposed observers can handle the uncertainties, external disturbance and nonlinear output equation, simultaneously.
Obviously, one can find that in comparison with the suggested UIO in [36], our proposed UIO works well for states estimation and the speed of the convergence is faster and better than [36]. Moreover, by the proposed R-UIO, the performance of the estimated states is improved significantly in both convergence speed and overshoot.    The 2 norm of the state estimation error ( ) and the convergence time of each state are illustrated in Table 1. Obviously, by the R-UIO, the performance of the observer is improved in both convergence time and 2 norm of the state estimation error in comparison with the suggested C-UIO in [36] and the proposed C-UIO. Furthermore, the results show the superiority of the proposed ∞ C-UIO compared with the C-UIO in [36]. In the second scenario, for the unmeasurable premise variables-based UIO/R-UIO, based on Theorems 2 and 3 and taking 1 = 1.5, 2 = 0.05, 3  The estimation of the (48) system states are depicted in Figure 3 for the estimated premise variables case. As the simulation results show, the estimation process under the proposed both UIO and R-UIO is satisfactory. For thoroughly checking the property of the proposed UIO, the results are compared with a newly published work in [18]. Figure 4 illustrates the control input signals under the proposed R-UIO-based controller with unmeasurable premise variables.   For the sake of evaluating the effectiveness of the devised observer, the 2 norm and convergence time are calculated. As the results show in Table 2, the proposed R-UIO outperform the C-UIO in [18] and our designed C-UIO in both 2 norm and convergence time. Moreover, our suggested ∞ C-UIO demonstrated the improvement performance compared with the one in [18].

V. CONCLUSION
This article was concerned with designing ∞ UIO/R-UIO for uncertain TSFM subject to external disturbance. To overcome the non-convex problem of nonlinear output equation, a new representation was presented. Based on Lyapunov theory, the stability of the state estimation error was ensured and new Off-line LMIs were generated. The performance of the proposed ∞ UIO was improved in both convergence time and 2 norm of the state estimation error by incorporating the reset mechanism. The case of unmeasurable states-based premise variables, i.e. asynchronous premise variables for TSFM-based nonlinear system and observer was also considered for the proposed UIO/R-UIO. A practical simulation example was shown the effectiveness of the proposed scheme. How to design the R-UIO for uncertain discrete-time TSFM-based nonlinear systems, study on the data-driven realization of the proposed method for unknown dynamic systems are interesting future research topics.