Fuzzy Model Predictive Control With Enhanced Robustness for Nonlinear System via a Discrete Disturbance Observer

This paper addresses the tracking accuracy and robustness enhancement problems of fuzzy model based predictive control (MPC) for a class of nonlinear systems subjecting to lumped disturbances composed of bounded unknown disturbances and a model-plant mismatch. Main features of the proposed method are: 1) A fuzzy disturbance observer and an auxiliary controller are jointly developed to meet a certain control objective that minimizes the peak bound of the errors caused by the lumped disturbances, which eventually leads to desired offset-free tracking performance. 2) A pre-computed robust positively invariant set whose central is the nominal state is derived with the premise of input-to-state stability. 3) Tightened constraints for the guarantee of recursive feasibility of MPC is computed off-line and the quasi-min-max fuzzy MPC is elaborately designed according to a piecewise Lyapunov function. Furthermore, characteristics of robustness enhancement and low on-line computational burden are obtained as compared with the existing offset-free MPCs, and further the impacts of estimation error arising from sampling time and admissible target set on the system performance are also discussed. Two simulation examples verify the effectiveness of the proposed approach ensuring the satisfaction of constraints.


I. INTRODUCTION
Model predictive control (MPC), which can predict the future process behavior and optimize the control input with the consideration of various constraints, has been extensively studied in the past decades [1]. However, for the general MPC, the control performance is significantly challenged when the industrial process characterizes large nonlinearity over a wide operating range and subjects to unknown strong disturbances and uncertainties [2], such familiar example of this sort of processes is the continuous stirred tank reactor (CSTR) in chemical plant [3], or boiler-turbine unit in power plant [4]. First, tracking offset is an unavoidable problem which will trouble the MPC and cause the tracking performance degradation. Although the disturbance rejection methods, such as disturbance estimation with feedforward compensation, The associate editor coordinating the review of this manuscript and approving it for publication was Radu-Emil Precup . have been regarded as effective strategies to counteract the disturbances, the recursive feasibility which is the intrinsic attribute of MPC for stability becomes a difficult issue to be managed since the separate frame design procedure. Second, for nonlinear systems in the presence of disturbances, the requirement of robustness becomes serious matters that the controller designed on a single nominal model is never probably representing the dynamics of the true plant. Third, within the framework of the existing disturbance observer based model predictive fuzzy control strategies, the original control input constraints for the model-based predictive control are undermined by the disturbance compensation in the composite control law.

A. LITERATURE REVIEW
It is well known that Takagi-Sugeno (T-S) model has been widely used to approximate the system's nonlinearity [5]- [7]. VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Many theoretical results on stability analysis and controller synthesis has been obtained T-S fuzzy models, in which sufficient conditions of stability are converted into a set of linear matrix inequalities (LMIs) based on a common Lyapunov function (CLF) [8] or a piecewise Lyapunov function (PLF) [9]. Considering the presence of uncertainty and disturbance in application, the stabilization with performance indexes of H 2 [10] and H ∞ [11] is achieved to satisfy the robust requirements. To solve the control problem of the nonlinear dynamic systems with persistent bounded disturbances, a fuzzy observer-based or filter-based fuzzy controller was developed to minimize the upper bound of L ∞ gain of the closed-loop system [12], [13]. The tracking accuracy was further enhanced in [14], in which the modeled disturbances were estimated by a novel fuzzy disturbance observer and then compensated in the compound control law, in term of the semiglobally input-to-state practically stability of the closedloop system. More recently, a disturbance observer-based integral slide-mode control scheme was proposed on the T-S fuzzy model to deal with the control problem of nonlinear system subjecting to non-periodic form of disturbance [15]. For more complicated chaotic processes, a fuzzy logic controller was proposed with stability analysis in [16]. Despite such great achievements, there are still some performance improvements to be made in industrial process with various constraints, such as transient control performance and/or economic efficiency. Among the advanced control methods, MPC has strong ability to address the issues of transient performance and deal with various constraints during design stage, appearing in [17]- [19]. T-S fuzzy model-based predictive control with the issues of stability and optimization has been designed on the basis of PLFs in [20]. Xia et al. developed new sufficient stability conditions for the fuzzy MPC through the technique of slack matrices [21]. Afterwards, T-S fuzzy modelbased predictive control has been successfully applied in many processes, such as energy-efficient office building [22] and power generation [23]. Although the nonlinear behavior or parameter variation of the system have been considered in the aforementioned methods, the appearance of disturbances, which is ubiquitous in the industry process, can cause severe degradation of the control performance [24]. Thus, for uncertain discrete-time T-S fuzzy systems with the consideration of input constraints and disturbance, a robust model predictive controller was developed in accordance of inputto-state stability (ISS) and the robust positively invariant (RPI) set for T-S systems was further investigated [25], [26].
It should be noted that the disturbances mentioned above are usually assumed to be smooth and centered around zero, regarded as general disturbances. When there are strong type disturbances intruding into the loop, field engineers will be confused by the problem of tracking offset. Pannocchia and Bemporad [27] proposed an offset-free MPC by estimating the integrated disturbances through a steady-state Kalman filter, in which the target set-point was optimally computed by formulating the estimated disturbance into the steady-state equations. Based on the T-S fuzzy model, Wu et al. [23] has extended the offset-free MPC to nonlinear system and applied it to a boiler-turbine unit. As an alternative, disturbance observer (DOB) provides another possibility for the estimation of disturbance [28]. In [29], a DOB based MPC for linear systems was proposed, in terms of ISS stability, to cancel out the disturbances effect from the control input via feedforward channel. Because the control design framework is on the basis of continuous time, the time interval reserved for control computation is expected to be as small as possible and the disturbance varying rate is required to be slow. In [30], the perturbations of a mobile robot treated as input addictive disturbances were estimated by an extended state observer (ESO) and compensated through the control law, thus the tracking performance of the distributed model predictive control was improved. The input disturbance compensation may violate the constraints of the model predictive control, which may lead to the performance degradation or even instability of the model predictive control. For nonlinear controlled plant, a fuzzy RMPC with ESO was proposed in [31], where the lumped disturbances could be estimated through the ESO and alleviated by an appropriate disturbance compensator. It was observed from the simulation that a better transient response in disturbance compensation could be achieved. However, due to the RMPC and ESO were designed individually, the disturbance compensation broke the optimality of the constrained predictive control system. Moreover, the recursive feasibility of the control scheme was not considered.
Recently, in [32], a novel method called tube-based MPC for the linear systems with bounded disturbances was provided, where the robustness can be enhanced by using an auxiliary controller. A disturbance RPI set was computed to approximate the adverse effect of disturbances, through which strong stability result can be obtained. Defined from the RPI set, the center of the ''tube'' is the trajectory of the corresponding nominal system driven by the conventional constrained MPC, while all possible trajectories of the perturbed system are constrained in it. The state of the nominal system, which is free from the disturbance, can guarantee the MPC to be recursively feasible. In [33], the linear tube-based MPC design approach was extended to nonlinear system. And in [34], a robust MPC was developed for a class of hybrid systems using the technique of RPI set, in which the ISS stability of the control system can be guaranteed. In [35], state observer was further considered in the tube-based MPC design.
Consequently, within the framework of conventional tube-based MPC, disturbances with features of finite energy or slow variation can be well handled. However, the tracking offset is still an urgent problem to be solved when the disturbances are strong and there is still a lot of research to be done to meet the challenges of large nonlinear systems.

B. NOVELTY AND CONTRIBUTION
In light of tube-based MPC, this paper proposes a disturbance observer based fuzzy model-based predictive control (DOBFMPC) approach, which is shown in FIG-URE 1 to address the aforementioned control challenges for industry process. Three controllers are involved in the DOBFMPC framework, namely, RMPC, discrete DOB and aux-controller. With the evolution of the system state, all the controllers and the fuzzy nominal model are updated according to the scheduling signal. The nominal inputū generated by the RMPC on the fuzzy nominal model fulfills a major role in tracking the set-point x s , which can guarantee the optimality of the system without disturbances. The estimated disturbanced, as a feed-forward compensating signal, is the output of discrete DOB designed with the consideration of sampling time. The remaining control input generated by the aux-controller is used to alleviate the deviation between real plant state x and nominal statex. It should be noted that the ultimate bound of the error state e x is determined by both the discrete DOB and the aux-controller, in which the impact factors are the disturbance variation and the selection of sampling time. For this reason, the DOB and aux-controller are jointly designed as shown in FIGURE 1 to meet a certain control objective that minimizes the peak bound of the error state. The RPI set used to constrain the error state and the tightened constraints acted on the RMPC optimization can be computed off-line. Consequently, DOBFMPC consisted of RMPC, discrete DOB and aux-controller is proposed for the fuzzy system, which improves the tracking performance and robustness of the control system.
To our best knowledge, a unified framework of discrete fuzzy DOB and fuzzy MPC for nonlinear systems subject to disturbances has never been reported in literature so far. Contributions and qualitative improvements of the proposed DOBFMPC approach compared with the previous literatures can be summarized as follows: 1) Compared with [30], [31], this paper proposes a novel offset-free MPC strategy with enhanced robustness in terms of aux-controller, and with the satisfactory of recursive feasibility that the global stability is achieved.
2) Unlike the tube-based MPC [32], [34] and the approach in [29], a discrete-time control strategy with full consideration of sampling time is developed for a nonlinear plant. The disturbance rejection's ability is improved and model-plant mismatch arising from the T-S fuzzy modeling is compensated.
3) Moreover, a novel RPI set solution for T-S fuzzy system is derived by constructing a new ISS-Lyapunov condition.
The rest of the paper is organized as follows. Section 2 presents relevant concepts, definitions and the control structure. Integrated design of the disturbance observer and aux-controller is provided in Section 3. The robust MPC design is presented in Section 4. Simulations are carried out in Section 5. Finally, some conclusions of the proposed approach are provided in Section 6.
Notation: Given two sets A and B, the Minkowski sumation is defined as A⊕B = {a+b|a ∈ A, b ∈ B} and the Pontryagin difference is defined as A B = {a|a ⊕ B ⊂ A}. ||w|| denotes the Euclidean norm and the norm ||w|| ∞ = sup 0≤i≤N −1 ||w(i||).

II. PRELIMINARIES A. DEFINITIONS
Based on the definition of K− functions, K ∞ − functions and KL− functions in [37], the input-to-state stability of a discrete nonlinear system is given as follows Definition 1 (ISS [37]): A discrete nonlinear system x(k + 1) = f (x(k), d(k)) is called input-to-state stable if there exist β ∈ KL and γ ∈ K such that for any bounded disturbance d(k) and any initial state x(0), the behavior of x(k) satisfies Definition 2 (ISS-Lyapunov Function [37]): For a discrete nonlinear system x(k + 1) = f (x(k), d(k)), a positive definite function V (x) is called an ISS-Lyapunov function if there exist α 1 , α 2 , α 3 ∈ K ∞ and γ ∈ K such that Lemma 1 [37]): A discrete nonlinear system x(k + 1) = f (x(k), d(k)) is ISS, if an ISS-Lyapunov function can be found for it.
Definition 3 (RPI): Consider a nonlinear system x(k +1) = f (x(k), d(k)). A set is called a RPI set for the closed-loop system with control law π , if ∀x(k) ∈ , x(k + 1) ∈ , ∀d ∈ D where D is a compact set.
where the matrix F d and the vector σ are assumed to be constant with σ > 0. The state and control input of the system are subject to the constraints x(k) ∈ X and u(k) ∈ U, respectively, where the origin is contained in the interior of X and U which are assumed to be compact.

B. T-S FUZZY MODEL
We consider a class of continuous nonlinear system with multiple disturbances described bẏ where x = [x 1 , . . . , x n ] ∈ R n is the state, u = [u 1 , . . . , u m ] ∈ R m is the control input and w = [w 1 , . . . , w m ] ∈ R m is the disturbance satisfying the matching condition that the disturbances enter the plant through the same input distribution matrix as the control input.
Using the approximation-based modeling method, a nonlinear system can be represented by the following discretetime T-S fuzzy model with the sampling time T s .
where l ∈ N L+ and L is the number of inference rules. M l 1 , . . . , M l υ are fuzzy set and ν := [ν 1 , ν 1 , . . . , ν υ ] are scheduling signals. x(k) ∈ R n , u(k) ∈ R m and d(k) ∈ R m represent the vector of state, input and lumped disturbances, respectively.
Remark 1: The lumped disturbances are mainly composed of plant uncertainties, model-plant mismatches, modeling errors and external disturbances.
The discrete-time perturbed dynamic fuzzy system used as the model for the controller design below can be rewritten as

III. LOCAL ROBUST CONTROLLER DESIGN WITH DISTURBANCE OBSERVER
To alleviate the influence of unknown disturbances on the model predictive control system, fuzzy DOB and auxcontroller are jointly designed in this section. The fuzzy DOB is designed to estimate the value of the disturbances and make a direct compensation for the MPC, while the auxcontroller is designed to limit the estimation error. To reduce the conservativeness of the MPC, the aux-controller and DOB are designed synthetically so that the estimation error RPI set is as small as possible, which will be used to calculate the tightened constraints for the RMPC

A. FUZZY DISCRETE DISTURBANCE OBSERVER
In this subsection, a discrete fuzzy disturbance observer (FDOB) is firstly developed to estimate the lumped disturbances. Extending the study in [26] for fuzzy system, the dynamics of the FDOB can be constructed as where θ (k) is the observer state variable, L µ is the observer gain and I n is an identity matrix. Denoting the estimation error e d (k) := d(k) −d(k), and then the estimation error dynamics can be given as Lemma 2: Suppose that the pair (I m , B µ ) is observable and the observer gain L µ are chosen such that the system (6) is stable, the estimation error will asymptotically converge to a bound in the order of O(T s ).
Proof: The proof can be referred to [37] and [38]. The stability of (6) can be guaranteed in condition of the pair (I m , B µ ) is observable which implies that B µ is of full column-rank, i.e., rank(B µ ) = m. In most control problem, such condition can be satisfied by choosing appropriate control inputs. Further, various methods can be applied here to choose the observer gain L µ , such as pole-placement or the method provided in [38]. In this paper, the observer gain is elaborately designed to meet a global goal, which is presented in the following section.
Corollary 1: Suppose lim k→∞ d(k) = 0, the lumped disturbances d(k) can be accurately estimated as k goes to infinity. Proof: , the corresponding eigenvalues are located strictly inside the unit disk), the system (6) is ISS from d(k) to state e d (k). Then, there exist β ∈ KL and γ ∈ K, such that (6), the estimation error will be e d (k → ∞) = 0. Therefore, the disturbance observer (5) can accurately estimate the disturbance. It should be noted that the smaller the sampling time T s , the smaller the estimation error, then the estimation error can be limited due to the assumption of the disturbance changing rate in Assumption 1. To prohibit the undesirable transient response, it is necessary to initialize the observer state to θ 0 = L µ x(0) where x(0) is the initial state of the plant, since the developed observer gain L µ may be large.
In [38], the approach of pole placement was used to determine the observer gain, which was simple but not suited for the fuzzy system. Moreover, as it is pointed out in Section I, both the aux-controller and the DOB have great influences on the performance of the control system. Thus, a novel design approach is proposed in the next subsection, which jointly determines the aux-controller and the DOB gain.

B. JOINT DESIGN OF DISCRETE DOB AND AUX-CONTROLLER
In this subsection, we first suppose that there are no disturbances appeared in the system, and the system (4) can be expressed in the following fuzzy nominal form wherex(k) is the nominal state satisfyingx(k) ∈X, andū(k) is the nominal input satisfyingū(k) ∈Ū. Remark 3: Due to the existence of d(k), the compact sets satisfyX ⊂ X,Ū ⊂ U, where the set (X,Ū) is prerequisite for the implementation of MPC. To force the system state x(k) close to the nominal statē x(k) in the presence of lumped disturbances d(k), a composite control law u(k) for the system (4) is proposed where K µ is the gain of the aux-controller, which can bring the enhanced robustness characteristic for the control system. Combining (4), (7) and (8), the dynamic error state e x can be expressed as Combining (6) and (10), the closed-loop of the augmented error system can be constructed as Theorem 1: Suppose that Assumption 1 is satisfied for the system (9a). If K µ and L µ are selected in such a way that A µ + B µ K µ and I m − L µ B µ are Schur matrices, the system (11) under the proposed control law (9b) is ISS and the error state e x (k) is bounded in the order of O(T s ).
Proof: 1) Since both A µ +B µ K µ and I m −L µ B µ are Schur matrices, is also a Schur matrix. Thus, the closed-loop system (11) is ISS [37].
2) Since the closed-loop system (11) is ISS, there exist β ∈ KL and γ ∈ K, such that Considering Assumption 1 and the Lemma 2, the state e x (k) Thus, we finish the proof. Remark 4: From Theorem 1, the upper bound of e x (k) is determined by the selection of K µ , L µ and d(k) ∞ . Furthermore, the sampling time T s is also a non-negligible factor as indicated in Lemma 2.
Lemma 3: Suppose that Assumption 1 is satisfied for system (9a) and lim k→∞ d(k) = 0. If K µ and L µ are calculated to ensure that A µ + B µ K µ and I m − L µ B µ are Schur matrices, the lumped disturbances d(k) can be attenuated from the state e x (k) under the control law (9b) as k goes to infinity. Proof: For system (10), because I m −L µ B µ is Schur matrix and lim k→∞ d(k) = 0, one can get e d (k → ∞) = 0 from the Corollary 1. Because A µ + B µ K µ is Schur matrix, then lim k→∞ e x (k) = 0. Thus, the impact of disturbances can be eliminated from the state e x (k).

C. MAIN RESULTS
Theorem 1 already obtained only provides a design criterion for making the system (9a) ISS stable. In this subsection, the problem of obtaining the aux-controller gains and observer gains will be solved with the technique of LMI. In [39], a minimal RPI was sought to express the minimal influence of the disturbances for the linear system. For the fuzzy system considered in this paper, the future behavior of the system is unknown, which brings difficulties for the solution of minimal RPI set using the reachable set method [40]. Thus, an alternative method is proposed here to find the minimal RPI set.
Reconsidering the disturbance deviation constraint D, one can find an outer ellipsoid E( Considering system (11) and choosing e x (k) as the controlled output, one can get Denoting the error state e(k) := e x (k) T e d (k) T T , the following lemma shows that there exists a controlled output bound for system (13) with the input bound E(P d ). Define is the piecewise Lyapunov matrix for the i-th local model of the fuzzy system, the following Lemma is derived.
Proof: Given in the Appendix. It should be emphasized that inequalities (14a), (14b), (14c) are a bilinear matrix inequality (BMI) problem which VOLUME 8, 2020 is difficult to solve. Through the analysis of the formula (28), we can find that the parameter τ plays the role of decay rate of the closed-loop system. To solve the BMI problem to reach the minimization of the upper bound χ, we can do a line search over 0 < τ < 1 or resort to PENBMI toolbox [41].
Remark 5: Lemma4 will be transformed to the result in [42] where an RPI set is identified for linear systems, if we add the condition τ + ρ = 1 into the LMIs. The value of ρ will be decreased during the minimization of χ , when considering the constraint of (14c).
To derive the feedback controller gains, the observer gains and the upper bound χ , the following Theorem 2 is given for the closed-loop system (13).
Theorem 2: Considering fuzzy system (13) and suppose e(0) = 0, if there exist positive-definite matrices X i (or X k ), positive-definite matrix Q, matrices G j , F j , H m , scalars ρ > 0, χ > 0 and 0 < τ < 1, such that the following minimization is feasible where i, j, k, m ∈ N L+ . Then, the system (13) is ultimate bounded corresponding to the control law (8b), where the aux-controller gain K µ = L j=1 µ j (ν)K j with K j = F j G −1 j , and the observer gain Proof: Given in the Appendix. Remark 6: The positive matrix P µ = diag{R µ , Q} is used to construct a partial piecewise Lyapunov function, which is less conservative than that using a common Lyapunov function. The reason for applying the common Lyapunov function Q is that the disturbance estimation error region can be constructed by a fixed ellipsoid set.
Remark 7: The upper bound χ we obtained is determined by the chosen of d(k) P d ≤ 1 whose volume is influenced by the weight matrix P d . In practice, the determination of P d should be in accordance to the plant characteristics and operation environment, where the larger the compact set volume, the more conservative the controller design will be, and vice versa.
In terms of Theorem 2, the aux-controller and the disturbance observer are elaborately designed, and system state x(k) can be regulated converging to the tube centered at the nominal statex(k) while k goes to infinity. The remaining work is to design a robust MPC to regulate the nominal statē x(k) to the set-point value.

IV. ROBUST FUZZY MPC
The remaining unsolved issue of the composite control law (8) is the nominal control inputū which mainly determines the tracking performance of the system. Since the scheduling signal can be measured at the sampling instant, quasi-minmax RMPC [19], acknowledged as an efficient controller whose first control action, can be designed with full consideration of the model, and it will be redesigned for the fuzzy nominal model (7) with piecewise Lyapunov function.

A. SOLUTION OF THE TIGHTENED CONSTRAINTS
In this subsection, the tightened constraints for the RMPC will be calculated. Obviously, it is difficult to achieve the optimal control performance by designing the predictive controller under the original constraints (X, U). As stated in Remark 3, the existence of lumped disturbances d(k) causes tighter constraints for the state X and control input U of the RMPC. Therefore, the tightened constraints (X,Ū) are required to be calculated first.
Denoting x := {e x ∈ R n : e x ∞ < χ 1/2 }, the tightened state constraint can be determined from the calculation of X = X x . The region x is a RPI set of state e x , which can be used to approximate the adverse effect of the disturbances [39].
The tightened input constraintŪ can be determined corresponding to the tightened state constraintX. Denoting d as the RPI set of the disturbance estimation error state e d , the constraint for the estimated disturbance can be defined as Reconsider the composite control law (8) for the nonlinear system (4). The nominal control law isū(k) = u(k) +d(k) − K µ e x (k), the original input constraints u ∈ U can be guaranteed by satisfying the tightened constraintū ∈Ū for the fuzzy nominal system. The tightened input constraint can be calculated fromŪ = U ⊕ (−W) where = K µ x , and can be solved following approach given in [27]: = Co{K i x , ∀i = 1, 2, . . . , L} where Co denotes the convex hull.

B. QUASI-MIN-MAX FMPC
The purpose of the fuzzy MPC is to regulate nominal fuzzy system (6) from an initial point (x(0),ū(0)) to a target point (x s , u s ). Suppose that the set point (x s , u s ) is admissible under the tightened constraints. The following infinite horizon objective function is considered in the MPC design where Q 0 > 0, R 0 > 0 are the state weighting matrix and input weight matrix, respectively. Considering a piecewise Lyapunov function where P µ := L l=1 µ l (ν)P l with positive matrices P l , the fuzzy nominal model (7) can be robustly stabilized by satisfying the following constraint Summing (19) Then, the minimization of the objective function is turned to optimization problem min ζ (20) subject to DenoteŪ 0 :=Ū u s as the input constraints andX 0 := X x s as the state constraints which both centered at the origin. The peak bound ofX 0 andŪ 0 can be defined as (x 0,min ,x 0,max ) and (ū 0,min ,ū 0,max ), respectively.
The control inputs of the quasi-min-max MPC can be split into two parts, i.e., ū 0 (k|k), F µ (x(k + i|k) − x s ) i≥1 , wherē u 0 (k|k) is a free control action constrained by (22c) and the remaining control laws rely on the feedback gain F µ which is constrained by (22b).
Using LMI-based approach, the optimization problem (19) can be solved through the following theorem.
Theorem 3: For the nominal fuzzy system (7), the control inputs ū 0 (k|k), F µ (x(k + i|k) − x s ) i≥1 minimize the worst case objective function (17) if there exist a decision variablē u 0,s , general matrices Y j , G j , positive matrices S i (or S l ) and symmetric matrices U, X , such that the following minimization problem is feasible where X tt and U tt are the tth diagonal element of the corresponding matrix, and i, j, l ∈ N L+ , then the feedback gain can be calculated from The LMIs (24a)-(24e) are equivalent to the expression of (21), (19), (22a), (22b), (22c), respectively. Proofs of Theorem 3 can be found in [6], [43] and thus are not repeated here.
Remark 8: After the optimization of the worst case infinite horizon objective, we only apply the free control action u 0 (k|k) to the plant and the existence of the feedback gain F µ can guarantee the stability of control strategy [19]. The system matrices (A µ (k|k), B µ (k|k)) of (24a) are then updated at the sampling instant according to the scheduling signal ν measured from the plant.
In conclusion, the algorithm of the proposed DOBFMPC is summarized as below Theorem 4: (Recursive feasibility and Stability) The proposed fuzzy MPC with disturbance rejection can asymptoti-

Algorithm 1
Offline: Calculate the disturbance observer gain L i=1,··· ,L and auxiliary feedback gain K i=1,··· ,L from inequalities (16a)-(16c). Compute the tightened constraints ofŪ,X. Online: Step 1: Initialize the system state x(0) and assign it to the fuzzy nominal model statex(0) such that the optimization problem of (23) is initially feasible for the target (x s , u s ). Also, initialize θ 0 = L µ x(0) for the fuzzy disturbance observer.
Step 2: Solve the optimization problem (23), with the inequalities (24a)-(24e), according to the current statē x(k|k) to obtain the control actionū 0 (k|k) and evolve the fuzzy disturbance observer (5) to get the current estimated lumped disturbanced(k).
Step 4: Measure the real state x(k + 1) at the next time instant and compute the subsequent statex(k + 1) of the nominal fuzzy system (7) under the control action u 0 (k|k) + u s .
Step 5: Replace (x(k),x(k|k)) with (x(k + 1),x(k + 1)) and set k = k + 1, then go to Step 2. cally steer the system state x(k) from a feasible initial state x(0) to the RPI set x whose center is the admissible target state x s . Proof: 1) The optimization problem (23) is solved considering the fuzzy nominal model (7) and the tightened constraints at each sampling instant, and the external disturbances are isolated from the solution. Therefore, the recursive feasibility can be guaranteed if the problem (23) is feasible at the initial state x(0) with the constraints x(0) ∈X [30].
Thus, the system state x(k) will be eventually driven to the set x s ⊕ x , where x = {x ∈ R n : x ∞ < χ 1/2 }.

C. SET-POINT TRACKING DISCUSSION
In this section, some treatments to improve the tracking performance of the system are given when the set-point of the system is not reachable due to tightened constraints. In the above strategy, the target set-point is assumed to be fixed and admissible. In case of operating point change, due to the existence of the tightened constraintsX andŪ, the target set-point may be unreachable which was revealed in [44]. To improve the feasibility region of the target set-point, the following approaches can be adopted: 1) Decrease the sampling time as Remark 2 stated. The RPI set x and d will be smaller than that designed under a larger sampling time since the deviation in disturbance will be smaller at two sampling instants.
2) Introduce an artificial steady-state point (x s , u s ) to evolve the perturbed system to the neighborhood of the desired steady point (x s , u s ). The artificial steady point, which can be determined through minimizing the deviation to the desired steady point through optimization computation, is an admissible tracking point [31].
Considering the fuzzy nominal system (7), (x s , u s ) can be obtained by solving the following quadratic programming: where (x min ,x max ) and (ū min ,ū max ) are the peak bound of tightened constraintsX andŪ, respectively, Q s = Q T s > 0 and R s = R T s > 0 are symmetric weighting matrices. 3) Note that the lumped disturbances of the system come from two aspects: internal disturbances and external disturbances. The amplitude and rate of external disturbances are difficult to be intervened, but the internal disturbances can be intervened by changing the variation of set-point from fast step variation to slow ramp variation. Both the disturbances d(k) and d(k) can be reduced since the disturbances coming from the modelling error or/and uncertainties can be limited to a smaller bound between the two neighboring setpoints.
Remark 9: Reducing the sampling time will inevitably bring an additional computational load, if the computation is limited, we can resort to other two approaches. For process control, engineers usually take the third approach, e.g., a ramping power reference input is adopted into the boilerturbine unit so that the internal disturbance derivation caused by model uncertainties is reduced.

V. SIMULATION RESULTS
Two simulation examples are given in this section to demonstrate the efficiency of the proposed DOBFMPC. The simulations are carried out under Matlab R2017a environment using the Yalmip toolbox [45].

A. NUMERICAL EXAMPLE
The first example is designed to validate the disturbance rejection performance of the DOBFMPC. A second-order perturbed discrete fuzzy system [25] is considered in this example: The membership functions are given as which are shown in Following the algorithm scheme in Section IV Part B, parameters of the proposed approach can be obtained. More details are presented here. By doing a rough line search with an step of ±0.1 around initial τ 0 = 0.5, scalar τ = 0.7 is obtained by solving the inequalities (16a)-(16c); scalar τ = 0.67 is finally chosen by carrying out same procedure with the step of ±0.01. Meanwhile, the controller gains are obtained; and the peak error state e x ∞ < χ 1/2 = 0.3093 and the estimation error RPI set d = {e d (k) T Qe d (k) ≤ 1.5282, Q = 400} can be determined. Then, the disturbance estimation bound |e d | < 0.0618 and auxiliary control bound K µ e x ∞ < 0.1079 can be obtained from the computation. Thus, a tightened input constraints |ū(k)| ≤ 0.8−0.2−0.1079 −0.0618 = 0.4303 can be found.
The control target is to drive the state from x(0) = [1, −2] T to the origin. A tube-based MPC (TMPC) developed in [33] is used for comparison and the control performance is shown in FIGURE 3. It can be observed that asymptotic stability can be achieved by the proposed DOBFMPC in the presence of disturbance, while the TMPC cannot achieve satisfactory  results. The input constraint is not violated during the simulation as shown in FIGURE 4, where the tightened input boundaries are marked with red region and the aux-controller input (abbreviated as u aux ) is constrained into the dashed mar-ked region. The disturbance observer estimates the disturbance with a relatively small error as shown in FIGURE 5, where the estimation error is bounded into a dashed marked region. The evolution of system states encircles nominal states with the peak norm whose boundaries are marked with a red region in FIGURE 6.

B. BOILER-TURBINE UNIT
The second simulation is done on a 160MWe oil-fired subcritical power plant model to show the strength of the proposed DOBFMPC in case of model-plant mismatches. The schematic diagram of the power plant is shown in FIGURE 7. The plant is mainly composed by two parts, i.e. the boiler and turbine. The basic working principle of the power plant is energy conversion. The chemical energy stored in the fossil fuel is transformed into thermal energy of the steam through the combustion and heat transferring in the boiler, the steam is then expanded through the turbine, converting its thermal VOLUME 8, 2020   energy into mechanical rotational energy, finally the energy is transformed into electric power through the coaxially connected turbo-generator.
For the boiler-turbine unit control, the presence of unknown disturbance may cause the deterioration of the tracking performance. In [46], a nonlinear disturbance rejec-tion ability, inherited from a sliding mode disturbance observer, was achieved for the control of the unit with the premise of slow-tracking demand. In recent works [31], disturbance observer and FMPC are designed individually. It was assumed that the disturbances were accurately estimated and then cancelled from true plant to obtain nominal model. The FMPC was designed on the nominal fuzzy model but considering the original input constraints coming from the valve physical limitation. It is obviously to see that the input constraints for the FMPC calculation should be tightened due to that the estimated disturbances are one part of the control compensation. Also, the observation error is not considered through the whole control design. In this section, we develop the proposed DOBFMPC into the control of boiler-turbine unit with the consideration of model-plant mismatch, and the sampling error causing from the selection of the sampling time is further considered.
The primary target of boiler-turbine unit control is to adjust the power output to meet the demand of the grid, meanwhile the steam pressure should be guaranteed within an appropriate range for the safe operation of the plant. During the operation, there are strict constraints due to the physical properties of the actuators.
The mathematic model of the boiler-turbine unit [47] is given as follows: (25) where state variables P, E, and ρ f denote drum pressure (kg/cm 2 ), electric power (MW ), and water-steam density (kg/m 3 ), respectively. The inputs variables u 1 , u 2 and u 3 represent the valve opening degree of fuel flow, steam control, and feed-water flow, respectively, which are constrained within the interval [0,1]. Several typical operating points of this boiler-turbine unit are shown in TABLE 1.
Denoting the state variables as x = [x 1 , x 2 , x 3 ] T := [P, E , ρ f T and considering the external disturbances, modeling errors, the dynamics of the boiler-turbine can be rewritten as:ẋ where d is the lumped disturbance, and Choosing the steam pressure as the scheduling signal, the membership functions of the fuzzy boiler-turbine system model can be expressed as [47]:  where P min and P max are set to be the steam pressure value from the operating conditions #1 and #7 in TABLE 1, respectively.
Replacing the steam pressure P with P min and P max , respectively, we achieve the final T-S fuzzy model respectively. After a long operation of the boiler-turbine unit, equipment wear and furnace ash will cause model mismatch. To show the robust performance of the proposed approach, a severe model-plant mismatch is considered in the simulation that at t = 360s, all coefficients of the plant model (24) are reduced to 70% of their original values. The parameters of the DOBFMPC are set as:     Choosing τ = 0.63, P d = diag{100, 2500, 100} and solving the inequalities (16a)-(16c), the peak error state e x ∞ < χ 1/2 = 1.66 can be obtained. Then, the tightened input constraints |ū(k)| < [0.85, 0.88, 0.84] T can be computed.
For a comparison, the robust fuzzy model control (RFC) method presented in [47] is carried out here in which an antiwindup strategy is adopted to prevent the windup caused by the saturation of the actuators.
The system response shown from FIGURE 8 illustrates that the target states can be tracked with no offset even in the presence of severe model-plant mismatch. In spite of the usage of anti-windup strategy, the RFC method performs large overshoot performance, while the proposed method shows the strong ability of dealing with input constraints. Through the simulation verification, without the control input constraints, the RFC exhibits outstanding performance. However, the controller designed in this paper incorporates input constraints in the design process which is also one of the advantages of model predictive control. We can also discover that the proposed approach designed with smaller sampling time has better control performance where the disturbances coming from the fuzzy modeling error and the model-plant mismatch. Also, the different disturbance changing rate in terms of sampling time influences the observer error and further the control performance. For the same step change of the set-point, the control inputs of the two cases shown in FIGURE 9 are different in cases of modeling mismatch and no mismatch, but the output responses of the boilerturbine system are similar owing to the help of disturbance compensator and the aux-controller. Comparatively, the RPC method obviously shows a greater overshoot output response when modelling mismatch occurs, which can be further observed from the control inputs. It should be stressed that input tightened constraints calculated offline keep the control input satisfying constraints during the whole simulation. The control inputs of the RFC are constrained by the limitation of the actuators.
The estimations of equivalent disturbances are illustrated in FIGURE 10, where the modeling errors shown before t = 360s are small and subsequently, the perturbations caused by model-plant mismatch increase significantly. Thus, modelplant mismatch, which poses a challenge to control performance, cannot be ignored in the field of process control. FIGURE 11 shows that due to the suppression effect of auxcontroller, the state errors (e x ) between nominal model states and true plant states are kept within a peak norm bound where the boundaries of case 1 are marked with yellow color region and the red color region for case 2. Furthermore, we can find that case 2 with larger sampling time behaves larger state error boundary than that in case 1. Therefore, reducing the sampling time helps improve the control performance which is in accordance to Remark 4.

VI. CONCLUSION
This paper proposes a novel DOBFMPC approach for nonlinear system in the presence of disturbance and various constraints. To reject the unknown disturbance, a fuzzy DOB is developed to estimate the disturbance where the disturbance estimation error is further considered to enhance the robustness by an aux-controller. The fuzzy DOB and the auxcontroller are jointly designed, so that the minimization of disturbance positively invariant set can be achieved. Tightened constraints are calculated to guarantee the recursive feasibility of RMPC in an optimal way. Simulation results show that the proposed DOBFMPC strategy can effectively drive the state of the system to the target set-point with satisfactory transient response.
From (14a), it hence holds that In conclusion, A cl µ is Schur stable with the satisfaction of V (k) = e(k) T P µ e(k) > 0, e(k) T A cl µ T P + µ A cl µ −P µ e(k) < 0. Then, the system (13) is ISS.
The proof of Theorem 2 is completed.