IAES Inter national J our nal of Robotics and A utomation (IJRA) V ol. 15, No. 2, June 2026, pp. 473 ∼ 487 ISSN: 2722-2586, DOI: 10.11591/ijra.v15i2.pp473-487 ❒ 473 Fuzzy integral fault-tolerant contr ol of an acti v ated sludge pr ocess Ahmed Sami Hamana, Mounir Bekaik, Messaoud Ramdani Laboratory of Automatic and Signals of Annaba (LASA), Department of Electronics, F aculty of T echnology , Uni v ersity Badji Mokhtar of Annaba, Annaba, Algeria Article Inf o Article history: Recei v ed Feb 25, 2026 Re vised Apr 21, 2026 Accepted May 12, 2026 K eyw ords: Acti v ated sludge process Dissolv ed oxygen control F ault-tolerant control Fuzzy PI observ er Linear matrix inequalities T akagi-Sugeno fuzzy model W aste w ater treatment ABSTRA CT This paper presents a fuzzy inte gral f ault-tolerant controller (FIFTC) for ro- b ust re gulation of substrate and dissolv ed oxygen in acti v ated sludge pro- cesses (ASP). The nonlinear dynamics of the process are represented using an augmented T akagi–Sugeno (TS) fuzzy model, which includes an additional v ector representing the inte gral state to impro v e tracking accurac y . A fuzzy proportional-inte gral (PI) observ er is emplo yed to estimate states and detect ac- tuator f aults, particularly in the aeration sys tem. Controller and observ er g ains are computed by solving linear matrix inequalities (LMIs), while an H ∞ perfor - mance criterion, dened by the parameter , ensures ef fecti v e dis turbance atten- uation and bounds the error ener gy . In the simulation, we considered actuator f aults of the loss of ef fecti v eness (LO E) type. Simulation results demonstrate that FIFTC signicantly outperforms classical linear quadratic re gulator (LQR) in terms of tracking accurac y , rob ustness, and f ault tolerance, e v en under par - tial actuator f ailures and e xternal disturbances. The proposed FIFTC control strate gy , which le v erages fuzzy modeling, rob ust observ ers, and LMI-based op- timization, pro vides signicant benets, primarily by impro ving ef cienc y , re- ducing ener gy consumption, and enhancing rob ustness. This is an open access article under the CC BY -SA license . Corresponding A uthor: Ahmed Sami Hamana Department of Electronics, Laboratory of Automatic and Signals of Annaba (LASA), F aculty of T echnology , Uni v ersity Badji Mokhtar of Annaba P .O. Box 12, 23000 Annaba, Algeria Email: ahmed-sami.hamana@uni v-annaba.dz 1. INTR ODUCTION W aste w ater treatment f acilities (WWTPs) are sophisticated systems characterized by comple x non- linearities and e xternal disturbances, which encompass all uncontrollable f actors impacting the process. The acti v ated sludge process (ASP) is the most intricate and demanding among the associated chemical, mechani- cal, and biological processes [1]. Managing and o v erseeing w aste w ater treatment f acilitie s (WWTP s) [ 2], [3] is dif cult due to the intricac y of the operations, system uncertainties, and inadequate measuring tools [4]. A fuzzy f ault-tolerant control (FTC) technique [5], [6] of fers an ef cient solution for preserving the stability and performance of an acti v ated sludge process, notwithstanding disturbances or actuator malfunc- tions. This me thod emplo ys fuzzy logic [7] to manage nonlinearities and uncertainti es, hence guaranteeing dependable performance. The acti v ated sludge process is a great case study due to the direct impact of impor - tant f actors, such as dilution rate and aeration o w , on reactor stability and treatment ef cienc y . Nonetheless, partial actuator f ailures, such as a defecti v e feed v alv e or v ariations in the air c o m pressor , J ournal homepage: http://ijr a.iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
474 ❒ ISSN: 2722-2586 might inuence these parameters, resulting in imbalances in dissolv ed oxygen le v els and substrate concentra- tion, both essential for the biologic al processes that decompose contaminants. It f acilitates the dynamic re- conguration of the control strate gy to ef fecti v ely addre ss anomalies when inte grated with fuzzy f ault-tolerant control. The inte gration of fuzzy control [8] and fuzzy observ er [9] ensures enhanced system resilience and maintains the continuity and quality of w aste w ater treatment. Recent studies on fuzzy f ault-tolerant control un- derscore the ef cac y of T akagi–Sugeno (TS) fuzzy models in conjunction with fuzzy or adapti v e observ ers for the estimation of internal states and the identication of actuator and sensor f aults [10]. These methodologies aim to maintain system performance and stability despite disruptions, measurement noise, or partial component f ailures. T o ensure strong closed-loop performance, control la ws are typically formulated by resolving linear matrix inequalities (LMIs) [11]. V arious strate gies ha v e been de v eloped to enhance estimation accurac y and f acilitate acti v e f ault com- pensation, including sliding mode observ ers, disturbance observ ers, and adapti v e fuzzy controllers [12], [13]. Findings on di v erse nonlinear systems (including manipulators, time-delay processes, and noise-af fected sys- tems) indicate that these techniques can attain rob ust trajectory tracking [14], [15] and reduce f ault sensiti vity . The y also enhance ener gy performance, render ing them particularly suitable for dissolv ed oxygen control in acti v ated sludge processes. This study seeks to de v elop a T akagi–Sugeno (TS) model representation [16], [17] of the ASP [18], inte grating actuator malfunctions and e xternal disturbances that may af fect system performance. An enhanced system incorporating both process states and the inte gral of the tracking error is presented to reframe the issue as a trajectory tracking challenge [19]. The object i v e of this technique is to implement the proposed fuzzy f ault-tolerant controller in real systems [20]. Furthermore, stability conditions articulated via LMIs can be inte grated utilizing optimization tools. This guarantees comprehensi v e e v aluation of disruptions and actuator malfunctions while maintaining the o v erall resilience and stability of the process. This presents a notable benet o v er parall el distrib uted compensation control (PDC) [21] and linear quadratic inte gral control (LQI) [22], which ne glect disturbances and actuator f ailures in their LMI formula- tions. Furthermore, numerous research ha v e compared the fuzzy proportional-inte gral (PI) controller [23] with the fuzzy PID controller [24]. W e focus the inquiry on the re gulation of dissolv ed oxygen and substrate to illustrate the originality and ef fecti v eness of our methods. A fuzzy inte gral f ault-tolerant controller (FIFTC) is designed to ensure that the system reliably follo ws the desired nominal beha vior despite f aults and disturbances. The H ∞ performance criterion, which mitig ates the ef fects of uncertainties and constrains error ener gy through an attenuation limit specied by the parameter γ , is emplo yed in the design of this controller . Additionally , an inte grated fuzzy observ er is designed to concurrently estimate the system’ s internal states and f aults, f acilitating accurate and dynamic f ault compensation. The achie v ed performance is subsequently compared with that of a LQR controller , illustrating that the proposed technique e xhibits signicantly enhanced beha vior under f ault situations, superior rob ustness, and impro v ed disturbance attenuation. This paper adv ances the domain of intelligent control for bioprocesses by: i) creating an enhanced TS fuzzy model that incorporates inte gral action for superior tracking precision in ASP; ii) introducing an inno v ati v e FIFTC stra te gy that enables concurrent state and f ault estimation through a fuzzy PI observ er; iii) establishing LMI-based stability criteria with H ∞ performance assurances; and i v) illustrating through com- parati v e simulations the enhanced ef cac y of FIFTC relati v e to traditional LQR in the presence of actuator f aults. 2. T AKA GI-SUGENO FUZZY MODEL WITH A CTU A T OR F A UL TS The ASP system is described using a nonlinear state model that captures ho w the actuator , the process, and the sensors interact. Equation (1) denes ho w the process e v olv es o v er time, while the sensor output reects the internal state through measurable quantities. This model clearly sho ws ho w actuator f aults and e xternal disturbances directly inuence the system’ s beha vior . It pro vides a solid basis for des igning f ault observ ers and f ault-tolerant control strate gies. ( ˙ x = h ( x, u + f , d ) y = g ( x ) (1) Let x ∈ R n represent the state and u ∈ R m denote the control; f ∈ R m signies the actuator f ault, while the IAES Int J Rob & Autom, V ol. 15, No. 2, June 2026: 473-487 Evaluation Warning : The document was created with Spire.PDF for Python.
IAES Int J Rob & Autom ISSN: 2722-2586 ❒ 475 output y ∈ R p is dened by the nonlinear functions h and g . Additionally , d ∈ R q corresponds to the e xternal disturbances. W e e xamine the T akagi-Sugeno representation (2), wherein the i-th rule in a fuzzy rule base is: R ULE i: IF z 1 is A i 1 and . . . and z r is A i r THEN ( ˙ x ( t ) = A i x ( t ) + B i u ( t ) + f ( t ) + R i d ( t ) y ( t ) = C i x ( t ) (2) The premise v ariable v ector z ∈ R r is a subset of x, u, θ , and y . A i j is a fuzzy subset characterized by the membership function µ A i j : R → [0 , 1] . The membership function µ A i j ( z j ) is delineated in the i-th rule pertaining to the j-th premise v ariable. A i = ∂ h ∂ x | ( x i ,u i ) , B i = ∂ h ∂ u | ( x i ,u i ) , C i = ∂ g ∂ x | ( x i ) , R i = ∂ h ∂ d | ( x i ,u i ) (3) where : d = d 1 d 2 = D O in S in , f = f 1 f 2 represents the v ector of actuator f aults i n dilution and aeration systems. The depiction of the TS system is pro vided as follo ws: ˙ x ( t ) = r X i =1 h i z ( t ) A i x ( t ) + B i u ( t ) + f ( t ) + R i d ( t ) y ( t ) = r X i =1 h i z ( t ) C i x ( t ) (4) h i ( z ) = µ i ( z ) P r k =1 µ k ( z ) ; µ i ( z ) = Q r j =1 µ i j ( z j ) (5) Gi v en that P r i =1 h i ( z ) = 1 , where r denotes the number of rules, u ( t ) signies the f ault-tolerant control la w , f ( t ) denotes the actuator f ault, d ( t ) represents the disturbance signal, and h i ( z ( t )) constitutes the v alidity model of the T akagi-Sugeno model. It is assumed that the time deri v ati v e of the f ault is bounded. 3. METHOD Our goal is to de v elop an intelligent control strate gy that maintains system performance e v en in the presence of f aults and e xternal disturbances. 3.1. Fuzzy integral fault-tolerant contr ol (FIFTC) The no v el fuzzy inte gral f ault-tolerant control strate gy is b uilt on three k e y components. At its c ore, the controller uses a control la w that combines inte gral action to ensure reference tracking, proportional action on the system states, and acti v e compensation of estimated f aults. The addition of an inte gral state forms an augmented system e x ( t ) = [ x ( t ) , x y ( t )] T , enhancing stability and accurac y . The diagram of fuzzy inte gral f ault-tolerant control of ASP is gi v en in Figure 1. The local state feedback controller with f ault compensation is designed as: u ( t ) = k 1 i x ( t ) + k 2 i x y ( t ) − b f ( t ) (6) where k 1 i ∈ R m × n , k 2 i ∈ R m × p are the control g ains and b f ( t ) ∈ R m is the estimated actuator f ault. ˙ x y ( t ) = y r ef − y ( t ) (7) where x y ( t ) ∈ R p represents an inte gral state v ariable. The fuzzy feedback controller with f ault tolerance is presented as follo ws: u ( t ) = P r i =1 h i ( z ( t )) h k 1 i x ( t ) + k 2 i x y ( t ) − b f ( t ) i (8) Fuzzy inte gr al fault-toler ant contr ol of an activated sludg e pr ocess (Ahmed Sami Hamana) Evaluation Warning : The document was created with Spire.PDF for Python.
476 ❒ ISSN: 2722-2586 Figure 1. Diagram of fuzzy inte gral f ault-tolerant control of acti v ated sludge process 3.1.1. A ugmented system f ormulation The fuzzy controller is subsequently incorporated into the state equation of the closed-loop system: ˙ x ( t ) = r X i =1 h i ( z ( t )) A i x ( t ) + B i h r X j =1 h j ( z ( t )) k 1 j x ( t ) + k 2 j x y ( t ) − b f ( t ) + f ( t ) i + R i d ( t ) (9) ˙ x ( t ) = r X i =1 Ω X j =1 h i ( z ( t )) h j ( z ( t ))[ A i x ( t ) + B i k 1 j x ( t ) + B i k 2 j x y ( t ) ) + B i ( f ( t ) − b f ( t )) + R i d ( t )] ) (10) ˙ x y ( t ) = r X i =1 h i ( z ( t )) ( y r ef − C i x ( t )) (11) we dene an augmented system such as: e x ( t ) = x ( t ) x y ( t ) (12) the augmented system is e xpressed as: ˙ x ( t ) = r X i =1 Ω X j =1 h i ( z ( t )) h j ( z ( t ))[ e A i e x ( t ) + e B i w ( t ) + e R i d ( t )]] (13) where e A ij = A i + B i k 1 j B i k 2 j − C i 0 , e B i = B i 0 0 − I , e R i = R i 0 0 0 , w ( t ) = f ( t ) − b f ( t ) y r ef ( t ) (14) IAES Int J Rob & Autom, V ol. 15, No. 2, June 2026: 473-487 Evaluation Warning : The document was created with Spire.PDF for Python.
IAES Int J Rob & Autom ISSN: 2722-2586 ❒ 477 In the augmented system, the state v ector e x ( t ) ∈ R n + p combines the original system state and the inte gral state, thus accounting for both the system dynamics and the inte gral action for control. The augmented state matrices e A ij ∈ R ( n + p ) × ( n + p ) describe the system dynamics for each pair of rules i and j , while the augmented input matrices e B i ∈ R ( n + p ) × ( m + p ) incorporate not onl y the control inputs b ut also the ef fects of f aults and references. The augmented dis turbance matrix e R i ∈ R ( n + p ) × l preserv es the contrib utions of the disturbance R i d ( t ) in the e xtended system. The e xternal input v ector w ( t ) ∈ R m + p groups the actuator f ault estimation error f ( t ) − b f ( t ) ∈ R m , the output reference y ref ( t ) ∈ R p , and the e xternal disturbances d ( t ) ∈ R q , which are treated separately , enabling better management of uncertainties and discrepancies between the model and the real system. 3.1.2. Fundamental Lemmas f or stability analysis Lemma 1 (Y oung’ s Inequality): F or an y matrices X , Y of appropriate dimensions and for an y positi v e scalar η , the follo wing inequality holds: X T Y + Y T X ≤ η X T X + η − 1 Y T Y (15) Lemma 2 (Schur complement): for a symmetric matrix: A B B T C with C > 0 , A B B T C < 0 ⇐ ⇒ A − B C − 1 B T < 0 (16) 3.2. LMI pr oblem f ormulation W e be gin by dening the L yapuno v function: V ( e x ( t )) = e x ( t ) T P e x ( t ) , P = P T > 0 (17) then, the deri v ati v e of this function along the trajectories of the system is: ˙ V ( e x ( t )) = ˙ e x ( t ) T P e x ( t ) + e x T P ˙ e x ( t ) (18) substituting the augmented system dynamics: ˙ V = r X i =1 Ω X j =1 h i h j [ e x T ( e A T ij P + P e A ij ) e x + w T e B T i P e x + e x T P e B i w + d T e R T i P d e x + e x T P e R i d ] (19) applying the H ∞ performance criterion to ensure disturbance and f ault attenuation: ˙ V ( e x ( t )) + e x ( t ) T Q e x ( t ) − γ 2 ( w ( t ) T w ( t ) + d ( t ) T d ( t )) < 0 (20) where Q = Q T > 0 ∈ R ( n + p ) × ( n + p ) : Positi v e denite weighting matrix for state performance. γ > 0 : H ∞ performance le v el (attenuation le v el from disturbances to outputs). w ( t ) T w ( t ) + d ( t ) T d ( t ) : Combined ener gy of f ault estimation errors and disturbances. According to Lemma 1, the follo wing inequalities are obtained for the cross terms: w T e B T i P e x + e x T P e B i w ≤ η 1 e x T P e B i e B T i P e x + η − 1 1 w T w d T e R T i P e x + e x T P e R i d ≤ η 2 e x T P e R i e R T i P e x + η − 1 2 d T d (21) Theor em 1 (F ault-tolerant contr ol stability) : There e xist positi v e denite matrices P ∈ R ( n + p ) × ( n + p ) and Q ∈ R ( n + p ) × ( n + p ) such that the system (10) is asymptotically stable with H ∞ performance le v el γ if: e A T ij P + P e A ij + Q + η 1 P e B i e B T i P + η 2 P e R i e R T i P + η − 1 1 + η − 1 2 − γ 2 I ≤ 0 (22) Fuzzy inte gr al fault-toler ant contr ol of an activated sludg e pr ocess (Ahmed Sami Hamana) Evaluation Warning : The document was created with Spire.PDF for Python.
478 ❒ ISSN: 2722-2586 Applying Lemma 2 (Schur complement), we obtain the LMI formulation: e A T ij P + P e A ij + Q P e B i P e R i e B T i P − γ 2 I 0 e R T i P 0 − γ 2 I < 0 (23) for Q = Q T > 0 , we can v erify that ˙ V ( e x ( t )) ≤ − e x T ( t ) Q e x ( t ) + γ 2 w T ( t ) w ( t ) + d T ( t ) d ( t ) the e xistence of the minimum eigen v alue λ min which v eries − λ min ∥ e x ( t ) ∥ 2 ≥ γ 2 ∥ w ( t ) ∥ 2 + ∥ d ( t ) ∥ 2 ˙ V ( e x ( t )) ≤ 0 (24) then the system is stable. 3.2.1. LMI solution pr ocedur e T o deri v e the LMIs form, we consider the subsequent v ariable transformation: X = P − 1 > 0 Y = X QX > 0 M 1 i = k 1 i X M 2 i = k 2 i X (25) where X ∈ R ( n + p ) × ( n + p ) : In v erse of L yapuno v matrix P , Y ∈ R ( n + p ) × ( n + p ) : T ransformed weighting matrix for LMI formulation, M 1 i ∈ R m × ( n + p ) : T ransformed state feedback g ain matrix for rule I , M 2 i ∈ R m × ( n + p ) : T ransformed inte gral g ain matrix for rule i . W e obtain the follo wing stability conditions: for i = j : Ψ ii e B i e R i X e B T i − γ 2 I 0 0 e R T i 0 − γ 2 I 0 X 0 0 − Y − 1 < 0 (26) for i ̸ = j : Ψ ij + Ψ j i e B i + e B j e R i + e R j X ( e B i + e B j ) T − 2 γ 2 I 0 0 ( e R i + e R j ) T 0 − 2 γ 2 I 0 X 0 0 − Y − 1 < 0 (27) where Ψ ij = e A i X + e B i M j + ( e A i X + e B i M j ) T + Y (28) Ψ ij ∈ R ( n + p ) × ( n + p ) : LMI matrix term for rules i and j combination. e A i X + e B i M j : T ransformed closed-loop system matrix. M j = [ M 1 j M 2 j ] : Combined g ain matrix for rule j . Y = Y 11 · · · Y 1 r . . . . . . . . . Y r 1 · · · Y r r > 0 (29) therefore, if the aforementioned conditions are met, the closed-loop system e xhibits asymptotic stability with f ault tolerance capabilities. IAES Int J Rob & Autom, V ol. 15, No. 2, June 2026: 473-487 Evaluation Warning : The document was created with Spire.PDF for Python.
IAES Int J Rob & Autom ISSN: 2722-2586 ❒ 479 4. FUZZY PI OBSER VER The goal is to design a PI observ er that can simultaneously estimate the system’ s internal states and actuator f aults while guaranteeing a H ∞ performance le v el in the presence of modeling uncertainties and e xternal disturbances. This is important to remember before presenting the stability conditions of the fuzzy observ er . 4.1. Fuzzy PI obser v er structur e Consider the estimated state of the f aulty system b x f ∈ R n , the estimated actuator f ault f ( t ) ∈ R m and the e xternal distur b a nces d ( t ) ∈ R q . In this study , L i ∈ R n × p represents the state observ er g ains, K I ∈ R m × p and K P ∈ R m × p denote respecti v ely the inte gral and proportional g ains of the f ault estimator , whil b Z f ( t ) corresponds to the estimated premise v ariables. The proportional-inte gral fuzzy observ er is designed for simultaneous state and f ault estimation: ˙ b x f ( t ) = P r i =1 h i ( b z f ( t )) A i b x f ( t ) + B i u ( t ) + b f ( t ) + L i ( y f ( t ) − b y f ( t )) ˙ b f ( t ) = K I ( y f ( t ) − b y f ( t )) + K P R t 0 ( y f ( τ ) − b y f ( τ )) dτ b y f ( t ) = P r i =1 h i ( b z f ( t )) C i b x f ( t ) (30) consider system (4) and (30), we dene the estimation errors: e x ( t ) = x f ( t ) − b x f ( t ) (state error) e f ( t ) = f ( t ) − b f ( t ) (f ault error) (31) the state error dynamics are gi v en by: ˙ e x ( t ) = r X j =1 h i ( b z f ( t ))[( A i − L C i ) ˆ e x ( t ) + B i e f ( t ) + ∆ h ( t ) + R i d ( t )] (32) where ∆ h ( t ) represents the uncertainty due to membership function dif ferences: ∆ h ( t ) = r X j =1 ( h i ( z f ( t ) − h i ( b z f ( t )) ( A i x f ( t ) + B i ( u ( t ) + f ( t ))) (33) consider the augmented state v ector: η ( t ) = e x ( t ) e f ( t ) ∈ R n + m (34) the augmented system dynamics become: ˙ η ( t ) = P r i =1 ( h i ( b z f ( t )) Aη ( t ) + E ξ ( t ) ) (35) with: A i = A i − L i C i B i − K P C i − K I C i B i E = I 0 R i − K P I − K I C i R i ξ ( t ) = ∆ h ( t ) ˙ f ( t ) d ( t ) Fuzzy inte gr al fault-toler ant contr ol of an activated sludg e pr ocess (Ahmed Sami Hamana) Evaluation Warning : The document was created with Spire.PDF for Python.
480 ❒ ISSN: 2722-2586 4.2. Stability analysis thr ough L yapuno v method The stability conditions of the fuzzy observ er are gi v en in the follo wing theorem. Theor em 2: The system (33) with fuzzy PI observ er is as ymptotically stable if there e xist symmetric positi v e denite matrices P 1 ∈ R n × n , P 2 ∈ R m × m , matrices J i ∈ R n × p , Y 1 ∈ R m × p , φ i ∈ R m × p , and a scala r γ > 0 such that the follo wing LMIs are satised for i = 1 , . . . , r : Φ 11 Φ 12 P 1 0 P 1 R i Φ T 12 Φ 22 − Y 1 C i P 2 − Y 1 C i R i P 1 − C T i Y T 1 − γ I 0 0 0 P 2 0 − γ I 0 R T i P 1 − R T i C T i Y T 1 0 0 − γ I < 0 (36) with: Φ 11 = P 1 A i − J i C i + ( P 1 A i − J i C i ) T + Q 1 , Φ 12 = P 1 B i + φ T i C i − Y 1 C i A i T , Φ 22 = − Y 1 C i B i + ( − Y 1 C i B i ) T + Q 2 . (37) Pr oof: Consider the L yapuno v function candidate: V ( η ( t )) = η ( t ) T P η ( t ) , P = P 1 0 0 P 2 > 0 (38) the time deri v ati v e along the system trajectories gi v es: ˙ V ( η ( t )) = ˙ η ( t ) T P η ( t ) + η ( t ) T P ˙ η ( t ) (39) applying the H ∞ performance criterion Z ∞ 0 η ( t ) T Qη ( t ) dt <γ 2 Z ∞ 0 ξ ( t ) T dt + V ( η (0)) Which guarantees the rob ustness of the observ er with respect to uncertainties and disturbances. ˙ V ( η ( t )) + η ( t ) T Qη ( t ) − γ 2 ξ ( t ) T ξ ( t ) < 0 (40) P r i =1 h i ( b z f ( t )) " η ( t ) T A T i P + P A i + Q η ( t ) + ξ ( t ) T E T P η ( t ) + η ( t ) T P E ξ ( t ) − γ 2 ξ ( t ) T ξ ( t ) # < 0 (41) This inequality can be written in matrix form: P r i =1 h i ( z ( t )) η ( t ) ξ ( t ) T " A T i P + P A i + Q P E E T P γ 2 I # η ( t ) ξ ( t ) < 0 (42) Using Lemma 2 (Schur Complement) and the v ariable changes: J i = P 1 L i Y 1 = P 2 K I φ i = P 2 ( K I C i L i − K P ) (43) then, we obtain LMI’ s conditions gi v en in Theorem 2. By solving these LMIs, we then obtain the observ er g ains in the follo wing form (44): L i = P − 1 1 J i , i = 1 , . . . , r K I = P − 1 2 Y 1 K P = 1 r P r i =1 K I C i L i − P − 1 2 φ i (44) where L i : State observ er g ains for each fuzzy rule, K I : Inte gral g ain of the f ault estimator , K P : Proportional g ain of the f ault estimator , γ is t he H ∞ performance le v el for the observ er . IAES Int J Rob & Autom, V ol. 15, No. 2, June 2026: 473-487 Evaluation Warning : The document was created with Spire.PDF for Python.
IAES Int J Rob & Autom ISSN: 2722-2586 ❒ 481 5. RESUL TS AND DISCUSSION 5.1. Pr ocess discr eption The ASP presented in Figure 2 is a biological w aste w ater treatment system based on aeration and the acti vity of microor g anisms responsible for de grading or g anic m atter . The chosen model in the simulat ion is the model of [25]. It ensures ef fecti v e puricati on through the continuous mixing of the substrate, dissolv ed oxygen, and biomass. Figure 2. Aerobic w aste w ater treatment process: ASP X represents the biomass, S denotes the substrate, while D O and X r signify the dissolv ed oxygen and the rec ycled biomass, respecti v ely , as the state v ariables. The comprehensi v e e xposition of the state equations and parameter v alues of the process w as delineated in our prior study [14]. The control v ector is specied as u = [ D W ] , where D signies the dilution rate and W indicates the aeration rate. The biomass X and rec ycled biomass X r v ariables are not directly accessible for real-time measur e- ment because of the sensors’ e xcessi v ely slo w response times and technological limitations. As a result, the control of the substrate and the dissolv ed oxygen, which are quantiable and pertinent f actors for controlling the ASP process, is the sole focus of our in v estig ation. The occurrence of actuator f aults that may af fect the system dynamics is considered in this study . In particular , a progressi v e loss f ault can lead to a gradual decrease in the ef fecti v eness of the dilution rate, resulting in a o w lo wer than the commanded v alue. Similarly , a gradual de gradation of the compressor can impact the aeration rate, causing the air supply to become increasingly lo wer than the desired setpoint. Equation (4) denes the state-space representation that is used to model these a wed beha viors. In order to construct the fuzzy model, we consider that the v ector of permise v ariables is composed of D ( t ) diluion rate and S in ( t ) Substrate in the inuent: z ( t ) = D ( t ) S in ( t ) T which leads to r = 2 2 = 4 sub- models corresponding to the minimum and maximum v alues of the premise v ariables. The resulting T akagi- Sugeno (TS) representation is therefore dened by r = 4 rules, yielding: A 1 = − 0 . 0432 0 . 0011 − 0 . 1202 − 0 . 1230 0 . 0003 0 . 0455 − 0 . 0005 0 − 0 . 0601 − 0 . 0008 0 . 1214 0 − 0 . 6205 0 0 − 0 . 0607 A 2 = 0 . 0192 0 . 0006 − 0 . 1171 − 0 . 0579 0 . 0002 0 . 0214 − 0 . 0003 0 − 0 . 0585 − 0 . 0005 0 . 0569 0 − 0 . 3007 0 0 − 0 . 0285 A 3 = 0 . 0068 0 . 0009 0 . 0127 − 0 . 0352 0 . 0339 0 . 0127 − 0 . 0522 0 − 0 . 0312 − 0 . 0007 − 0 . 0007 0 − 0 . 1210 0 0 − 0 . 0169 A 4 = − 0 . 0012 0 . 0009 − 0 . 0286 − 0 . 0212 0 . 1196 0 . 0074 − 0 . 1841 0 − 0 . 0143 − 0 . 0007 0 . 0198 0 − 0 . 1706 0 0 − 0 . 0099 Fuzzy inte gr al fault-toler ant contr ol of an activated sludg e pr ocess (Ahmed Sami Hamana) Evaluation Warning : The document was created with Spire.PDF for Python.
482 ❒ ISSN: 2722-2586 B 1 = 0 0 − 0 . 5592 0 140 . 7595 − 192 . 6664 − 65 . 7072 − 189 . 6118 B 2 = 0 0 − 0 . 5532 0 140 . 7359 − 182 . 1245 − 65 . 1720 − 188 . 7495 B 3 = 0 0 0 . 1375 0 126 . 6169 − 159 . 6921 − 3 . 7803 − 171 . 1206 B 4 = 0 0 0 . 1661 0 101 . 1574 − 127 . 8967 − 1 . 2369 − 138 . 5586 R 1 = 0 0 . 0758 0 0 0 0 0 . 0758 0 R 2 = 0 0 . 0356 0 0 0 0 0 . 0356 0 R 3 = 0 0 . 0211 0 0 0 0 0 . 0211 0 R 4 = 0 0 . 0124 0 0 0 0 0 . 0124 0 The PI observ er g ains K I and K P respecti v ely are gi v en by: K I = 0 . 1 0 0 0 . 1 K P = 0 . 05 0 0 0 . 05 The inputs of the system with actuator f aults is gi v en in Figure 3 and 4: Figure 3. Input signal (Dilution rate) with actuator f ault 5.2. Fuzzy integral fault-tolerant contr ol The follo wing controller g ains were deri v ed from the resolution of the LMIs associated with Theorem 1: K 1 i 1 = − 926 . 263 − 13 . 673 − 0 . 324 − 673 . 121 0 . 1576 0 . 0762 − 0 . 00006 0 . 0339 K 1 i 2 = − 1268 . 587 49 . 687 − 0 . 897 − 993 . 657 0 . 367 0 . 244 − 0 . 00001 0 . 031 IAES Int J Rob & Autom, V ol. 15, No. 2, June 2026: 473-487 Evaluation Warning : The document was created with Spire.PDF for Python.