(Luận án tiến sĩ) bài toán ước lượng trạng thái cho một số lớp hệ động lực và áp dụng

MINISTRY OF EDUCATION AND TRAINING QUY NHON UNIVERSITY DAO THI HAI YEN ON THE STATE ESTIMATION PROBLEM FOR SOME CLASSES OF DYNAMICAL SYSTEMS AND ITS APPLICATION DOCTORAL THESIS IN MATHEMATICS Binh Dinh - 2023 i MINISTRY OF EDUCATION AND TRAINING QUY NHON UNIVERSITY DAO THI HAI YEN ON THE STATE ESTIMATION PROBLEM FOR SOME CLASSES OF DYNAMICAL SYSTEMS AND ITS APPLICATION Speciality: Mathematical Analysis Code: 46 01 02 Reviewer 1: Prof Dr Vu Ngoc Phat Reviewer 2: Assoc Prof Dr Nguyen Dinh Phu Reviewer 3: Assoc Prof Dr Pham Quy Muoi Board of Supervisors: Assoc Prof Dr Dinh Cong Huong Binh Dinh - 2023 i Declaration This thesis was completed at the Department of Mathematics and Statistics, Quy Nhon University under the guidance of Assoc Prof Dinh Cong Huong I hereby declare that the results presented in here are new and original All of them were published in peer-reviewed journals and conferences For using results from joint papers I have gotten permissions from my co-authors Binh Dinh, 2023 PhD student Dao Thi Hai Yen i i Acknowledgments First, I would like to express my gratitude to the two offices Phu Yen University, my workplace, and Quy Nhon University, where I have learnt and studied for a long time I would also like to express my gratitude to all people working at the Department of Postgraduate Training and the Department of Mathematics and Statistics of Quy Nhon University Next, the thesis was carried out during the years I have been a PhD student at the Department of Mathematics and Statistics, Quy Nhon University On the occasion of completing the thesis, I would like to express the deep gratitude to Assoc Prof Dr Dinh Cong Huong not only for his teaching and scientific leadership, but also for the helping me access to the academic environment through the workshops, mini courses that assist me in broadening my thinking to get the entire view on the related issues in my research Finally, I wish to acknowledge my father, my parents in law for supporting me in every decision And, my enormous gratitude goes to my husband and sons for their love and patience during the time I was working intensively to complete my PhD program Finally, my sincere thank goes to my mother for guiding me to math and this thesis is dedicated to her ii i Contents Table of Notations List of drawings Introduction Preliminaries 1.1 1.2 Some basic concepts 1.1.1 Stability criteria of some classes of dynamic systems 1.1.2 Additional lemmas 14 State observer design problems 15 1.2.1 Full order state observer 15 1.2.2 Reduced order state observer 17 1.2.3 Linear functional state observer 17 1.2.4 Interval observer 19 A new method for designing observers of a nonlinear time-delay Glucose-Insulin system 21 2.1 A novel state transformation 22 2.2 Application to the GI model 28 2.2.1 State transformation for the GI model 28 2.2.2 State observer design for the GI model 29 2.2.3 Simulation results 30 A new observer for interconnected time-delay systems and its applications to fault detection problem 34 iii i 3.1 New state transformation 37 3.2 Fault detection observer 45 3.3 A numerical example 46 Distributed functional interval observer design for large-scale networks impulsive systems 52 4.1 Designing distributed linear functional interval observers 53 4.2 Existence conditions of distributed linear functional interval observers 55 4.3 Solving unknown matrices 57 4.4 Numerical examples 60 iv i Table of Notations Rn+ Rn×m Z+ A  () A, B ∈ Rn×m , A  () B In 1n : The set of all n-dimensional real non-negative vectors : The set of all real n × m dimensional matrices : Z ∩ R+ : Real part of complex number s : Transpose matrix of the matrix A : Matrix inverse of matrix A : The Moore-Penrose-inverse of A : Rank of the matrix A : The determinant of the matrix A : The matrix A is a positively definite symmetric matrix (non-negative) : All the elements of the matrix A are positive (non-negative) : aij > (>) bij , i ∈ {1, , n}, j ∈ {1, , m} : Identity matrix of size n × n : Vector in Rn with all elements equal to one  T i.e 1 | {z } n ? Lr∞ M = max(A, B) A+ A− |A| x(t¯i+1 ) n S>0 P K K∞ L KL , Nullity A : Symmetric term in a symmetric matrix : The set of all inputs v(t) with the property ||v|| < ∞ : The matrix where each entry is mij = max(aij , bij ) : max(A, 0) : A+ − A : A+ + A− : The left-sided limit of x(t) for t → ti+1 : The set of n × n positive definite matrices : {γ : R+ → R+ | γ is continuous, γ(0) = 0, γ(r) > 0, r > 0} : {γ ∈ P | γ is strictly increasing} : {γ ∈ K | γ is unbounded} : {γ : R+ → R+ | γ is continuous and strictly decreasing with limt→∞ γ(t) = 0} : {β : R+ × R+ → R+ | β is continuous, β(., t) ∈ K ∀t > 0, β(r, ) ∈ L ∀r > 0} : Equal to by definition : The dimension of the N (A) = {x ∈ Rn |Ax = 0} i List of drawings Trang Figure 1.1 Responses of x(t) and its estimation 11 Figure 1.2 Responses of x(t), x− (t) and x+ (t) 18 Figure 2.1 Responses of x2 (t) and x ˆ2 (t − 3) 33 Figure 2.2 Responses of x ˆ2 (t − 3) and x2 (t − 3) 34 Figure 3.1 Residual generator using third-order observer effectively triggers fault in the 1-th subsystem 53 Residual generator using third-order observer effectively triggers fault in the 2-th subsystem 53 Figure 4.1 Responses of x11 (t), x12 (t), x13 (t), x21 (t), x22 (t) and x23 (t) 68 Figure 4.2 Responses of z1 (t) = x13 (t), z1− (t) and z1+ (t) 69 Figure 4.3 Responses of z2 (t) = x23 (t), z2− (t) and z2+ (t) 69 Figure 4.4 Responses of xi1 (t) and xi2 (t) 70 Figure 4.5 Responses of zi (t) = xi1 (t), zi − (t) and zi + (t) 71 Figure 3.2 i Introduction Many real-world physical systems such as electricity, water, communication networks, etc are often modeled by systems of differential equations (see [4], [19], [20], [50], [57], [58]) The qualitative theory of these dynamical systems will bring many useful applications In recent decades, along with the development of qualitative theory, the control theory has become one of important research directions To meet the actual requirements, control systems (biology, medicine, electricity, water, communication networks ,) are becoming more and more diverse and complex These control systems are often very large, in which the controls, devices, and subsystems are linked together through transmission lines and connection bands Therefore, the information received about the state of the system is often delayed (i.e no instantaneous state information is received, but only delayed information is received), missing (i.e not received full information of the whole system but only some information is received), lost (i.e information is provided intermittently), interference (information received is incorrect) Consequences inevitably face failures, disturbances, and many other complications The operation of control systems is usually based on the state information of the systems However, in some cases, these outputs often not provide enough state information for the operation of systems Therefore, the problem of designing functional state observes or designing interval observers for functions of state variables to estimate some more unknown states, which are enough to operate the system at a cost of operation, has become known as an important control problem This problem has many applications in other problems such as fault detection problems, fault estimation problems, identification problems, and signal processing problems, In control theory, a state observer is an auxiliary dynamical system that mirrors the behavior of a physical system, and it is driven by input and output measurements of the physical system in order to provide an estimate of the internal states of the physical system The primary consideration in the design of an observer is that the estimate of the states should be close to the actual value of the system states The next requirement is to provide conditions for the existence of the state observers Some direct applications of the state observer problem include: using estimated state information to design controllers for the considered dynamic system; using estimation information to solve problems of detecting, isolating, and estimating actuator faults as well as sensor faults Since the 60s of the twentieth century, along with the development of control problem, the state estimation problem has attracted a lot of research attention due to the increase in its practical applications (see, for example, [60], [12], [16], [22], [13], [79], [38], [42], [43], [57], [10], [48]) In particular, Luenberger (1966) (see [60]) studied the following control system with input u(t) ∈ Rm and output y(t) ∈ Rp : x(t) ˙ = Ax(t) + Bu(t), t ≥ 0, (1) y(t) = Cx(t), (2) where x(t) ∈ Rn is the state vector, A ∈ Rn×n , B ∈ Rn×m and C ∈ Rp×n are known constant matrices He proposed the following full order state observers for this system: x ˆ˙ (t) = Aˆ x(t) + Bu(t) + L(y(t) − yˆ(t)), t ≥ 0, yˆ(t) = Cx ˆ(t), i where L ∈ Rn×p is the observer gain matrix, x ˆ(t) ∈ Rn is an estimate of the state x(t) Darouach et al (1994) (see [12]) developed full order state observers reported in [60] to deal with system (1)-(2), where an extra term Dω(t) is added in the equation (1) The authors proposed the following observer: z(t) ˙ = N z(t) + Ly(t) + Gu(t) t ≥ 0, x ˆ(t) = z(t) − Ey(t), where z(t) ∈ Rn is the observer state vector, N , L, G and E are the observer gain matrices and x ˆ(t) ∈ Rn is an estimate of the state x(t) Deng et al (2004) (see [16]) extended the results of Darouach et al (1994) to a class of systems of differential equations with two unknown inputs Eω(t) and Dω(t) appear in equations (1) and (2), respectively (i.e., noise appears in the state and output information of control system) The authors transformed the systems (1) and (2) (in case there are two more unknown inputs) to the following form: x ˆ˙ (t) = Aˆ x(t) + B¯ u(t) + E1 ω1 (t), t ≥ 0, = Cx ˆ(t) y1 (t) Then, they proposed state observers as follows: ˙ ζ(t) = N ζ(t) + Ly1 (t) + (K − M C)¯ u(t) t ≥ 0, zˆ(t) = ζ(t) + M y1 (t), where ζ(t) ∈ Rr , zˆ(t) is the estimated vector of z(t) = Kx(t), N , L and M are gain observer matrices Fairman et al (1986); Darouach et al (1999) (see [22], [13]) considered the following class of delayed linear systems: x(t) ˙ = Ax(t) + Ad x(t − τ ) + Bu(t), t ≥ 0, (3) x(θ) = φ(θ), θ ∈ [−τ, 0], (4) y(t) = (5) Cx(t), where φ(θ) is the initial condition function, τ > is the time delay, x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the control vector, y(t) ∈ Rp is the output vector, matrices A, Ad , B and C are constant and of appropriate dimensions The authors proposed the following reduced order observer without time delays to estimate the (n − p) components of the state vector: x ˆ(t) = Dw(t) + Ey(t), w(t) ˙ = N w(t) + Jy(t) + Jd y(t − τ ) + Hu(t), where x ˆ(t) is the estimated vector of x(t), and D, E, N , J, Jd and H are gain observer matrices Trinh et al (2010) (see [79]) considered the time-delayed term in the equation (3) as an unknown input, i.e, Ad x(t − τ ) = W ω(t), and moved back to the state estimation problem for systems without i (4.50) zi− (tik+1 ) = Li zi− (t¯ik+1 ) + Fi Di ui (tik+1 ) − |Ki |1pi Vi + Ki yi (tik+1 ) + Fi gi− (tik+1 ), (4.51) z˙i+ (t) = Ni zi+ (t) + Ji yi (t) + |Ji |1pi Vi + Fi Bi ui (t) + Fi d+ i (t), t ∈ [tik , tik+1 ), = Li zi+ (t¯ik+1 ) (4.52) zi+ (tik+1 ) + Ki yi (tik+1 ) + |Ki |1pi Vi + Fi gi+ (tik+1 ) + Fi Di ui (tik+1 ), (4.53) where zi+ (t) ∈ Rri , zi− (t) ∈ Rri Matrices Ni , Ji , Li and Ki are unknown observer parameters zi− (t¯ik+1 ) and zi+ (t¯ik+1 ) are the left-sided limit of zi− (t) and zi+ (t) for t → tik+1 , respectively By following the proof of Theorem 4.2.1, we obtain the following corollary Corollary 4.3.1 Systems (4.50)-(4.51) and (4.52)-(4.53) are an interval observer of linear function zi (t) of system (4.1)-(4.3) (in case Aij = 0, j = 1, , N, j 6= i) if the following conditions are satisfied Ni is Metzler, (4.54) Li > 0, (4.55) Ni Fi + Ji Ci − Fi Aii = 0, (4.56) Li Fi + Ki Ci − Fi Gi = 0, (4.57) ni There exist matrices Pi , Qi ∈ S>0 such that T LTi eNi θ Pi eNi θ Li − Pi = −Qi (4.58) hold for all θ ∈ [Timin , Timax ] Remark 4.3.2 Matrices Ni , Ji , Li and Ki satisfying conditions of Corollary 4.3.1 can be obtained by using Theorem 4.3.1, where matrices Xi and Yi in (4.35) and (4.36) are replaced by    Xi =   4.4 Fi Ci 0 0 Fi Ci   h   , Yi = Fi Aii  Fi Gi i Numerical examples Example 4.4.1 We now consider the large-scale networks of impulsive system (4.1)-(4.3) with N = 2, impulse events are as tik = 10 + 20k, i = 1, 2, k ∈ Z+ , where h i h i xT1 (t) = x11 (t) x12 (t) x13 (t) , xT2 (t) = x21 (t) x22 (t) x23 (t) ,       −1.5 0.1 −0.5 0.2       A11 =  −10  , C1T =   , A22 =  −5  , 1 −5 0 −6 60 i   C2T =  0   0    , A12 =  0      , A21 =  2        , B1 =   ,    0     B2 =   , F1T = F2T =   , 1     2.1 0 1.8 0     G1 =  1.2 0.7 1.1  , G2 =  1.3 0.6 0.1  0.1 0.2 1.3  Note that, for this example, the robust observer reported in [46] and [17] cannot be applied Let us now use the method in this chapter to design distributed functional interval observers for the impulsive interconnected system in this example First, we consider the 1- subsystem: According to"Step #1 of X1 Algorithm 4.1, we obtain matrices X1 and Y1 from equations (4.35)-(4.36) Since rank = Y1 h i = rank X1 , condition (4.39) is satisfied By solving the LP problem (4.49) with the constraint 0.1 ≤ λ1 ≤ 1.2 we obtain a solution λ1 = 0.1, Z1 = 01,8 (4.59) Now, taking (4.59) into 4.1,i the observer gains are h account i for Step and Step h of i Algorithm h obtained as N1 = −5, J1 = 1 , L1 = 1.3, K1 = , H12 = Next, we found that the LMI conditions (4.29)-(4.30) hold for all θ ∈ [0.21, ∞] Therefore all conditions of Corollary 4.2.1 are satisfied and this completes the design of a first-order interval observer to estimate z1 (t) = F1 x1 (t) = x13 (t) Next, we consider the 2- subsystem: According to#Step of Algorithm 4.1, we obtain matrices X2 " h i X2 and Y2 from equations (4.35)-(4.36) Since rank = = rank X2 , condition (4.39) is satisfied Y2 By solving the LP problem (4.49) with the constraint 0.15 ≤ λ1 ≤ 2.5 we obtain a solution λ1 = 0.15, Z1 = 01,8 (4.60) Now, taking (4.60) into account fori Step and Step hof Algorithm 4.1,h the observer gains h i i are obtained as N2 = −6, J2 = , L2 = 0.2, K2 = 0.1 , H21 = Next, we found that the LMI conditions (4.29)-(4.30) holds for all θ ∈ [0.16, ∞] Therefore all conditions of Corollary 4.2.1 are satisfied and this completes the design of a first-order interval observer to estimate z2 (t) = F2 x2 (t) = x23 (t) Simulation results of Example 4.4.1 We consider the input u1 (t) = u2 (t) = sin t, t 100, the signals d1 (t), g1 (t) and v1 (t), d2 (t), g2 (t) and v2 (t) are " # 0.1 sin(2t) d1 (t)= d2 (t) = , 0.1 sin(2t) 61 i " # 0.1 sin(t) + 0.3 g1 (t)= g2 (t) = , 0.1 sin(t) + 0.3 # " # " 0.1| sin(t)| t+1 v1 (t) = , v2 (t) = 0.2| sin(t)| t+1 for t 100 and the initial conditions are x11 (0) = 1, x12 (0) = 2, x13 (0) = 3, x21 (0) = 4, x22 (0) = 5, x23 (0) = 6, z1− (0) = 0, z1+ (0) = 1, z2− (0) = 1, z2+ (0) = Figure 4.1 shows the responses of x11 (t), x12 (t), x13 (t), x21 (t), x22 (t) and x23 (t) With the above data, Figure 4.2 shows the response of z1 (t) = x13 (t), z1− (t), z1+ (t), while Figure 4.3 shows the response of z2 (t) = x23 (t), z2− (t), z2+ (t), respectively 35 30 25 20 15 10 0 10 20 30 40 50 60 70 80 90 100 Time(sec) Figure 4.1: Responses of x11 (t), x12 (t), x13 (t), x21 (t), x22 (t) and x23 (t) Example 4.4.2 (A practical example) Consider the commercial electric vehicle equipped with a low power range extender fuel cell system which can be represented by system (4.1)-(4.3) (in case Aij = 0, j = 1, , N, j 6= i), where " # " # " # xi1 (t) iLf c −1.5 T xi (t) = = , Aii = , xi2 (t) vveh −0.1 " # h i 1.8 Gi = , Ci = , Bi = 02,1 , Di = 02,1 , 1.2 0.7 " di (t) = 0.1 sin(2t) 0.1 sin(2t) # " , gi (t) = 0.1 sin t + 0.3 0.1 sin t + 0.3 # , vi (t) = 0.1 sin t For this example, in [17], the authors provided a method to derive a full-order interval observer for state vectors of a single impulsive system However, they did not consider the issue of designing functional interval observers as well as showing how to choose matrices L and M such that A − LC is 62 i 10 -2 10 20 30 40 50 60 70 80 90 100 Time(sec) Figure 4.2: Responses of z1 (t) = x13 (t), z1− (t) and z1+ (t) -1 10 20 30 40 50 60 70 80 Time(sec) Figure 4.3: Responses of z2 (t) = x23 (t), z2− (t) and z2+ (t) 63 i 90 100 Metzler and G − M C is nonnegative In the following, we will apply Corollary 4.3.1 and Remark 4.3.2 in this chapter to design an iinterval observer of the form (4.50)-(4.51) and (4.52)-(4.53) for linear function h zi (t) = Fi xi (t) = xi (t) = xi1 (t) By following steps (Step 1-Step 4) of Algorithm 4.1 we obtain Ni = −1.5, Ji = 0, Li = 1.8, Ki = In addition, we see that the LMI conditions (4.29)-(4.30) hold for all θ ∈ [0.41, ∞] Therefore all conditions of Corollary 4.3.1 are satisfied and we obtain a first-order interval observer to estimate function zi (t) = Fi xi (t) = xi1 (t) Simulation results of Example 4.4.2 " # " # −0.1 0.1 − + − + − + We consider bounds di (t), di (t), gi (t), gi (t) and Vi are di (t) = , di (t) = , gi− (t) = −0.1 0.1 " # " # 0.2 0.4 + , gi (t) = , for t 100 and Vi = 0.1 The initial conditions are chosen as xi1 (0) = 1, 0.2 0.4 xi2 (0) = 2, zi− (0) = 0, zi+ (0) = Figure 4.4 shows the responses of xi1 (t) and xi2 (t) while Figure 4.5 shows the response of zi (t) = xi1 (t), zi− (t), zi+ (t) 1.5 0.5 -0.5 10 20 30 40 50 60 70 80 90 100 Time(sec) Figure 4.4: Responses of xi1 (t) and xi2 (t) Conclusion of Chapter In this chapter, we have proposed a new method for deriving interval observers for linear functions of each subsystem of a class of large-scale network impulsive systems with bounded uncertainties Conditions for the existence of the interval observer and an effective algorithm for solving the unknown observer matrices have been presented in this chapter Two examples with simulation results have been presented to show the simplicity of our design method Limitations in this chapter are that in the case where the system matrices of large-scale impulsive network systems are uncertain, the issue of time delays in the state and output vectors has not been investigated 64 i 1.5 0.5 -0.5 10 20 30 40 50 60 70 80 Time(sec) Figure 4.5: Responses of zi (t) = xi1 (t), zi − (t) and zi + (t) 65 i 90 100 Conclusions This thesis obtained the following main results: • We have received a novel procedure for designing a state observer of a general nonlinear time-delay G.I model - Result of Theorem 2.1.1 is significant as the two-stage design process transforms a nonlinear time-delay model into a new observable form which allows a third-order delayed state observer to be easily designed - Simulation results have been given to illustrate the effectiveness of our results (see Subsection 2.2.3) - Since the matrix pair (A, C) is not observable, we used the transformation containing the ¯ C) ¯ Therefore, delayed state variable to return to a system with the observable matrix pair (A, the designed state observer will have been able to estimate a delayed version of the state vector instead of the instantaneous state vector • We have received the design of distributed functional observers to detect actuator faults of interconnected time-delay systems with unknown inputs - We have developed the state transformations in [23] and [24] to transform each subsystem of time-delay interconnected systems into an observable canonical form (see Theorem 3.1.1) - In the new coordinate system, we have designed a functional observer to construct a residual function that can trigger the i-th subsystem faults - Conditions for the existence of such state transformation and a new method for determining unknown parameters are provided (see Theorem 3.2.1) - A numerical example with simulations is given to illustrate the effectiveness of the proposed design method (see Section 3.3) • We have received a new method for deriving interval observers for linear functions of each subsystem of a class of large-scale networks impulsive systems with bounded uncertainties - Conditions for the existence of the interval observer (see Theorem 4.2.1) and an effective algorithm for solving the unknown observer matrices (see Algorithm 4.1) - Two examples with simulation results have been presented to show the simplicity of our design method (see Example 4.4.1, Example 4.4.2) 66 i Future investigation In the future, we intend to continue the investigation in the following directions: • Extend the results of Chapter to design dynamic event-triggered state observers to estimate the state vectors of nonlinear time-delay systems • Extend the results of Chapter to solve fault detection problems for interconnected time-delay systems where the system matrices of large-scale impulsive network systems are uncertain, and unknown time-varying delays are in the output vectors • Extend the results of Chapter to the case where the system matrices of large-scale impulsive network systems are uncertain, and unknown time-varying delays are in the state and output vectors 67 i List of Author’s Related Publication Yen D.T.H., Huong D.C (2020), “A new method for designing observers of a nonlinear time-delay Glucose-Insulin system”, Quy Nhon Journal of Science, 14(1), pp 37-46 Huong D.C., Yen D.T.H., Sau N.H (2021), “A new observer for interconnected time-delay systems and its applications to fault detection problem”, The international conference on differential equations, functional equations and some related problems at Dong Thap University, pp 126–148 Huong D.C., Yen D.T.H., Thuan M.V (2021), “Design of distributed functional interval observers for large-scale networks impulsive systems”, Transactions of the Institute of Measurement and Control, 43(14), pp 3233–3243 68 i Bibliography [1] Ahmed-Ali T., Cherrier E., Lamnabhi-Lagarrigue F (2012), “Cascade high gain predictors for a class of nonlinear systems”, IEEE Transactions on Automatic Control, 57(1), pp 221–226 [2] Bai L.S., Tian Z.H., Shi S.R (2017), “Robust fault detection for a class of nonlinear time-delay systems”, Journal of The Franklin Institute, 344(6), pp 837–888 [3] Boutat D., Darouach M., Voos H (2014), “Observer design for a nonlinear minimal model of glucose disappearance and insulin kinetics”, Communications in Computer and Information Science, 511, pp 261–273 [4] Briat C., Verriest E (2009), “A new delay-SIR model for pulse vaccination”, Biomedical Signal Processing and Control, 4(4), pp 272–277 [5] Briat C (2013), “Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints”, Automatica, 49(11), pp 3449– 3457 [6] Briat C., Khammash M (2017), “Simple interval observers for linear impulsive systems with applications to sampled-data and switched systems”, IFAC-Papers OnLine, 50(1), pp 5079–5084 [7] Chan J.C.L., Lee T.H (2020), “Sliding mode observer-based fault-tolerant secondary control of microgrids”, Electronics, 9(9), pp 1417 [8] Chan J.C.L., Lee T.H., Tan C.P., Trinh H., Park J.H (2021), “A Nonlinear observer for robust fault reconstruction in one-sided Lipschitz and quadratically inner-bounded nonlinear descriptor systems”, IEEE Access, 9, pp 22455–22469 [9] Che H., Huang J., Zhao X., Ma X., Xu N (2020), “Functional interval observer for discrete-time systems with disturbances”, Applied Mathematics and Computation, 383(15), pp 125352 [10] Chen J., R., Patton R.J (1999), “Robust model-based fault diagnosis for dynamic systems”, Massachusetts: Kluwer Academic Publishers [11] Cheng P., Wang J., He S., Luan X., Liu F (2020), “Observer-based asynchronous fault detection for conic-type nonlinear jumping systems and its application to separately excited DC motor”, IEEE Transactions on Circuits and Systems I: Regular Papers, 67(3), pp 951–962 69 i [12] Darouach M., Zasadzinski M., Xu S.J (1994), “Full-order observers for linear systems with unknown inputs”, IEEE Transactions on Automatic Control, 39(3), pp 606-609 [13] Darouach M., Pierrot P., Richard E (1999), “Design of reduced-order observers without internal delays”, IEEE Transactions on Automatic Control, 44 (9), pp 1711-1713 [14] Darouach M (2001), “Linear functional observers for systems with delays in state variables”, IEEE Transactions on Automatic Control, 46(3), pp 491–496 [15] Dashkovskiy S., Mironchenko A (2013), “Input-to-state stability of nonlinear impulsive systems”, SIAM Journal on Control and Optimization, 51(3), pp 1962–1987 [16] Deng H., Li H.X (2004), “Functional observers for linear systems with unknown inputs”, IEEE Transactions on Automatic Control, 6(4), pp 462–468 [17] Degue K.H., Efimov D., Richard J.P (2018), “Stabilization of linear impulsive systems under dwelltime constraints: Interval observer-based framework”, European Journal of Control, 42, pp 1–14 [18] Efimov D., Perruquetti W., Zolghadri A (2013), “Interval observers for time-varying discrete-time systems”, IEEE Transactions on Automatic Control, 58(12), pp 3218–3224 [19] Emami K., Nener B., Sreeram V., Trinh H., Fernando T (2013), “A fault detection technique for dynamical systems”, International Conference on Industrial and Information Systems, pp 201–206 [20] Emami K., Fernando T., Nener B., Trinh H., Zhang Y., “A functional observer based fault detection technique for dynamical systems”, Journal of the Franklin Institute, 352(5), pp 2113–2128 [21] Ellouze I., Hammami M.A., Vivalda J.C (2014), “A separation principle for linear impulsive systems”, European Journal of Control, 20(3), pp 105–110 [22] Fairman F.W., Kumar A (1986), “Delayless observers for systems with delay”, IEEE Transactions on Automatic Control, 31(3), pp 258–259 [23] Farina L., Rinaldi S (2000), “Positive Linear Systems: Theory and Applications”, John Wiley & Sons, New York [24] Fichera F., Prieur C., Tarbouriech S., Zaccarian L (2013), “Using Luenberger observers and dwelltime logic for feedback hybrid loops in continuous-time control systems”, International Journal of Robust and Nonlinear Control, 23(10), pp 1065–1086 [25] Gao Z.W., Breikin T., Wang H (2008), “Reliable observer-based control against sensor failures for systems with time delays in both state and input”, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, 38(5), pp 1018–1029 [26] Germani A., Pepe P.A (2005), “State observer for a class of nonlinear systems with multiple discrete and distributed time delays”, European Journal of Control, 11(3), pp 196–205 [27] Goebel R., Sanfelice R., Teel A (2012), “Hybrid Dynamical Systems: Modeling, Stability, and Robustness”, Princeton University Press 70 i [28] Gouz´e J.L., Rapaport A., Hadj-Sadok M.Z (2000), “Interval observers for uncertain biological systems”, Ecological Modelling, 133(1-2), pp 45–56 [29] Gu D.K., Liu L.W., Duan G.R (2018), “Functional interval observer for the linear systems with disturbances”, IET Control Theory and Applications, 12, pp 2562–2568 [30] Gou L., Zhang Y.M., Wang H., Fang J.C (2006), “Observer-based optimal fault detection and diagnosis using conditional probability distributions”, IEEE Transactions on Signal Processing, 54(10), pp 3712–3719 [31] Haddad W.M., Chellaboina V.S., Nersesov S.G (2006), “Impulsive and hybrid dynamical systems”, Princeton University Press Princeton N.J [32] Han J., Zhang H., Wang Y., Sun X (2018), “Robust fault detection for switched fuzzy systems with unknown input”, IEEE Transactions on Cybernetics, 48(11), pp 3056–3066 [33] He Z., Xie W (2016), “Control of nonlinear switched systems with average dwell time: Interval observer-based framework”, IET Control Theory and Applications, 10, pp 10–16 [34] Hespanha J., Liberzon D.,Teel A (2008), “Lyapunov conditions for input-to-state stability of impulsive systems”, Automatica, 44(11), pp 2735–2744 [35] Hetel L., Fiter C., Omran H., Seuret A., Fridman E., Richard J.P., Niculescu S.I (2017), “Recent developments on the stability of systems with aperiodic sampling: an overview”, Automatica, 76, pp 309–335 [36] Hirsch M.W., Smith H.L (2005), “Monotone maps: a review”, Journal of Difference Equations and Applications, 11, pp 379–398 [37] Hou M., Muller P.C (1992), “Design of observers for linear systems with unknown inputs”, IEEE Transactions on Automatic Control, 37(6), pp 871–875 [38] Hou M., Zitek P., Patton R.J (2002), “An observer design for linear time-delay systems”, IEEE Transactions on Automatic Control, 47(1), pp 121–125 [39] Huang J., Ma X., Zhao X., Che H., Chen L (2020), “Interval observer design method for asynchronous switched systems”, IET Control Theory and Applications, 14(12), pp 1082-1090 [40] Huang J., Ma X., Che H., Han Z (2020), “Further result on interval observer design for discrete-time switched systems and application to circuit systems”, IEEE Transactions on Circuits and Systems II: Express Briefs, 67(11), pp 2542–2546 [41] Huong D.C., Trinh H., Tran H.M., Fernando T (2017), “Approach to fault detection of time-delay systems using functional observers”, Electronics Letters, 50, pp 1132–1134 [42] Huong D.C., Trinh H (2015), “Method for computing state transformations of time-delay systems”, IET Control Theory and Applications, 9(16), pp 2405–2413 71 i [43] Huong D.C., Trinh H (2016), “New state transformations of time-delay systems with multiple delays and their applications to state observer design”, Journal of the Franklin Institute, 353(14), pp 3487– 3523 [44] Huong D.C (2018), “A fresh approach to the design of observers for time-delay systems”, Transactions of the Institute of Measurement and Control, 40(2), pp 477–503 [45] Huong D C., Thuan M V (2018), “Design of unknown input reduced-order observers for a class of nonlinear fractional-order time-delay systems”, International Journal of Adaptive Control and Signal Processing, 32(3), pp 412–423 [46] Huong D.C (2019), “Design of functional interval observers for nonlinear fractional-order interconnected systems”, International Journal of Systems Science, 50(15), pp 2802–2814 [47] Huong D.C., Trinh H (2016), “New state transformations of time-delay systems with multiple delays and their applications to state observer design”, Journal of the Franklin Institute, 353(14), pp 3487–3523 [48] Huong D.C., Trinh H., Tran H.M., Fernando T (2014), “Approach to fault detection of time-delay systems using functional observers”, Electronics Letters, 50, pp 1132–1134 [49] Huong D.C., Huynh V.T., Trinh H (2019), “Integral outputs-based robust state observers design for time-delay systems”, SIAM Journal on Control and Optimization, 57(3), pp 2214–2239 [50] Huong D.C., Huynh V.T., Thao N.G.M (2019.), “Design of functional interval observers for a battery and motor circuit model of Electric vehicle”, IEEE Vehicle Power and Propulsion Conference, 10.1109/VPPC46532.2019.8952280 [51] Huong D.C., Huynh V.T., Trinh H (2020), “Dynamic event-triggered state observers for a class of nonlinear systems with time delays and disturbances”, IEEE Transactions on Circuits and Systems-II: Express Briefs, 12, pp 3457–3461 [52] Huong D.C., Yen D.T.H., Sau N.H (2021), “A new observer for interconnected time-delay systems and its applications to fault detection problem”, The International Conference on Differential Equations, Functional Equations and Some Related Problems at Dong Thap University, pp 126–148 [53] Huong D.C., Yen D.T.H., Thuan M.V (2021), “Design of distributed functional interval observers for large-scale networks impulsive systems”, Transactions of the Institute of Measurement and Control, 43(14), pp 3233–3243 [54] Jiang C., Zhou D.H (2005), “Fault detection and identification for uncertain linear time-delay systems”, Computers and Chemical Engineering, 30(2), pp 228–242 [55] Khan A., Xie W (2019), “Interval state Estimator design using the observability matrix for multiple input multiple output linear time-varying discrete-time systems”, IEEE Access, 7, pp 167566–167576 [56] Lee T.H., Park M.J., Park J.H (2021), “An improved stability criterion of neural networks with timevarying delays in the form of quadratic function using novel geometry-based conditions”, Applied Mathematics and Computation, 404, pp 126226 72 i [57] Li W., Gui W.H., Xie Y.F., Ding S.X (2010), “Decentralised fault detection of large-scale systems with limited network communications”, IET Control Theory and Applications, 4(9), pp 1867–1876 [58] Li H., Chen Z., Wu L., Lam H.K., Du H (2017), “Event-triggered fault detection of nonlinear networked systems”, IEEE Transactions on Cybernetics, 47(4), pp 1041–1052 [59] Loxton R.C., Teo K.L., Rehbock V., Ling W.K (2009), “Optimal switching instants for a switchedcapacitor DC/DC power converter”, Automatica, 45(4), pp 973–980 [60] Luenberger D.G (1966), “Observers for multivariate systems”, IEEE Transactions on Automatic Control, 11(2), pp 190–197 [61] Luenberger D (1971), “An introduction to observers”, IEEE Transactions on Automatic Control, 16(6), pp 596–602 [62] Luenberger D G (1979), “Introduction to Dynamic Systems: Theory, Models and Applications”, John Wiley & Sons, New York [63] Mazenc F., Niculescu S.L., Bernard O (2012), “Exponentially stable interval observers for linear systems with delay”, SIAM Journal on Control and Optimization, 50(1), pp 286–305 [64] Medina E., Lawrence D (2009), “State estimation for linear impulsive systems”, in: Proceedings of the American Control Conference, pp 1183-–1188 [65] Meskin N., Khorasani K (2009), “Robust fault detection and isolation of time-delay systems using a geometric approach”, Automatica., 45(6), pp 1567–1537 [66] Mohajerpoor R., Shanmugam L., Abdi H., Saif M (2017), “Functional observer design for retarded system with interval time-varying delays”, International Journal of Systems Science, 48(5), pp 1060– 1070 [67] Nam P.T., Pathirana P.N., Trinh H (2014), “-bounded state estimation for time-delay systems with bounded disturbances”, International Journal of Control, 87 (9), pp 1747–1756 [68] Naghshtabrizi P., Hespanha J.P., Teel A.R (2008), “Exponential stability of impulsive systems with application to uncertain sampled-data systems”, Systems and Control Letters, 57(5), pp 378–385 [69] Palumbo P., Pepe P., Panunzi P., De Gaetano A (2012), “Time-delay model-based control of the glucose -insulin system, by means of a state observer”, European Journal of Control, 18(6), pp 591–606 [70] Palumbo P., Pepe P., Panunzi S., De Gaetano A (2011), “Glucose control by subcutaneous insulin administration: a DDE modelling approach”, In: Proceedings the 18th IFAC World Congress Milano (Italy), pp 1471–1476 [71] Palumbo P., Panunzi P., De Gaetano A (2007), “Qualitative behavior of a family of delay differential models of the glucose insulin system”, Discrete and Continuous Dynamical Systems-series B, 7(2), pp 399–424 73 i [72] Pereira F., Silva G (2002), “Stability for impulsive control systems”, Systems and Control Letters, 17(4), pp 421–434 [73] Phat V.N (2000), “Introduction to mathematical control theory” (in Vietnamese), Vietnam National University Press, Hanoi [74] Shanmugam L., Joo Y.H (2020), “Stability and stabilization for T-S fuzzy large-scale interconnected power system with wind farm via sampled-data control”, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 51(4), pp 2134–2144 [75] Shen J., Liu J., Cui Y (2020), “An exact characterization of the L1 /L-index of positive systems and its application to fault detection filter design”, IEEE Transactions on Circuits and Systems II: Express Briefs, 67(12), pp 3262–3266 [76] Stone B., Shulgin B., Agur Z (2000), “Theoretical examination of the pulse vaccination policy in the SIR epidemic model”, Mathematical and Computer Modelling, 31(4-5), pp 201–215 [77] Tanwani A., Shim H., Liberzon D (2013), “Observability for switched linear systems: characterization and observer design”, IEEE Transactions on Automatic Control, 58(4), pp 891–904 [78] Tran H.M., Trinh H (2017), “Distributed functional observer based fault detection for interconnected time-delay systems”, IEEE Systems Journal, 13(1), pp 940–951 [79] Trinh H.M., Teh P.S., Fernando T.L (2010), “Time-delay systems: design of delay-free and low-order observers”, IEEE Transactions on Automatic Control, 55 (10), pp 2434-2438 [80] Trinh H., Fernando T (2012), “Functional observers for dynamical ystems”, Springer-Verlag, Berlin Heidelberg [81] Trinh H., Huong D.C (2018), “A new method for designing distributed reduced-order functional observers of interconnected time-delay systems”, Journal of the Franklin Institute, 355(3), pp 1411– 1451 [82] Yang T., Chua L.O (1997), “Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication”, IEEE Transactions on Circuits and Systems Part I: Fundamental Theory and Applications, 44(10), pp 976–988 [83] Yen D.T.H., Huong D.C (2020), “A new method for designing observers of a nonlinear time-delay Glucose-Insulin system”, Quy hon Journal of Science, 14(1), pp 37–46 [84] Zhang K., Jiang B., Yan X.G., Shen J (2019), “Interval sliding mode observer based incipient sensor fault detection with application to a traction device in China railway high-speed”, IEEE Transactions on Vehicular Technology, 68(3), pp 2585–2597 [85] Zhao Y., Lam J., Gao H (2009), “Fault detection for fuzzy systems with intermittent measurements”, IEEE Transactions on Fuzzy Systems, 17(2), pp 398–410 [86] Zhong M., Ding S.X., Lam J., Zhang C (2003), “Fault detection filter design for LTI system with time delays”, In Proceedings of the IEEE Conference on Decision and Control, 2, pp 1467–1472 74 i