ON CESTIMATION PROBLEM FOR SOME CLASSES OF DYNAMICAL SYSTEMS AND ITS APPLICATION

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 Din.

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 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 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 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 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 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 Table of Notations Rn+ Rn×m Z+ (s) AT A−1 A‡ rank(A) det(A) A A ( )0 A, B ∈ Rn×m , A In 1n ( )B : 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 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} 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 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), 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 (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 = i) if the following conditions are satisfied Ni is Metzler, Li (4.54) 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     , Yi =  Fi Aii Fi Gi 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 xT1 (t) =  A11 x11 (t) x12 (t) x13 (t) , xT2 (t) = x21 (t) x22 (t) x23 (t) ,      −1.5 0.1 −0.5 0.2       = −10  , C1T =   , A22 =  −5  , 1 −5 0 −6 60   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 of X1 Algorithm 4.1, we obtain matrices X1 and Y1 from equations (4.35)-(4.36) Since rank = Y1 = 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 account for Step and Step of Algorithm 4.1, the observer gains are 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 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 for Step and Step of Algorithm 4.1, the observer gains 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 d1 (t)= d2 (t) = 0.1 sin(2t) 0.1 sin(2t) 61 , 0.1 sin(t) + 0.3 0.1 sin(t) + 0.3 g1 (t)= g2 (t) = v1 (t) = 0.1| sin(t)| 0.2| sin(t)| , t+1 t+1 , v2 (t) = 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 = i), where xTi (t) = Gi = di (t) = xi1 (t) xi2 (t) 1.8 1.2 0.7 0.1 sin(2t) 0.1 sin(2t) = iLf c vveh , Ci = , Aii = , gi (t) = −1.5 0 −0.1 , , Bi = 02,1 , Di = 02,1 , 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 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 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 interval observer of the form (4.50)-(4.51) and (4.52)-(4.53) for linear function 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 d− , d+ , gi− (t) = i (t), di (t), gi (t), gi (t) and Vi are di (t) = i (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 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 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 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 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