Luận án on the state estimation 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 SOMECLASSES OF DYNAMICAL SYSTEMS AND ITS APPLICATION

Speciality: Mathematical Analysis

Code: 9 46 01 02

Reviewer 1: Prof Dr Vu Ngoc PhatReviewer 2: Assoc Prof Dr Nguyen Dinh PhuReviewer 3: Assoc Prof Dr Pham Quy Muoi

Board of Supervisors:

Assoc Prof Dr Dinh Cong Huong

Binh Dinh - 2023

This thesis was completed at the Department of Mathematics and Statistics, Quy Nhon Universityunder the guidance of Assoc Prof Dinh Cong Huong I hereby declare that the results presented in hereare new and original All of them were published in peer-reviewed journals and conferences For usingresults from joint papers I have gotten permissions from my co-authors

Binh Dinh, 2023PhD student

Dao Thi Hai Yen

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 wouldlike to express the deep gratitude to Assoc Prof Dr Dinh Cong Huong not only for his teachingand scientific leadership, but also for the helping me access to the academic environment through theworkshops, mini courses that assist me in broadening my thinking to get the entire view on the relatedissues 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 Iwas working intensively to complete my PhD program Finally, my sincere thank goes to my mother forguiding me to math and this thesis is dedicated to her

1.1 Some basic concepts 9

1.1.1 Stability criteria of some classes of dynamic systems 9

1.1.2 Additional lemmas 14

1.2 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

2 A new method for designing observers of a nonlinear time-delay Glucose-Insulin sys-tem 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

3 A new observer for interconnected time-delay systems and its applications to fault

3.1 New state transformation 37

3.2 Fault detection observer 45

3.3 A numerical example 46

4 Distributed functional interval observer design for large-scale networks impulsive sys-tems 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

<(s) : Real part of complex number s

AT : Transpose matrix of the matrix A

A−1 : Matrix inverse of matrix A

A‡ : The Moore-Penrose-inverse of A

rank(A) : Rank of the matrix A

det(A) : The determinant of the matrix A

A > 0 : The matrix A is a positively definite

symmetric matrix (non-negative)

A  () 0 : All the elements of the matrix A are positive

(non-negative)

A, B ∈ Rn×m, A  () B : aij > (>) bij, i ∈ {1, , n}, j ∈ {1, , m}

In : Identity matrix of size n × n

1n : Vector in Rn with all elements equal to one

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 ˆx2(t − 3) 33

Figure 2.2 Responses of ˆx2(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

Figure 3.2 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

Many real-world physical systems such as electricity, water, communication networks, etc areoften modeled by systems of differential equations (see [4], [19], [20], [50], [57], [58]) The qualitativetheory of these dynamical systems will bring many useful applications In recent decades, along with thedevelopment of qualitative theory, the control theory has become one of important research directions

To meet the actual requirements, control systems (biology, medicine, electricity, water, communicationnetworks ,) are becoming more and more diverse and complex These control systems are often verylarge, in which the controls, devices, and subsystems are linked together through transmission lines andconnection 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) Consequencesinevitably 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 do 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 thedesign of an observer is that the estimate of the states should be close to the actual value of the systemstates The next requirement is to provide conditions for the existence of the state observers Some directapplications of the state observer problem include: using estimated state information to design controllersfor 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 stateestimation 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)

where x(t) ∈ Rn

is the state vector, A ∈ Rn×n

, B ∈ Rn×m

and C ∈ Rp×nare known constant matrices

He proposed the following full order state observers for this system:

˙ˆx(t) = Aˆx(t) + Bu(t) + L(y(t) − ˆy(t)), t ≥ 0,ˆ

y(t) = C ˆx(t),

where L ∈ R is the observer gain matrix, ˆx(t) ∈ R is an estimate of the state x(t).

Darouach et al (1994) (see [12]) developed full order state observers reported in [60] to deal withsystem (1)-(2), where an extra term Dω(t) is added in the equation (1) The authors proposed thefollowing observer:

˙z(t) = N z(t) + Ly(t) + Gu(t) t ≥ 0,ˆ

x(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 ˆx(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 authorstransformed the systems (1) and (2) (in case there are two more unknown inputs) to the following form:

˙ˆx(t) = Aˆx(t) + B¯u(t) + E1ω1(t), t ≥ 0,

y1(t) = C ˆx(t)

Then, they proposed state observers as follows:

˙ζ(t) = N ζ(t) + Ly1(t) + (K − M C)¯u(t) t ≥ 0,ˆ

z(t) = ζ(t) + M y1(t),where ζ(t) ∈ Rr, ˆz(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 delayedlinear systems:

˙x(t) = Ax(t) + Adx(t − τ ) + Bu(t), t ≥ 0, (3)x(θ) = φ(θ), θ ∈ [−τ, 0], (4)

where φ(θ) is the initial condition function, τ > 0 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 constantand of appropriate dimensions The authors proposed the following reduced order observer without timedelays to estimate the (n − p) components of the state vector:

ˆx(t) = Dw(t) + Ey(t),

˙w(t) = N w(t) + J y(t) + Jdy(t − τ ) + Hu(t),where ˆx(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 unknowninput, i.e, Adx(t − τ ) = W ω(t), and moved back to the state estimation problem for systems without

time delays Darouach et al (2001) (see [14]) proposed a linear function observer for the system (3)-(4)

as follows:

˙ζ(t) = N ζ(t) + Ndζ(t − τ ) + Dy(t) + Ddy(t − τ ) + Eu(t),ˆ

z(t) = ζ(t) + F y(t),where ζ(θ) = ρ(θ), ∀ θ ∈ [−τ, 0], is the initial condition function, ζ(t) ∈ Rris the vector to be evaluated,ˆ

z(t) ∈ Rris the estimated vector of z(t), N , Nd, D, Dd, E and F are matrices to be determined Nam et

al (2014) (see [67]) extended the method proposed in [14] to deal with (3) -(4) where an unknown inputHω(t) is added into the equation (3) Namely, the authors proposed the following state observer:

˙ζ(t) = N ζ(t) + Ndζ(t − τ ) + Dy(t) + D1y(t − τ )

+D2y(t − 2τ ) + Eu(t) + E1u(t − τ ),ˆ

z(t) = ζ(t) + F y(t) + Gy(t − τ ),where ζ(θ) = ρ(θ), ∀ θ ∈ [−τ, 0], is the initial conditional function, ζ(t) ∈ Rris the vector to be evaluated,ˆ

z ∈ Rris the estimated vector of z(t), N , Nd, D, D1, D2, E, E1, F , and G are dimensionalally appropriateconstant matrices to be determined

Note that the above results only apply to systems where only a time delay occurs in the statevariable Many systems have a lot of time delays in the state variable and the output These systems can

be described as below:

˙x(t) = A(d)x(t),y(t) = C(d)x(t),x(t) ∈ Rn

and y(t) ∈ Rm are the state vector and the output vector, respectively A(d) and C(d) arethe polynomial matrices of the time delay operator d = {di} = {d1, d2, , dq} (q delay) For each i, the

di operator is defined by dix(t) = x(t − τi) where τi > 0 is the time delay The higher order operator issimilarly defined For example d2idj is defined as d2idjx(t) = x(t − 2τi− τj) Hou et al (2002) proposed acoordinate transformation z(t) = T (d)x(t), z(t) ∈ Rnz, where T (d) is a polynomial of the operator d totransform the class under consideration to the following form:

˙z(t) = Az(t) + ¯¯ F (d)y(t),y(t) = Cz(t),¯

where ( ¯A, ¯C) is the observability of the matrix pair (i.e., the pair matrix ( ¯A, ¯C) is called completely

¯

C ¯A

is of rank n) With this new structure, it is easy to design a

Luenberger-type observer to estimate full order and reduce order as well as a linear function of the state variable.However, the method proposed in Hou et al (2002) is an operator method; the conditions for the existence

of such a state transformation are very tight To overcome this limitation, Huong et al (2015, 2016)(see [42], [43]) proposed a new method to transform the time-delay systems where the time delays are in

the state vector to the ones where time delays in the state variable are moved to the input and outputinformation Therefore, in the new coordinate system, a Luenberger-type state observer can be readilydesigned Subsequently, of the three possible versions of the original state vector, namely, instantaneous,delayed, and a mixed of instantaneous and delayed, a state observer which estimates one of these versionscan be obtained This new finding allows the authors to design state observers for a wider class oftime-delay systems.

Today’s requirements for engineering systems (reducing the complexity of operating equipment,lowering control levels, resisting abnormal failures, operating multiple systems, increasing the stability

of the system, etc.) are getting higher and higher Therefore, solving state estimation problems, errordetection problems, and error estimation problems for dynamic systems ar not only meaningful in theorybut also offer many potential practical applications with high efficiency and low cost (see [57], [10]).Note that most state observation design methods for time-delay systems require the observable matrixpair (A, C) Therefore, if the matrix pair (A, C) is not observable, then state observation-based errordetection schemes are not applicable To overcome this difficulty, Huong et al (2014) (see [48]) proposed

a new method to design an observer to detect faults of the following time-delay systems where the pair(A, C) is not required to be observable

˙x(t) = Ax(t) + Adx(t − τ ) + Bu(t) + Df (t), t ≥ 0,x(θ) = φ(θ), θ ∈ [−τ, 0],

y(t) = Cx(t),where φ(θ) is the initial value function, τ > 0 is the known delay, x(t) ∈ Rn

is the state vector, u(t) ∈ Rm

is the input vector, y(t) ∈ Rp is the output vector, f (t) ∈ R is the unpredictable error signal, A, Ad,

B, D and C are dimensionally appropriate constant matrices For this, they transformed the consideredsystems into the following form

˙z(t) = Az(t) + ¯¯ Bu(t) + ¯Bdu(t − τ ) + ¯Df (t) + ¯Ddf (t − τ )

+Γy(t) + Γdy(t − τ ) + Ψdy(t − 2τ ), t > τ,y(t) = Cz(t),¯

where the matrix pair ( ¯A, ¯C) is observable In the new coordinate system, the authors designed a functionalobserver with order q > 1 (q is a natural number) to detect the fault f (t) This method is significantsince it has contributed to reducing the cost and complexity of the system operation

As the above discussion, there is a wide range of techniques for the state estimation problem inthe literature However, the existing observer design techniques may not be suitable for systems withmany types of uncertainties [18], [28], [29], [33], [39], [40], [55] In this situation, the problem of pointwiseestimation can be substituted by the interval one, then using input-output measurements, an observerhas to estimate the set of admissible values for the state at each instant of time

On the other hand, since the upper and lower estimate of the state interval are generated by

an observer, the interval observers have an enlarged dimension with respect to the system dimension;see, for example, [63] Therefore it is essential to consider the problem of designing functional intervalobservers for dynamical systems, i.e problem of designing only interval observers for some functions of

the state vector and therefore reduce the interval observer dimension To solve this problem, in [9], theauthors considered the design problem of the functional interval observer for discrete-time systems withdisturbances, while, the authors of the work [29] proposed a Luenberger-like functional interval observerfor the linear systems with time varying disturbances In [29], the sufficient conditions for the existence

of the functional interval observer were introduced and the parametric design method of the intervalobserver for the linear functional of the states in the systems was presented by employing the solution ofthe Sylvester equations The results of the work [29] was extended to a class of non-linear fractional-orderinterconnected systems with bounded uncertainties [46]

Although the problem of designing state observers and interval state observers has been intensivelystudied in the literature, there are still many problems (for example, the fault detection problem ofnonlinear time-delay systems and time-delay interconnected systems; the design of distributed linearfunctional interval observers for large-scale interconnected impulsive systems) that have not been solvedand they are currently topical challenges in the research direction of control theory and estimation theory.The main purpose of this thesis is to research and propose new methods to design state observersand interval observers to estimate the state vectors of some classes of dynamical systems and to find theirapplications

The thesis has four chapters and a list of references

Chapter 1 provides some necessary mathematical knowledge used in the thesis Namely, Section1.1 presents some basic concepts about Lyapunov functions, stabilization problems, and some auxiliarylemmas, while Section 1.2 introduces in detail the problem of functional state observers

Chapter 2, presents a novel procedure for designing a state observer of a class of nonlinear delay Glucose-Insulin (GI) systems A general GI model with two time delays is considered Based onthe positivity of its solutions, we utilize the concept of diffeomorphism on the output and derive a newstate transformation to transform the model into a new observable form In this new form, the nonlineartime-delay term in the system description is associated with the output and input only As a result, astate observer can be easily established Simulation results are given to illustrate the design procedure.Chapter 3 studies the design of distributed functional observers to detect actuator faults of inter-connected systems with time delays and unknown inputs First, we extend the state transformations oftime-delay systems to the general structure in order to deal with time-delay interconnected systems withunknown inputs Second, we provide existence conditions of the state transformation Third, by using theobtained state transformations, we transform each subsystem of the time-delay interconnected systemsinto an observable canonical form Fourth, in the new coordinate system, we design a functional observer

time-to construct a residual function that can trigger the i-th subsystem faults Finally, a numerical exampleand simulation results are provided to demonstrate the theoretical results

Chapter 4 considers the problem of designing distributed functional interval observers for a class

of large-scale networks impulsive systems with bounded uncertainties Given bounds of uncertainties inthe state and the output, we first design novel interval observers for linear functions of the state vector

of each system of the considered system We then provide conditions for the existence of such intervalobservers and an effective algorithm for computing unknown observer matrices Finally, two examples

and simulation results are given to illustrate the effectiveness of the proposed design method In contrast

to existing interval observers in the literature, each functional interval observer proposed in this chapterconsists of two bounds (upper and lower bound) which are impulsive dynamical systems Therefore theycan deal with interconnected impulsive dynamical systems

The thesis is written based on the results of the papers [52,53,83] The results of this thesis havebeen presented at:

1 The 20th Workshop on “Optimization and Scientific Computing”, Institute of Mathematics,Vietnam Academy of Science and Technology (April 21–23, 2022, Ba Vi, Hanoi)

2 Seminar at the Department of Mathematics and Statistics, Quy Nhon University

3 The forth Mathematical Conference of Central and Highland of Vietnam, Hue University, August2022

Chapter 1

Preliminaries

This chapter provides some necessary mathematical knowledge to be used in the thesis Namely,Section 1.1 presents some basic concepts about Lyapunov functions, stabilization problems, and someauxiliary lemmas, while Section 1.2 introduces in detail the problem of functional state observers

1.1.1 Stability criteria of some classes of dynamic systems

Based on the known results of Lyapunov stability, people have researched, developed, and applied

to solve the problem of stabilization of control systems Stability is one of the important properties ofthe qualitative theory of dynamical systems Figuratively speaking, a system is said to be stable at someequilibrium if small disturbances of the data or the initial structure of the system do not cause the system

to change much from its state balance

Consider a system described by the differential equation







˙x(t) = f (t, x), t> t0,x(t0) = x0,

(1.1)

where x(t) ∈ Rn

is the state vector of the system, f : R+

× Rn

→ Rnis the given vector function Suppose

f (t, x) is a function satisfying the conditions such that the solution of the Cauchy problem of system (1.1)always has a unique solution Then the integral form of the solution is given by the formula

x(t) = x0+

Z t t0

f (s, x(s))ds (1.2)Definition 1.1.1 ([73], Definition 3.1) The solution x(t) of the system (1.1) is said to be stable if forevery number ε > 0, t0 > 0 there will be a number δ > 0 (depends on ε, t0) such that any solutiony(t), y(t0) = y0 of the system such that ky0− x0k < δ will be true any equality

ky(t) − x(t)k < ε, ∀t > t0 (1.3)

Definition 1.1.2 ([73], Definition 3.2) The solution x(t) of the system (1.1) is said to be asymptoticallystable if it is stable and there is some δ > 0 such that for ky0− x0k < δ then

Definition 1.1.3 ([73], Definition 3.3) The system (1.1) is exponentially stable if there exist numbers

M > 0, δ > 0 such that every solution of system (1.1) with x(t0) = x0 satisfies

kx(t)k 6 Me−δ(t−t0)kx0k, ∀t > t0, (1.6)that is, the zero solution of a system that is not only asymptotically stable, but also all solutions of (1.1)approach zero rapidly at an exponential rate

Stabilizing Linear Systems

Consider the linear system

˙x(t) = Ax(t), t> t0, (1.7)where A is a n × n- matrix The solution of the system (1.7) comes from the initial state x(t0) = x0givenby

x(t) = x0eA(t−t0), t > t0.The following theorem gives a first criterion for the stability of the system (1.7), often called the Lyapunovalgebraic stability criterion

Theorem 1.1.1 ([73], Theorem 3.1) The system (1.7) is exponentially stable if and only if the real part

of all eigenvalues of A is negative, i.e

<(λ) < 0, ∀λ ∈ λ(A)

The following will introduce another Routh-Hurwitz method to determine the stability of the system

in more convenient cases

Theorem 1.1.2 ([73], Theorem 3.2) Suppose the characteristic polynomial of the given differential tion (1.7) is

equa-f (z) = zn+ a1zn−1+ · · · + an.Then if the determinant of all submatrices Dk, k = 1, 2, , n, is positive, then the real part of allsolutions of f (z) is negative, i.e the system is asymptotically stable, where

det D1 = a1, det D2= det a1 a3

1 a2

!,

where X, Y are the (n × n)- dimensional matrices and are called the pairs of solutions of (1.8)

Considering the system (1.7), from now on, we will refer to the stable of matrix A if real part ofall eigenvalues of A a negative

Theorem 1.1.3 ([73], Theorem 3.3) The matrix A is stable if and only if, for any positive definitesymmetric matrix Y , the equation (1.8) has a solution that is a symmetric, positive definite matrix X

Lyapunov function method

Consider a system of stationary nonlinear equations

˙

x = f (x), f (0) = 0, t ∈ R+ (1.9)The function V : Rn

→ R is positive definite if

i) V (x) > 0 for all x ∈ Rn,

ii) V (x) = 0 if and only if x = 0

Definition 1.1.4 ([73], Definition 3.5) A function V : D → R, where D ⊆ Rn is an arbitrary openneighborhood of 0, called the Lyapunov function of the system (1.9) if

i) V (x) is a continuously differentiable function on D,

ii) V (x) is a positive definite function,

iii) DfV (x) := ∂V∂xf (x) 6 0, ∀x ∈ D

A function V (x) is called a tight Lyapunov function if the inequality iii) is replaced by

iv) ∃c > 0 : DfV (x) 6 −ckxk < 0, ∀x ∈ D \ {0}

Theorem 1.1.4 ([73], Theorem 3.15) If the system (1.9) has a Lyapunov function, then it is stable.Furthermore, if the Lyapunov function is tight, then the system is uniformly asymptotically stable

Stabilizing impulsive system

Consider the following system

˙x(t) = Ax(t) + f (x(t), u(t)), t ∈ [t0, ∞) \ T, (1.10)x(t) = g(x−(t), u−(t)), t ∈ T, (1.11)where x(t) ∈ X, u(t) ∈ U , A is an infinitesimal generator of a C0-semigroup on X, f, g : X × U → X,

T = {t1, t2, t3, , } is a strictly increasing sequence of impulse times

Definition 1.1.5 ([15]) For a given sequence T of impulse times, system (1.10)-(1.11) is called locallyinput-to-state stable (LISS) if there exist ρ > 0 and β ∈ KL, γ ∈ K∞, such that ∀x ∈ X, ||x||X 6 ρ ∀u ∈

Uc, ||u||Uc 6 ρ ∀t > t0 it holds that

||φ(t, t0, x, u)||X 6 β(||x||X, t − t0) + γ(||u||Uc), (1.12)where φ(t, t0, x, u) is the state of (1.10)-(1.11) corresponding to the initial value x ∈ X, the initial time

t0, and to the input u ∈ Uc at time t > t0

System (1.10)-(1.11) is input-to-state stable (ISS) if (1.12) holds ∀x ∈ X, u ∈ Uc

Definition 1.1.6 ([62]) A square real matrix M is called a Metzler matrix if its off-diagonal elementsare nonnegative, i.e mij> 0, i 6= j

Lemma 1.1.1 ([23]) Any solution of the linear system

˙x(t) = Ax(t) + Bϕ(t), ϕ : R+→ Rq+,y(t) = Cx(t) + Dϕ(t),

with x(t) ∈ Rn

, y(t) ∈ Rp

and a Metzler matrix A ∈ Rn×n

, is elementwise nonnegative for all t > 0provided that x(0) > 0 and B ∈ Rn×q+

Lemma 1.1.2 ([36]) Any solution of the system

xt+1 = Axt+ Bϕt, ϕ : Z+→ Rm

+,with xt ∈ Rn

and a nonnegative matrices A ∈ Rn×n+ and B ∈ Rn×m+ , is elementwise nonnegative for all

t ∈ Z+ provided that x(0) > 0

Consider an impulsive linear system with external inputs

˙x(t) = Ax(t) + f (t), t ∈ [ti, ti+1), i ∈ Z+, (1.13)

x(ti+1) = Gx(¯ti+1) + g(ti+1), ∀i > 1, (1.14)where x(t) ∈ Rn is the state vector, A, G ∈ Rn×n : R+ → Rn×n, d ∈ L∞ is the input for t ∈ [ti, ti+1);

g : R+→ Rn, g ∈ C1∩ L∞is the input at time instants ti+1 for all i > 1 The sequence of impulse events

ti is assumed to be positively incremental, i.e θi= ti+1− ti> 0 and t0= 0

Lemma 1.1.3 ([17]) Consider system (1.13)-(1.14) with a ranged dwell-time

θi∈ [Tmin, Tmax], ∀i ∈ Z+,where 0 < Tmin6 Tmax< ∞ are given constants If there exist matrices P ∈ Sn

>0 and Q ∈ Sn

>0 such thatfor all θ ∈ [Tmin, Tmax]:

GTeATθP eAθG − P = −Q, (1.15)then (1.13)-(1.14) is ISS and the following asymptotic gain is guaranteed

lim

t→∞|x(t)| 6 [ρP,Q,W||g||∞+ Tmax(1 + ρP,Q,W|G|ρ(A))||f ||∞]%(A),where ρP,Q,W =

qλmax(W ) λmin(P )

for µ(A) = maxi=1,nλ(A+A2 T) being a logarithmic norm of the matrix A

Proof From the system equations we can obtain for all i ∈ Z+:

x(t) = eA(t−ti)x(ti) +

Z t ti

eA(t−s)f (s)ds ∀t ∈ [tt, ti+1)

and

x(ti+1) = GeATix(ti) + r(ti) ∀i > 1,where r(ti) = GRti+1

ti eA(ti+1−sf (s)ds + g(ti+1) Next,

|eA(t−ti)x(ti)| 6 |eA(t−ti )||x(ti)| 6 eµ(A)(t−ti )|x(ti)|

Let us denote φ = |GRti+1

ti eA(t−sf (s)ds| One can write

eµ(A)(ti+1−s)|f (s)|ds

6 Ti|G||%(A)|kf k∞

|r(ti)| 6 Ti|G||%(A)|kf k∞+ kgk∞and

krk∞6 Tmax|G||ρ(A)|kf k∞+ kgk∞.Consider a Lyapunov function ν(x) = xT

P x, where P ∈ Rn×n is given in LMI (1.15), thenν(x(ti+1)) − ν(x(ti)) = xT(ti+1)P x(ti+1) − xT(ti)P x(ti)

= xT(ti)[eATTiGTP GeATi− P ]x(ti)+2rT(ti)P GeATix(ti) + rT(ti)P r(ti)

6 −0.5xT(ti)Qx(ti) + rT(ti)[P+2P GeATiQeATTiGTP ]r(ti)

6 −2λλmin(Q))

max(P )V (x(ti)) + r

T(ti)W r(ti)

From this expression we obtain

λmin(P )|x(ti+1)|26 (λmax(P ) −1

On any interval [ti, ti+1)the following estimate is satisfied

|x(t)| 6 eµ(A)(t−t)|x(ti)| + Tmax%(A)kf k∞

6 [|x(ti)| + Tmaxkf k∞]%(A)for all t ∈ [ti, ti+1) then asymptotically the trajectories of (1.15) enter into the ball

lim

t→∞|x(t)| 6 [ρP,Q,W||g||∞+ Tmax(1 + ρP,Q,W|G|ρ(A))||f ||∞]%(A)

The proof is completed

(ii) Assuming N ∈ R is a positive definite symmetric matrix, we have:

±2xT

y 6 xTN x + yTN−1y, ∀x, y ∈ Rn.Lemma 1.1.5 (Schur’s Lemma) Given the matrix X, Y, Z, where Y = YT > 0, X = XT, we have

1.2.1 Full order state observer

As the name suggests, the full order state observer estimates all the states of the system Luenberger(see [60]) defined this observer and introduced the conditions which must be satisfied for the observer toexist It follows that a system with n state variables requires an observer of order n for full state estimation.Consider the following linear time-invariant system

˙x(t) = Ax(t) + Bu(t), t > 0, (1.16)

where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the control input vector, y(t) ∈ Rp is the measurementoutput vector, matrices A ∈ Rn×n

, B ∈ Rn×m

and C ∈ Rp×n are constant and known

For this system, it is now assumed that all state variables contain in x(t) cannot be measureddirectly and that an observer to reconstruct the state variables must be formed To reconstruct all thestate variables, ˆx(t) is introduced (where ˆx(t) denotes an approximation of the state x(t) as opposed tothe actual state) Define an observer with the following dynamics

˙ˆx(t) = Aˆx(t) + Bu(t) + L(y(t) − ˆy(t)), t > 0, (1.18)ˆ

where L ∈ Rn×p is the observer gain matrix, ˆx(t) ∈ Rn is an estimate of the state x(t)

By denoting error e(t) as

e(t) = x(t) − ˆx(t), t > 0, (1.20)then the error dynamic system is obtained as follows

˙e(t) = (A − LC) e(t), t > 0 (1.21)The solution of the differential equation has the form

e(t) = e(A−LC)te(0) (1.22)This important result determines the existence of a full order state observer It emphasizes that if theoriginal system is unobservable, it is not possible to construct a full order state observer

Figure 1.1: Responses of x(t) and its estimation

Remark 1.2.1 Combining the controller and observer, we obtain

ˆ

˙ˆx(t) = Aˆx(t) + Bu(t) + L(y(t) − ˆy(t)) (1.24)

and a combination system of x(t) and ˆx(t) can be expressed as below:

"

˙x(t)

#

1.2.2 Reduced order state observer

The full order state observer (1.18) is of order n, being equal to the number of state variables inthe original system (1.16) Although simple both conceptually and in its construction, there are someinherent redundancies in its design Recall our objective of reconstructing the state vector x(t) ∈ Rn andthat the system output contains p linear combinations of the state variables Intuitively, the remaining(n − p) state variables should be reconstructed by an observer of order (n − p)

Let us consider the following state transformation

"

A11 A12

A21 A22

#, P−1B =

"

B1

B2

#, then thesystem (1.16)-(1.17) can be written into the following form

"

B1

B2

#u(t), t > 0, (1.29)w(0) = P−1x0∈ Rn, (1.30)y(t) = h Ip 0n−p

˙

wu(t) = A22wu(t) + A21wp(t) + B2u(t), (1.32)

¯y(t) = A12wu(t) = ˙wp(t) − A11wp(t) − B1u(t)

= ˙y(t) − A11y(t) − B1u(t) (1.33)

Let us now design a Luenberger-type state observer for system (1.32)-(1.33) as follows:

˙ˆ

wu(t) = A22wˆu(t) + A21wp(t) + B2u(t)

+L(¯y(t) − A12wˆu(t)), (1.34)where L ∈ R(n−p)×pis a gain matrix needed to be determined such that the error vector eu(t) = wu(t) −ˆ

wu(t) converges asymptotically to zero

1.2.3 Linear functional state observer

Linear functional observers take advantage of a recurring theme in state feedback control That is,frequently in feedback control applications, only a linear combination of the state variables, i.e., Kx(t), isrequired, rather than complete knowledge of the entire state vector x(t) In the previous subsections, thefocus was on either the reconstruction of the whole state vector or (n − p) unmeasurable state variables,

highlighting a redundant feature The question therefore, arises as to whether a less complex observercan be constructed to estimate a linear combination of some of the unmeasurable state variables This ispossible and this subsection is concerned with the theory and construction of linear functional observers.The primary aim is to produce observers that are of further reduced order, simpler structure, and stable.The main result for this problem was first presented by Luenberger (see [61]) Consider a system

(1.35)

where x(t) ∈ Rn

is the state vector, u(t) ∈ Rm

is the control vector, y(t) ∈ Rpis the measurement outputvector, p 6 n; A, B, C are constant matrices of appropriate dimensions

Without loss of generality, we assume that the matrix C has a full degree (i.e rank(C)=p) and thatthe system is observable Let z(t) ∈ Rr be a vector that is required to be reconstructed (or estimated),where

and F ∈ Rr×n is the known matrix Assume that rank(F ) = r and rank

"

FC

#

= r + p To reconstructthe state function, z(t) = F x(t), F ∈ Rr×n

(1 6 r 6 n − p) is known, we consider a functional observer

of order q as follows:

ˆz(t) = Dω(t) + Ey(t), (1.37)

˙ω(t) = N ω(t) + J y(t) + LBu(t), (1.38)where ω(t) ∈ Rq, ˆz(t) ∈ Rq is the estimate of z(t), D, E, N , J and L are observer parameters to bedetermined.The error e(t) = ˆz(t) − z(t) need to converge asymptotically to zero as t → ∞

In (1.37), the output ˆz(t) provides an asymptotic estimate of F x(t) if

lim

t→∞[ˆz(t) − F x(t)] = 0 (1.39)

It stands to reason that if ˆz(t) estimates F x(t), then w(t) defined in (1.38) estimates other linearcombination of x(t), let it be called Lx(t) This gives rise to the following theorem [80] for the existence

of the proposed linear functional observer (1.37), (1.38)

Theorem 1.2.1 ([80, Theorem 3.1]) Estimation ˆz(t) converges asymptotically to F x(t) for any initialcondition w(0) and any u(t) if and only if the following conditions hold

N L + J C − LA = 0, (1.41)

F − DL − EC = 0 (1.43)Remark 1.2.2 Consider a closed-loop feedback system with a control input u(t) = F x(t), which isestimated by the observer (1.37), (1.38), ˙x(t) and ˙e(t) can be derived as follows

˙x(t) = Ax(t) + B ˆu(t) = Ax(t) + B(Dw(t) + Ey(t))

Remark 1.2.3 We now present a method for solving observer matrices D, E, N, J, L and H From(1.42), H is obtained from H = LB so the problem of reconstructing the state function revolves aroundthe design of the observed parameters D, E, J, L, N to satisfy the condition (1.40), (1.41), (1.43) ofthe Theorem1.2.1 Before proceeding to determine the observed parameters, we simplify both equations(1.41), (1.43) This can be done as follows Define ˜F ∈ Rr×n, ˜L ∈ Rq×nand partition them as follows

where P = [C‡, C⊥] ∈ Rn×n , C‡ ∈ Rn×p is the Moore-Penrose inverse of C, (i.e, CC‡ = Ip) and

C⊥ ∈ Rn×(n−p) are denoted by the orthogonal basis of the zero space of C (i.e., CC⊥ = 0) From thedefinition of P above, it is clear that CP = [Ip, 0] , A = P−1AP =

"

A11 A12

A21 A22

#, then (1.41) and (1.43)are shortened as follows

J = L1A11+ L2A21− N L1, (1.50)

E = F1− DL1, (1.51)

N L2− L1A12− L2A22= 0, (1.52)

F2= DL2 (1.53)

L1, L2, N (Hurwitz) and D are solved from a pair of matrix equations (1.52) and (1.53) After finding

L1, L2, N and D, from (1.50) and (1.51) we get J and E

1.2.4 Interval observer

Consider the following dynamical system:

˙x(t) = f (t, x(t)), x(0) = x0, x ∈ X, (1.54)

x−(t) 6 x(t) 6 x+(t), t > 0for any initial condition x(0) ∈ X0.

Time(sec) -4

Chapter 2

A new method for designing

observers of a nonlinear time-delay Glucose-Insulin system

Diabetes is a world-wide epidemic In the treatment of diabetes, it is essential to monitor glucoseand insulin levels in diabetic patients so that appropriate treatment such as insulin injections can beimplemented to maintain satisfactory blood glucose levels Blood glucose levels can be readily measured

by using a glucose-oxidase-based amperometric sensor The sensor utilizes glucose in interstitial fluidunder the skin to indirectly reflect the blood sugar level Whereas, insulin measurements are slower,harder to obtain and less accurate than glucose measurements Thus, model-based state observers havebeen proposed in order to estimate insulin levels [26], [69] The contribution of this chapter is in thedesign of a novel state observer to estimate insulin levels in diabetic patients

In this chapter, we consider a general nonlinear time-delay GI model of the following form

˙x(t) = f (x(t), x(t − τ )) + Bu(t) + g(y(t), y(t − τg)), t > 0, (2.1)x(θ) = φ(θ), θ ∈ [−τmax, 0], τmax= max{τ, τg}, (2.2)







, C = [1 0 0 0],

ai (i = 1, 2, , 7) are positive parameters, φ(θ) = h φ1(θ) φ2(θ) φ3(θ) φ4(θ) i

T

is a continuousinitial function, φi(θ) (i = 1, 2, 3, 4) are positive functions, lim

VI



x1(t−τg ) G∗

γ 1+



x1(t−τg ) G∗

γ

00

as there are some difficulties in dealing with the nonlinear delayed term x1(t)x2(t − τ ) in the model

In this section, we present a novel procedure for designing a state observer of the nonlinear delay model (2.1)-(2.3) In our design procedure, we propose a two-stage process to transform (2.1)-(2.3)into a new observable form where the nonlinear term x1(t)x2(t − τ ) is injected into the output and input ofthe system To achieve this, we first utilize the concept of diffeomorphism on the output [3] by defining anew output ¯y = − ln(y(t)) for the system (2.1)-(2.3) To ensure such a diffeomorphism can take place, weshow that x1(t) > 0 for all t > 0 for the model (2.1)-(2.3) (i.e., in order for ln(y(t)) to exist, it is necessarythat y(t) > 0, ∀t > 0) In the second stage of the process, we introduce a novel state transformation totransform the system into a novel observable form where a state observer can be easily designed.First, let us prove that x1(t) > 0 for all t > 0 Indeed, note that x1(0) > 0 According to thecontinuous solution of a diffrenetial equation, x1(t) would become non-positive if there existed a t0 > 0

time-such that x1(t0) = 0 for any 0 6 t0< t Then, necessarily ˙x1(t0) 6 0, which is a contradiction since wehave

˙

x1(t0) = −a1x1(t0)x2(t0− τ ) + g1(y(t0), y(t0− τg))

= g1(0, y(t0− τg)) > 0 (2.4)Hence, we can conclude that x1(t) > 0 for all t > 0 With this fact, we can now utilize the concept ofdiffeomorphism on the output [3] For this, let us divide both sides of the first equation of (2.1) by −x1(t)and let a new output be defined as ¯y(t) = ξ(t) = − ln(y(t)) Then (2.1)-(2.3) is equivalent to the following

˙˜

x(t) = A˜x(t) + Adx(t − τ ) + Bu(t) + ¯˜ µ(¯y(t), ¯y(t − τg)), t > 0, (2.5)

˜x(θ) = φ(θ), θ ∈ [−τ˜ max, 0], (2.6)

is due to the fact that the matrix pair (A, C) is not observable [38] as well as there are some fixed poles inthe observer error dynamics [37], [80] These stable fixed poles are very close to zeros and thus resulted

in a very slow convergent rate for the designed state observers Recognizing this difficulty, in chapter,

we present a new type of state observer, and referred to it as a “delayed” state observer In this regard,the designed state observer will be able to estimate a delayed version of the state vector instead of theinstantaneous state vector, which is impossible based on existing observer design methods [38], [80].Accordingly, in the following, we consider the general form of system (2.5)-(2.7), where ˜x =h

For m, n ∈ N, n > 1 and an arbitrary matrix M ∈ R1×n, MT denotes the transpose of M , 0m,n

denotes the m × n zero matrix, M = h [M ]L [M ]R

i, where [M ]L ∈ R and [M]R ∈ R1×(n−1) aresub-matrices of M

We define a new change of coordinates as follows

˜x(t − τ )

Theorem 2.1.1 For some scalars γi (i = 2, 3, , n), αj and βj (j = 1, 2, , n − 1), if the followingequations hold

[NiAd]R= 0, i = 2, 3, , n, (2.12)[MnA −

+Γ2y(t − 2τ ) + Γ¯ 3µ(¯¯ y(t), ¯y(t − τg)),+Γ4µ(¯¯ y(t − τ ), ¯y(t − τ − τg)), t ≥ τ, (2.15)

.0

N2Ad i

L

.h

Proof For i = 1, 2, , n − 1, by taking the derivatives of (2.8) and using (2.10)-(2.12), we obtain

= zi+1(t) + MiBu(t) + NiBu(t − τ )+αiy(t) + β¯ iy(t − τ ) + [N¯ iAd]Ly(t − 2τ )¯+Miµ(¯¯ y(t), ¯y(t − τg)) + Niµ(¯¯ y(t − τ ), ¯y(t − τ − τg)) (2.17)

Next, for i = n from (2.13)-(2.14), we have

˙

zn(t) = MnA˜x(t) + (MnAd+ NnA)˜x(t − τ ) + MnBu(t)

+ NnBu(t − τ ) + [NnAd]Ly(t − 2τ )¯+Mnµ(¯¯ y(t), ¯y(t − τg)) + Nnµ(¯¯ y(t − τ ), ¯y(t − τ − τg))

+Mnµ(¯¯ y(t), ¯y(t − τg)) + Nnµ(¯¯ y(t − τ ), ¯y(t − τ − τg)) (2.18)

Finally, note that N1 = 0, therefore (2.17) and (2.18) can now be expressed in the form (2.15(2.16) This completes the proof of Theorem 2.1.1

)-An algorithm for solving unknown parameters according to Theorem 2.1.1

In the following development, we will provide a procedure for solving the unknowns γi (i =

2, 3, , n), αj and βj (j = 1, 2, , n − 1) as defined in Theorem 2.1.1 Let us denote the followingrecursive matrices

X1i = [M1Ai]R, ¯X1i = [M1Ai−1Ad]R, (2.19)

χn=h χ1

n χ2 n

i,

Xn=

"

X1 n

X2 n

#, Yn=h Y1 Y2 Y3 Y1n−1 i,

χ2n =h α1 α2 αn−2

i,

From (2.22), a solution for χn exists if and only if

Next, to determine the remaining unknowns αn−1 and γj (j = 2, 3, , n), let us look at thesolvability of equations (2.13)-(2.14) Substituting (2.10)-(2.11) into (2.13)-(2.14), using (2.19)-(2.20) andafter some rearranging, we obtain the following equation expressed in a compact vector-matrix form

ζnZn= Tn, (2.24)where

Step 1: Obtain matrices Xn and Yn according to (2.22) Check if condition (2.23) is satisfied or not If

so, obtain χn where χn= YnX+

Remark 2.1.1 Once a transformed system as described by (2.15)-(2.16), we can easily apply anyLuenberger-typed state observers design method (see, for example, [80]) to design a state observer toestimate z(t) since the matrix pair ( ¯A, ¯C) is now observable After a satisfactory state observer z(t) hasbeen designed, we can use the method of backward state transformations reported in [47].

2.2.1 State transformation for the GI model

In this section, we will apply the results obtained in the previous section to the GI model (2.5)-(2.7)

By following the steps (Step 1-Step 3) of Algorithm 2.1, we obtain

M1=h 1 0 0 0

i, N1= 01,4,

M2= 01,4, N2=h 0 a1 0 0

i,

M3= 01,4, N3=h 0 −a1a2 0 a1a3

i,

M4=h a2 a4a6

2 0 0 0 i, N4=h 0 a1a2 a1a3a5 −a1a3(a2+ a4) i.Hence, we obtain the following state transformations

−a2 a4a62 a2a4a6(a2+a4+a6) 2

2.2.2 State observer design for the GI model

Since the matrix pair ( ¯A, ¯C) is observable, it is easy to design a state observer to estimate anylinear function of the state vector z(t) Let

˙ω(t) = N ω(t) + J ¯y(t) + Hu(t − τ )

+L¯µ(¯y(t), ¯y(t − τ ), ¯y(t − τ − τg)), t > τmax, (2.34)where ω(t) ∈ R3, ˆh(t) ∈ R3 is the estimate of h(t), E, N , J , H and L are observer parameters to bedetermined Let us define the following error vectors (t) and e(t) as

(t) = ω(t) − Lz(t), (2.35)e(t) = h(t) − F z(t).ˆ (2.36)

Based on [80, Theorem 3.1], ˆh(t) converges asymptotically to F z(t) if the following conditions aresatisfied

N L + J ¯C − L ¯A = 0, (2.38)

H − L ¯B = 0, (2.39)

F − E ¯C − L = 0 (2.40)Accordingly, for the given matrices ¯A, ¯C, and ¯B as above, we can easily solve (2.37)-(2.40) to obtainthe following matrices: N = ¯A22+ L1A¯12, L1 is chosen such that N is Hurwitz, E = −L1, J = −N L1,

H = L ¯B, where L =h L1 L2

i, ¯A12=h 1 0 0 i, ¯A22 =

Upon ˆh(t) has been obtained, then based the method of backward state transformations (Case 2)reported in [47], we obtain

ˆ

x4(t − τ ) = 1

a1a3[a2ˆ

h1(t) + ˆh2(t)] (2.43)

1+



x1(t−τg ) 9

u(t) =

(sin t + 50, 0 6 t 6 100,sin t + 3, 100 < t 6 180

Let us now apply the reduced-order state observer (2.33)-(2.34) for this example The eigenvalues

of matrix N are chosen as, say, λ1 = −0.05, λ2 = −0.07, λ3 = −0.08, hence by using LMI Toolbox

 Figure 2.1 shows the responses of x2(t) and its delayed-estimation, i.e., ˆx2(t − 3),

while, Figure 2.2 shows the responses of x2(t − 3) and its estimation, i.e., ˆx2(t − 3) It is clear from Figure2.2 that the designed observer able to track the delayed version of the state vector, as expected

0 20 40 60 80 100 120 140 160 180 0

x 2 (t − 3)

Figure 2.1: Responses of x2(t) and ˆx2(t − 3)

0 20 40 60 80 100 120 140 160 180 0

Conclusion of Chapter 2 In this chapter, we have proposed a novel procedure for designing a stateobserver of a general nonlinear time-delay Glucose-Insulin model The reported result is significant as thetwo-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 illustratethe effectiveness of our results The main advantage of the proposed method in this chapter is that thestate transformations can be used to transform nonlinear time-delay systems into new coordinates whereall the time-delay terms in the system description are associated with the output and input only Therefore,

in the new coordinate system, a Luenberger-type state observer can be readily designed However, theadvantage of this method is that there are some cases instead of an instantaneous state observer; we onlyobtain a delayed or a mixed instantaneous and delayed state observer of nonlinear time-delay systems

Chapter 3

A new observer for interconnected

time-delay systems and its

applications to fault detection

problem

Faults of control systems may very often lead to a drastic reduction of system performance or aloss of stability, which even cause damages to the physical system [8], [56], [74] Therefore, increasingattention has recently been devoted to the problem of fault detection of control systems (see, for example,[2], [10], [25], [58], [32], [19], [20])

To date and to the best of knowledge, the problem of designing functional observers to detectactuator faults of interconnected time-delay systems has not yet received adequate attention In [78], theauthors presented an approach to the design of fault detection scheme to detect actuator faults for a class

of interconnected time-delay systems consisting of N subsystems with unknown input vector di(t) ∈ Rnid

and actuator fault vector fi(t) ∈ Rnif entering from the i-th subsystem input defined as follows:

Di∈ Rn i×nid, Ei∈ Rni×nif and Ci∈ Rp i×ni are real known system matrices Without loss of generality,