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.
Some basic concepts
Stability criteria of some classes of dynamic systems
Research and development based on Lyapunov stability principles have been instrumental in addressing the stabilization of control systems Stability is a crucial aspect of the qualitative theory of dynamical systems In essence, a system is considered stable at a given equilibrium if minor disturbances in the data or initial conditions do not significantly alter its balanced state.
Consider a system described by the differential equation
In the context of the Cauchy problem for a system defined by the state vector \( x(t) \in \mathbb{R}^n \) and a vector function \( f: \mathbb{R}^+ \times \mathbb{R}^n \to \mathbb{R}^n \), it is assumed that \( f(t, x) \) meets specific conditions that guarantee a unique solution exists The integral form of the solution can be expressed as \( x(t) = x_0 + \int_{t_0}^{t} f(s, x(s)) ds \).
Definition 1.1.1 ([73], Definition 3.1) The solutionx(t) of the system (1.1) is said to be stable if for every number ε > 0, t0 ⩾ 0 there will be a number δ > 0 (depends on ε, t0) such that any solution y(t), y(t0) =y0 of the system such thatky0−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 solutionx(t) of the system (1.1) is said to be asymptotically stable if it is stable and there is someδ >0 such that forky0−x0k< δ then t→∞lim ky(t)−x(t)k= 0 (1.4)
To be concise, from now on, we will refer to system (1.1) as asymptotically stable instead of the solution of system (1.1) being asymptotically stable
If δ >0 in the above definitions does not depend on timet >0, then the stability (or asymptotic stability) is said to be uniformly stable (or uniformly asymptotically stable).
Remark 1.1.1 By the transformation (x−y)7→z, (t−t0)7→τ the system (1.1) will be returned to the form ˙ z=F(τ, z), (1.5) whereF(τ,0) = 0, and then the stability of a certain solutionx(t) of the system (1.1) will be brought to study the stability of the zero solution of the system (1.5).
Definition 1.1.3 ([73], Definition 3.3) The system (1.1) is exponentially stable if there exist numbers
For every positive constant \( M > 0 \) and \( \delta > 0 \), every solution of the system defined by equation (1.1) with initial condition \( x(t_0) = x_0 \) satisfies the inequality \( \|x(t)\| \leq M e^{-\delta(t - t_0)} \|x_0\| \) for all \( t \geq t_0 \) This indicates that the zero solution of the system is not only asymptotically stable, but all solutions of (1.1) converge to zero at a rapid exponential rate.
Consider the linear system ˙ x(t) =Ax(t), t⩾t0, (1.7) where A is an×n- matrix The solution of the system (1.7) comes from the initial statex(t 0 ) =x 0 given by x(t) =x0e A(t−t 0 ) , t⩾t0.
The following theorem gives a first criterion for the stability of the system (1.7), often called the Lyapunov algebraic 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.
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 equa- tion (1.7) is f(z) =z n +a 1 z n−1 +ã ã ã+a n
Then if the determinant of all submatrices Dk, k = 1, 2, , n, is positive, then the real part of all solutions off(z) is negative, i.e the system is asymptotically stable, where detD1 = a1,detD2= det a 1 a 3
The stability of a stationary linear system (1.7) is equivalent to the existence of a solution of a matrix equation, often called a Lyapunov equation of the form
A T X+XA=−Y, (1.8) whereX, 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 matrixA if real part of all eigenvalues ofAa negative.
Theorem 1.1.3 ([73], Theorem 3.3) The matrix A is stable if and only if, for any positive definite symmetric matrix Y, the equation (1.8) has a solution that is a symmetric, positive definite matrixX.
Consider a system of stationary nonlinear equations ˙ x=f(x), f(0) = 0, t∈R + (1.9)
The functionV :R n →Ris positive definite if i) V(x)⩾0 for allx∈R n , ii) V(x) = 0 if and only ifx= 0.
Definition 1.1.4 ([73], Definition 3.5) A function V : D → R, where D ⊆ R n is an arbitrary open neighborhood of 0, called the Lyapunov function of the system (1.9) if i) V(x) is a continuously differentiable function onD, ii) V(x) is a positive definite function, iii) DfV(x) := ∂V ∂x f(x)⩽0,∀x∈D.
A functionV(x) is called a tight Lyapunov function if the inequality iii) is replaced by iv) ∃c >0 :D f V(x)⩽−ckxk0 andβ∈ KL,γ∈ K∞, such that∀x∈X,||x||X ⩽ρ ∀u∈
||φ(t, t0, x, u)||X ⩽β(||x||X, t−t 0 ) +γ(||u||U c), (1.12) where φ(t, t0, x, u) is the state of (1.10)-(1.11) corresponding to the initial valuex∈X, the initial time t 0 , and to the inputu∈U c at timet⩾t 0
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 matrixM is called a Metzler matrix if its off-diagonal elements are nonnegative, i.e m ij ⩾0, i6=j.
Lemma 1.1.1 ([23]) Any solution of the linear system ˙ x(t) = Ax(t) +Bϕ(t), ϕ:R+→R q +, y(t) = Cx(t) +Dϕ(t), with x(t) ∈ R n , y(t) ∈ R p and a Metzler matrix A ∈ R n×n , is elementwise nonnegative for all t ⩾ 0 provided thatx(0)>0 andB∈R n×q +
Lemma 1.1.2 ([36]) Any solution of the system xt+1 = Axt+Bϕt, ϕ:Z+→R m +, with x t ∈R n and a nonnegative matrices A ∈R n×n + and B ∈R n×m + , is elementwise nonnegative for all t∈Z+ provided thatx(0)⩾0.
This article discusses an impulsive linear system characterized by external inputs, described by the equations \$\dot{x}(t) = Ax(t) + f(t)\$ for \$t \in [t_i, t_{i+1})\$ and \$x(t_{i+1}) = Gx(\bar{t}_{i+1}) + g(t_{i+1})\$ for all \$i \geq 1\$ Here, \$x(t) \in \mathbb{R}^n\$ represents the state vector, while matrices \$A\$ and \$G\$ belong to \$\mathbb{R}^{n \times n}\$ The input function \$f(t)\$ is bounded, and the function \$g(t)\$ is continuously differentiable and also bounded at the time instants \$t_{i+1}\$ The sequence of impulse events \$t_i\$ is defined to be positively incremental, ensuring that the time intervals \$\theta_i = t_{i+1} - t_i > 0\$ with the initial time set at \$t_0 = 0\$.
Lemma 1.1.3 ([17]) Consider system (1.13)-(1.14) with a ranged dwell-time θi∈[Tmin, Tmax], ∀i∈Z+, where0< Tmin⩽Tmax0 n andQ∈ S >0 n such that for allθ∈[T min , T max ]:
G T e A T θ P e Aθ G−P=−Q, (1.15) then (1.13)-(1.14) is ISS and the following asymptotic gain is guaranteed t→∞lim |x(t)|⩽[ρ P,Q,W ||g||∞+T max (1 +ρ P,Q,W |G|ρ(A))||f||∞]%(A), where ρ P,Q,W q λ max (W ) λ min (P)
W =P+ sup θ∈[T min ,T max ]2P Ge Aθ Qe A T θ G T P and
1 à(A)⩽0 forà(A) = max i=1,n λ( A+A 2 T )being a logarithmic norm of the matrix A.
Proof From the system equations we can obtain for alli∈Z+: x(t) =e A(t−t i ) x(ti) +
Z t t i e A(t−s) f(s)ds∀t∈[tt, ti+1) and x(t i+1 ) =Ge AT i x(t i ) +r(t i )∀i⩾1, wherer(ti) =GRt i+1 t i e A(t i+1 −s f(s)ds+g(ti+1) Next,
Let us denoteφ=|GRt i+1 t i e A(t−s f(s)ds| One can write φ ⩽ |G|
|r(ti)|⩽Ti|G||%(A)|kfk ∞ +kgk ∞ and krk ∞ ⩽Tmax|G||ρ(A)|kfk ∞ +kgk ∞ Consider a Lyapunov functionν(x) =x T P x, whereP ∈R n×n is given in LMI (1.15), then ν(x(ti+1))−ν(x(ti)) =x T (ti+1)P x(ti+1)−x T (ti)P x(ti)
=x T (ti)[e A T T i G T P Ge AT i −P]x(ti) +2r T (ti)P Ge AT i x(ti) +r T (ti)P r(ti)
⩽−0.5x T (ti)Qx(ti) +r T (ti)[P +2P Ge AT i Qe A T T i G T P]r(ti)
⩽−λmin(Q)) 2λ max (P)V(x(ti)) +r T (ti)W r(ti).
From this expression we obtain λmin(P)|x(ti+1)| 2 ⩽(λmax(P)−1
2λmin(Q))|x(ti)| 2 +λmax(W)krk 2 ∞ or equivalently for alli⩾1
|x(ti+1)|⩽ s λmax(P)− 1 2 λmin(Q) λmin(P) |x(ti)|+ s λmax(W) λmin(P)krk ∞
Fori→+∞, sinceq λ max (P)− 1 2 λ min (Q) λ min (P) 0, X=X T , we have
State observer design problems
Full order state observer
The full order state observer, as defined by Luenberger, estimates all states of a system and requires an observer of order \( n \) for full state estimation in a system with \( n \) state variables For a linear time-invariant system described by the equations \( \dot{x}(t) = Ax(t) + Bu(t) \) and \( y(t) = Cx(t) \), where \( x(t) \in \mathbb{R}^n \) is the state vector, \( u(t) \in \mathbb{R}^m \) is the control input vector, and \( y(t) \in \mathbb{R}^p \) is the measurement output vector, the matrices \( A \in \mathbb{R}^{n \times n} \), \( B \in \mathbb{R}^{n \times m} \), and \( C \in \mathbb{R}^{p \times n} \) are constant and known.
In this system, it is assumed that the state variables represented by \( x(t) \) cannot be measured directly, necessitating the formation of an observer to reconstruct these variables To achieve this, an approximation \( \hat{x}(t) \) is introduced, which represents the estimated state as opposed to the actual state The observer is defined by the dynamics \( \dot{\hat{x}}(t) = A\hat{x}(t) + Bu(t) + L(y(t) - \hat{y}(t)) \) for \( t \geq 0 \), and \( \hat{y}(t) = C\hat{x}(t) \), where \( L \in \mathbb{R}^{n \times p} \) is the observer gain matrix and \( \hat{x}(t) \in \mathbb{R}^n \) serves as the estimate of the state \( x(t) \).
By denoting errore(t) as e(t) =x(t)−x(t), tˆ ⩾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)t e(0) (1.22)
The existence of a full order state observer is contingent upon the observability of the original system; if the system is unobservable, constructing a full order state observer is impossible.
Figure 1.1: Responses of x(t) and its estimation
Remark 1.2.1 Combining the controller and observer, we obtain ˆ u(t) = −Fx(t),ˆ (1.23) ˙ˆ x(t) = Aˆx(t) +Bu(t) +L(y(t)−y(t))ˆ (1.24) and a combination system ofx(t) and ˆx(t) can be expressed as below:
Therefore, the following combination system of x(t) ande(t) is obtained:
Reduced order state observer
The full order state observer, as described in equation (1.18), matches the order of the original system (1.16) and is equal to the number of state variables While its design is conceptually straightforward, it contains inherent redundancies Our goal is to reconstruct the state vector \$x(t) \in \mathbb{R}^n\$, and since the system output consists of \$p\$ linear combinations of the state variables, it follows that the remaining \$n - p\$ state variables can be effectively reconstructed using an observer of order \$n - p\$.
Let us consider the following state transformation x(t) =P w(t), (1.28) whereP is an invertible matrix,w(t) " wp(t) w u (t)
# , then the system (1.16)-(1.17) can be written into the following form
In the given equations, \( w_p(t) \) represents measurable states, while \( w_u(t) \) signifies unknown states Since \( w_p(t) \) is derived directly from the output \( y(t) \), it is necessary to estimate the remaining \( n - p \) state variables of \( w_u(t) \) From the equations (1.29) to (1.31), we derive the relationship \( \dot{w_u}(t) = A_{22}w_u(t) + A_{21}w_p(t) + B_2u(t) \) and \( \bar{y}(t) = A_{12}w_u(t) = \dot{w_p}(t) - A_{11}w_p(t) - B_1u(t) \).
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) whereL∈R (n−p)×p is a gain matrix needed to be determined such that the error vectore u (t) =w u (t)− ˆ wu(t) converges asymptotically to zero.
Linear functional state observer
Linear functional observers leverage a common principle in state feedback control, where only a linear combination of state variables, represented as \$Kx(t)\$, is often necessary instead of the complete state vector \$x(t)\$ Previous discussions focused on reconstructing the entire state vector or the unmeasurable state variables, revealing a redundant aspect This raises the question of whether a simpler observer can be designed to estimate a linear combination of certain unmeasurable state variables This subsection addresses the theory and development of linear functional observers, aiming to create observers that are of reduced order, simpler in structure, and stable.
The main result for this problem was first presented by Luenberger (see [61]) Consider a system of linear equations
(1.35) wherex(t)∈R n is the state vector,u(t)∈R m is the control vector,y(t)∈R p is the measurement output vector,p⩽n; A, B, C are constant matrices of appropriate dimensions.
Assuming the matrix \( C \) has full rank (i.e., \( \text{rank}(C) = p \)) and the system is observable, we consider the vector \( z(t) \in \mathbb{R}^r \) that needs to be reconstructed or estimated This vector is defined by the equation \( z(t) = F x(t) \), where \( F \in \mathbb{R}^{r \times n} \) is a known matrix, and it is given that \( \text{rank}(F) = r \).
To reconstruct the state function \( z(t) = F x(t) \), where \( F \in \mathbb{R}^{r \times n} \) and \( 1 \leq r \leq n - p \), we consider a functional observer of order \( q \) The estimated state function is given by \( \hat{z}(t) = D\omega(t) + Ey(t) \), while the dynamics of the observer are described by \( \dot{\omega}(t) = N \omega(t) + J y(t) + LBu(t) \) Here, \( \omega(t) \in \mathbb{R}^q \) and \( \hat{z}(t) \in \mathbb{R}^q \) represent the estimate of \( z(t) \), with \( D, E, N, J, \) and \( L \) being observer parameters that need to be determined The error \( e(t) = \hat{z}(t) - z(t) \) must converge asymptotically to zero as \( t \to \infty \).
In (1.37), the output ˆz(t) provides an asymptotic estimate ofF x(t) if t→∞lim[ˆz(t)−F x(t)] = 0 (1.39)
If \$\hat{z}(t)\$ estimates \$F x(t)\$, then \$w(t)\$, as defined in (1.38), can estimate other linear combinations of \$x(t)\$, referred to as \$Lx(t)\$ This leads to the formulation of a theorem regarding the existence of the proposed linear functional observer as described in (1.37) and (1.38).
Theorem 1.2.1 ([80, Theorem 3.1]) Estimation z(t)ˆ converges asymptotically to F x(t) for any initial conditionw(0) and anyu(t)if and only if the following conditions hold
Remark 1.2.2 Consider a closed-loop feedback system with a control input u(t) = F x(t), which is estimated by the observer (1.37), (1.38), ˙x(t) and ˙e(t) can be derived as follows ˙ x(t) = Ax(t) +Bu(t) =ˆ Ax(t) +B(Dw(t) +Ey(t))
This yields the following composite closed-loop system
Remark 1.2.3 We now present a method for solving observer matrices D, E, N, J, Land H From (1.42),H is obtained from H =LB so the problem of reconstructing the state function revolves around the design of the observed parametersD, E, J, L, N to satisfy the condition (1.40), (1.41), (1.43) of the 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 ∈R r×n ,L˜∈R q×n and partition them as follows
, (1.48) whereP is an invertible matrix defined as follows x(t) =P x(t), (1.49) where P = [C ‡ , C ⊥ ] ∈ R n×n , C ‡ ∈ R n×p is the Moore-Penrose inverse of C, (i.e, CC ‡ = Ip) and
C ⊥ ∈ R n×(n−p) are denoted by the orthogonal basis of the zero space ofC (i.e., CC ⊥ = 0) From the definition ofP above, it is clear that CP = [Ip,0] , A=P −1 AP "
# , then (1.41) and (1.43) are shortened as follows
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 getJ andE.
Interval observer
Consider the following dynamical system: ˙ x(t) = f(t, x(t)), x(0) =x0, x∈X, (1.54) y(t) = h(t, x(t)), (1.55) ˙ z − (t) = g − (t, z − (t), y(t)), (1.56) x − (t) = ` − (t, z − (t), z + (t), y(t)), (1.57) ˙ z + (t) = g + (t, z + (t), y(t)), (1.58) x + (t) = ` + (t, z − (t), z + (t), y(t)) (1.59)
A pair of systems (1.56)-(1.57), (1.58)-(1.59) is an interval estimator for system (1.54)-(1.55) if for any (relatively) compact set X 0 ⊂X, there exist (z − (0), z + (0)) such that the coupled system (1.54)-(1.55), (1.56)-(1.57), (1.58)-(1.59) verifies x − (t)⩽x(t)⩽x + (t), t⩾0 for any initial conditionx(0)∈X0.
A new method for designing observers of a nonlinear time-delay Glucose-Insulin system
Diabetes is a global epidemic that necessitates careful monitoring of glucose and insulin levels in patients to ensure effective treatment, such as insulin injections, for maintaining optimal blood glucose levels A glucose-oxidase-based amperometric sensor can easily measure blood glucose levels by utilizing glucose in interstitial fluid, providing an indirect reflection of blood sugar levels In contrast, insulin measurements are more challenging, slower, and less accurate To address this issue, model-based state observers have been developed to estimate insulin levels This chapter introduces a novel state observer designed specifically for estimating insulin levels in diabetic patients.
This chapter presents a general nonlinear time-delay GI model described by the equation \$\dot{x}(t) = f(x(t), x(t−\tau)) + Bu(t) + g(y(t), y(t−\tau_g))\$, where \$t \geq 0\$ The initial conditions are defined by \$x(\theta) = \phi(\theta)\$ for \$\theta \in [-\tau_{max}, 0]\$, with \$\tau_{max} = \max\{\tau, \tau_g\}\$ The output variable, representing the measured glucose levels, is given by \$y(t) = Cx(t) = x_1(t)\$ The state vector \$x(t) = \begin{pmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \\ x_4(t) \end{pmatrix}\$ includes blood glucose levels (\$x_1(t)\$), insulin levels (\$x_2(t)\$), and insulin mass in both accessible (\$x_3(t)\$) and non-accessible (\$x_4(t)\$) subcutaneous depots The control input \$u(t)\$ represents the subcutaneous insulin delivery rate, while the functions \$f(., )\$ and \$g(., )\$ are positive, bounded on \$[0, \infty) \times [0, \infty)\$, and continuously differentiable with respect to their arguments.
A novel state transformation
This article introduces a novel procedure for designing a state observer for a nonlinear time-delay model The proposed two-stage process transforms the model into a new observable form by incorporating the nonlinear term \(x_1(t)x_2(t-\tau)\) into the system's output and input Initially, we apply the concept of diffeomorphism to define a new output \(\bar{y} = -\ln(y(t))\), ensuring that \(x_1(t) > 0\) for all \(t \geq 0\) to maintain the existence of \(\ln(y(t))\) In the second stage, we introduce a state transformation that facilitates the design of a state observer in this new observable form We demonstrate that \(x_1(t) > 0\) for all \(t \geq 0\) by showing that if \(x_1(0) > 0\), it cannot become non-positive, thus confirming the validity of our approach.
We conclude that \( x_1(t) > 0 \) for all \( t \geq 0 \) This allows us to apply the concept of diffeomorphism to the output By dividing both sides of the first equation of (2.1) by \(-x_1(t)\), we define a new output as \( \bar{y}(t) = \xi(t) = -\ln(y(t)) \) Consequently, equations (2.1)-(2.3) can be rewritten as follows: \[\dot{\tilde{x}}(t) = A\tilde{x}(t) + A_d\tilde{x}(t - \tau) + B u(t) + \bar{a}(\bar{y}(t), y(t - \tau_g)), \quad t \geq 0,\]\[\tilde{x}(\theta) = \phi(\theta), \quad \theta \in [-\tau_{\text{max}}, 0],\]\[\bar{y}(t) = Cx(t) = \tilde{\xi}(t).\]
In designing a state observer to estimate the unknown state vector, existing methods for time-delay systems have limitations due to the non-observable matrix pair (A, C) and the presence of stable fixed poles in the observer error dynamics These issues lead to slow convergence rates for traditional state observers To address this challenge, we introduce a novel "delayed" state observer that estimates a delayed version of the state vector, overcoming the constraints of conventional observer design methods.
Accordingly, in the following, we consider the general form of system (2.5)-(2.7), where ˜x h ξ(t) ˜x2(t) x˜n(t) iT
We introduce a novel state transformation that converts the system into an observable format, incorporating the time-delay term \( Adx(t - \tau) \) into both the input and output of the transformed system This approach facilitates the straightforward design of a delayed state observer.
For m, n∈N, n >1 and an arbitrary matrixM ∈R 1×n , M T denotes the transpose ofM, 0m,n denotes the m×n zero matrix, M = h
[M]L [M]R i, where [M]L ∈ R and [M]R ∈ R 1×(n−1) are sub-matrices ofM.
We define a new change of coordinates as follows z(t)
, (2.8) where matricesM i , N i (i= 1,2, , n) are generated by the following algorithm
Ni+1=MiAd+NiA−βiM1, i= 1,2, , n−1, (2.11) whereα i andβ i are scalars to be determined later.
Theorem 2.1.1 For some scalars γi (i = 2,3, , n), αj andβj (j = 1,2, , n−1), if the following equations hold
X i=2 γiNi]R= 0, (2.14) then the change of coordinate (2.8) transforms the system (2.5)-(2.7) into the following form ˙ z(t) = Az(t) + ¯¯ Bu(t) + ¯B 1 u(t−τ) + Γ¯y(t) + Γ 1 y(t¯ −τ)
Proof Fori= 1,2, , n−1, by taking the derivatives of (2.8) and using (2.10)-(2.12), we obtain ˙ z i (t) = h
Next, fori=nfrom (2.13)-(2.14), we have ˙ z n (t) = M n A˜x(t) + (M n A d +N n A)˜x(t−τ) +M n Bu(t)
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 following recursive matrices
First, we consider (2.12) and by using (2.19)-(2.20), we obtain the following recursive equations
Equation (2.21) can be expressed in the following compact form χnXn =Yn, (2.22) where χn=h χ 1 n χ 2 n i
, withχ 1 n ,χ 2 n ,X n 1 andX n 2 are as defined below χ 1 n =h β1 β2 β3 β n−1 i
From (2.22), a solution forχn exists if and only if rank
To find the unknowns α n−1 and γ j (for j = 2, 3, , n), we analyze the solvability of equations (2.13) and (2.14) By substituting equations (2.10) and (2.11) into (2.13) and (2.14), and applying equations (2.19) and (2.20), we can rearrange the terms to derive a compact vector-matrix equation: ζnZn = Tn, where ζn is defined as the vector \( h \, \alpha_{n-1}, \, \gamma_2, \, \gamma_3, \, \gamma_4, \, \ldots, \, \gamma_n \, \).
In (2.26)-(2.27),T n 1 ,T n 2 ,Z n 1 (k, n−1) andZ n 2 (k, n−1) (k= 3,4, , n) are defined as follows
It is clear from (2.28)-(2.31),Z n andT n are two known constant matrices sinceβ k (k= 1,2, , n−
1) andα`(`= 1,2, , n−2) have already been derived from the solution to equation (2.22) From (2.24), a solution forζ n always exists if and only if rank
Accordingly, we present an effective algorithm to transform a general n-order time-delay system (n⩾3) with single output into the observable form (2.15)-(2.16).
Step 1: Obtain matricesXn and Yn according to (2.22) Check if condition (2.23) is satisfied or not If so, obtainχ n whereχ n =Y n X n + , where X n + denotes the Moore-Penrose inverse ofX n
Step 2: Substitute βk (k = 1,2, , n−1) andα` (`= 1,2, , n−2) into (2.26)-(2.27) and obtain Zn and Tn Check if condition (2.24) is satisfied or not If so, obtain ζn = TnZ n + , where Z n + denotes the Moore-Penrose inverse ofZn.
Step 3: From (2.9)-(2.11), obtain matricesMiandNi (i= 1,2, , n) and hence the state transformation(2.8) Finally, obtain a transformed system according to (2.15)-(2.16).
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 to estimatez(t) since the matrix pair ( ¯A,C) is now observable After a satisfactory state observer¯ z(t) has been designed, we can use the method of backward state transformations reported in [47].
Application to the GI model
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
. Hence, we obtain the following state transformations z 1 (t) = ξ(t), z 2 (t) = a 1 x 2 (t−τ), z 3 (t) = −a1a 2 x 2 (t−τ) +a 1 a 3 x 4 (t−τ), z 4 (t) = a 2 a 4 a 6
−a1a 3 (a 2 +a 6 )x 4 (t−τ) and a transformed system of the forms (2.15)-(2.16), where
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 any¯ linear function of the state vectorz(t) Let h(t)
z(t) be a vector that is required to be estimated To reconstruct the state function, h(t), we consider a functional observer of order 3 as follows: ˆh(t) = ω(t) +Ey(t),¯ (2.33) ˙ ω(t) = N ω(t) +Jy(t) +¯ Hu(t−τ)
+L¯à(¯y(t),y(t¯ −τ),y(t¯ −τ−τ g )), t⩾τ max , (2.34) where ω(t) ∈ R 3 , ˆh(t) ∈ R 3 is the estimate of h(t), E, N, J, H and L are observer parameters to be determined Let us define the following error vectors(t) ande(t) as
Based on [80, Theorem 3.1], ˆh(t) converges asymptotically toF z(t) if the following conditions are satisfied
Accordingly, for the given matrices ¯A, ¯C, and ¯B as above, we can easily solve (2.37)-(2.40) to obtain the following matrices: N = ¯A22+L1A¯12, L1 is chosen such that N is Hurwitz, E =−L1,J =−N L1,
Note that the matrix pair ( ¯A22,A¯12) is observable and thusL1 can be easily found to ensure that N is stable with any prescribed eigenvalues.
Upon ˆh(t) has been obtained, then based the method of backward state transformations (Case 2) reported in [47], we obtain ˆ x2(t−τ) = 1 a1 ˆh1(t), (2.41) ˆ x3(t−τ) = 1 a 1 a 3 a 5 [a2a6hˆ1(t) + (a2+a6)ˆh2(t)
In order to obtain simulation results, we consider the nonlinear time-delay GI model (2.1)-(2.3) with a set of parameters, the initial conditions and the input u(t) are as follows: a1 = 3.11×10 −5 , a2 = 1.211 ×10 −2 , a3 = 0.25×55 1 , a4 = a5 = a6 = 55 1 , a7 = 1, τ = 3min, g(y(t), y(t −τg))
, τg= 4min, x1(θ) = 10.66, x2(θ) = 49.29, x3(θ) = 0,x4(θ) = 0 for allθ∈[−4,0],ω 1 (ζ) = 20e −0.07t ,ω 2 (ζ) = 2e −t ,ω 3 (ζ) = 5e −t for allt⩾0,ζ∈[−8,0] and u(t) ( sint+ 50, 0⩽t⩽100, sint+ 3, 100< t⩽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 in Matlab, we obtain L
Figure 2.1 illustrates the responses of \(x_2(t)\) and its delayed estimation, \(\hat{x}_2(t-3)\) Meanwhile, Figure 2.2 presents the responses of \(x_2(t-3)\) alongside its estimation, \(\hat{x}_2(t-3)\) The results in Figure 2.2 clearly demonstrate that the designed observer successfully tracks the delayed version of the state vector, confirming the expected performance.
In Chapter 2, we introduced an innovative procedure for designing a state observer for a general nonlinear time-delay Glucose-Insulin model This significant two-stage design process converts a nonlinear time-delay model into an observable form, facilitating the straightforward design of a third-order delayed state observer Our simulation results demonstrate the effectiveness of this approach A key advantage of our method is its ability to transform nonlinear time-delay systems into new coordinates, where all time-delay terms are linked solely to the output and input This transformation enables the easy design of a Luenberger-type state observer However, it is important to note that in some instances, the method yields a delayed or a mixed instantaneous and delayed state observer instead of an instantaneous state observer for nonlinear time-delay systems.
A new observer for interconnected time-delay systems and its applications to fault detection problem
Faults in control systems can significantly impair performance and stability, potentially resulting in physical damage Consequently, there has been a growing focus on the issue of fault detection in control systems.
The design of functional observers for detecting actuator faults in interconnected time-delay systems remains an underexplored area A notable contribution to this field is presented in [78], where the authors propose a fault detection scheme tailored for a class of interconnected time-delay systems comprising \(N\) subsystems This approach addresses the challenges posed by unknown input vectors \(d_i(t) \in \mathbb{R}^n\) and actuator fault vectors \(f_i(t) \in \mathbb{R}^n\) that originate from the input of the \(i\)-th subsystem, defined by the equation \(\dot{x}_i(t) = A_{ii}x_i(t) + \ldots\).
Aijxj(t−τji) +Ad ii xi(t−τii) +Biui(t)
The equations presented describe the dynamics of a system comprising multiple subsystems Specifically, for each subsystem \(i\), the local state vector \(x_i(t)\) and the remote state vector \(x_j(t)\) are defined in their respective dimensions, while the control input vector \(u_i(t)\) and the measured output vector \(y_i(t)\) are also specified The initial function for each subsystem is denoted as \(\phi_i(\theta)\), where \(\theta\) ranges from \(-\tau_{\text{max}}\) to \(0\) Additionally, the matrices \(A_{ii}\), \(A_{dii}\), \(A_{ij}\), and \(B_i\) represent the system's structural relationships and interactions among the subsystems, with dimensions corresponding to the state and control inputs.
Di∈R n i ×n id ,Ei∈R n i ×n if andCi∈R p i ×n i are real known system matrices Without loss of generality, it is assumed that rankC i =p i In (3.2), τ max is defined as τ max = max
1≤i,j≤K,i6=j{τii, τ ji } d i (t)∈R n id is the unknown disturbance andfi(t)∈R n if is an unpredictable fault signal.
To detect the faultsf i (t) in thei-st subsystem, the authors in [78] proposed the following residual generator: r i (t) = T i ω i (t) +F i y i (t), (3.4) ˙ ω i (t) = N i ω i (t) +N di ω i (t−τ ii ) +G i y i (t) +G di y i (t−τ ii )
The local residual generator, denoted as \( r_i(t) \in \mathbb{R} \), is derived from the functional observer state vector \( \omega_i(t) \in \mathbb{R}^{q_i} \) and the local output vector \( y_i(t) \) The observer parameters, including \( N_i, N_{di}, H_i, G_i, G_{di}, \) and \( G_{ij} \), must be determined to ensure that \( \omega_i(t) \) converges asymptotically to a linear function of the local state vector \( L_i x_i(t) \) in the absence of faults However, there are cases where such a residual generator may not exist, as illustrated by the example \( \dot{x}_1(t) = A_{11} x_1(t) + A_{d11} x_1(t - \tau_{11}) + A_{12} x_2(t - \tau_{21}) \).
+B2u2(t) +D2d2(t) +E2f2(t), t⩾0, (3.10) x2(θ) = φ2(θ), θ∈[−τmax,0], (3.11) y2(t) = C2x2(t), (3.12) whereτ max = max{τ 11 , τ 22 , τ 12 , τ 21 }, and x 1 (t)
For 1-st subsystem, there exists a residual generator r 1 (t) that is constructed from the following reduced-order unknown input functional observer: r 1 (t) = T 1 ω 1 (t) +F 1 y 1 (t), (3.13) ˙ ω 1 (t) = N 1 ω 1 (t) +N d1 ω 1 (t−τ 11 ) +G 1 y 1 (t) +G d1 y 1 (t−τ 11 ) +H 1 u 1 (t)
In this article, we discuss the application of Design Algorithm 1 from reference [78] to a specific example We note that the condition outlined in Step 1 of the algorithm is satisfied, as indicated by the relationship \$\nu_1 = \text{rank}(\Psi_1) = 4 < p_2 + n_1 = 5\$ However, we will demonstrate that the method presented in [78] is not applicable to this case.
G12 L1 i6= 0, to (11) exist From (14),X1 is obtained, where
According to Step 2 of Design Algorithm 1 in [78], we search for the minimum order q1, where
1 ⩽q 1 ⩽nullityΨ 1 = 2 First, we assign q 1 = 1 and matrix L 1 , extracted from X 1
Note that condition (23) does not hold since rank
The design method referenced in [78] fails to meet the conditions for subsystem 2, as evidenced by the verification process However, this chapter will demonstrate that our proposed method remains effective for this particular example.
To date, there has been no existing observer design method in the literature that facilitates the creation of two independent distributed functional observers for fault detection in this context Consequently, this chapter aims to introduce a novel observer design method We will enhance the state transformation presented in [50] to derive a general approach that simplifies the fault detection process for such systems.
For m, n∈ N, n >1 and arbitrary matrices M ∈ R n×n and N ∈R 1×n , the following notations will be used throughout this chapter:
•n ζ =n+ 2p−1,M T denotes the transpose ofM, 0 m,n denotes them×nzero matrix,I n denotes the identity matrix of dimensionn×n; and
[N]O [N]U i, where [N] O ∈R 1×p and [N] U ∈R 1×n are sub-matrices of N.
New state transformation
Without loss of generality, we assume that matrixCi takes the canonical form, where
Let us define the following new state transformation vector ζ i (t)
(3.18) and the row matricesT σ i ,T σ d ii andT σ `i (σ= 1,2, , ni ζ ) are obtained according to the following
T σ−p i i Ad ii −Pp i s=1β σi s T s i , σ=pi+ 1, ,2pi,
(3.21) whereα s σ ,β σi s ,β σ` s (σ=pi+ 1, , ni ζ ,s= 1, , pi) are scalars to be determined later We now denote the following matrices
H k i1 =T k i Aii, H k i2 =T k i Ad ii +T dk i Aii, H k i3 =T dk i Ad ii , H k i4 =T dk i Ai`, (3.22)
H k i5 =T k i Ai`+T k `i A``, H k i6 =T k `i Ad `` , H k i7 =T k `i A`i, H k if =T dk i Ei, (3.23)
H k id =T dk i Di, H k `f =T k `i Ei, H k `d =T k `i Di (3.24)
Remark 3.1.1 In [50], the authors proposed a state transformation method (state transformation (11) in [50]) to transform a class of single systems with a time delay (system (5)-(7) in [50]) into an observable canonical form Note that ifA ij = 0, ∀j = 1, , K, j 6=iand T `i = 0, ∀`= 1, , K, `6=i, then the system (3.1)-(3.3) and the state transform (3.17) in this chapter are reduced to the system (5)-(7) and the state transformation (11) in [50], respectively Hence, the state transformation considered in [50] can be regarded as a special case of the one in this chapter.
Remark 3.1.2 In contrast to the state transformations in [81], the state transformation (3.17) in this chapter utilizes multiple delayed statesx`(t−τ`i), `= 1, , K, `6=iand has a more general structure. This trade-off feature overcomes some drawbacks in [81] and enables distributed functional observers to be designed for a wider class of time-delay interconnected systems.
The following theorem presents conditions for the existence of the state transformation vector (3.16) to transform each subsystem of system (3.1)-(3.3) into a novel canonical form.
Theorem 3.1.1 For some scalarsα j i ,β i1 j ,β j i2 (i=p i + 1, p i + 2, , n i ζ , j= 1,2, , p i ), if the following equations hold hH k i1 i
H k if = 0 1×n if , k=pi+ 1, pi+ 2, , ni ζ , (3.32)
H k `d = 0 1×n id , k=p i + 1, p i + 2, , n i ζ , (3.35) where H k i1 , H k i2 ,H k i3 , H k i4 , H k i5 ,H k i6 , H k i7 , H k id , H k `f and H k `d are as defined in (3.22)-(3.24), then the transformation vector (3.17) transforms each subsystem of system (3.1)-(3.3) into the following observable form ζ˙ i (t) = ¯A i ζ i (t) + ¯B i u i (t) + ¯B i 1 u i (t−τ ii ) +
+ ¯Didi(t) + ¯Eifi(t) + Γiyi(t) + Γ 1 i yi(t−τii) + Γ 2 i yi(t−2τii) +
`=1(`6=i) Γ¯ ` si ys(t−τs`−τ`i), t⩾2τmax, (3.36) yi(t) = ¯Ciζi(t), (3.37) where
R n iζ ×p i ,Γ¯ ` si ∈R n iζ ×p i ,Γ 1 `i ∈R n iζ ×p ` ,Γ 2 `i ∈R n iζ ×p i ,Γ 3 `i ∈R n iζ ×p ` are defined as below
Proof By taking the derivatives of (3.17), we obtain ζ˙ i (t) = h
Forσ= 1,2, , pi, from (3.19)-(3.21), we have ζ˙ σ i (t) = T σ i x˙i(t) =T σ i (Aiixi(t) +Ad ii xi(t−τii) +
= T σ i Aiixi(t) +T σ i Ad ii xi(t−τii) +
(3.39) Forσ=pi+ 1, ,2pi−1, from (3.19)-(3.21), we obtain ζ˙ σ i (t) = T σ i Aiixi(t) + (T σ i Ad ii +T dσ i Aii)xi(t−τii) +T dσ i Ad ii xi(t−2τii)
Oy`(t−τ``−τ`i) +T σ i Biui(t) +T σ i Didi(t) +T σ i Eifi(t) +T dσ i Biui(t−τii) +
Forσ= 2pi, from (3.19)-(3.21), we obtain ζ˙ σ i (t) = T σ i (A ii x i (t) +A d ii x i (t−τ ii )
A i` x ` (t−τ ii −τ `i ) +Biui(t−τii) +Didi(t−τii) +Eifi(t−τii))
= T σ i Aiixi(t) + (T σ i Ad ii +T dσ i Aii)xi(t−τii)
Forσ= 2pi+ 1, , ni ζ −1, from (3.38), we obtain ζ˙ σ i (t) = T σ i (Aiixi(t) +Ad ii xi(t−τii) +
From (3.19)-(3.21), we have forσ= 2pi,2pi+ 1, , ni ζ −1 and`= 1, , N, `6=i,
(T σ i Ad ii+T dσ i Aii)xi(t−τii) =T d(σ+1) i xi(t−τii) +
Substituting (3.43)-(3.45) into (3.42) and using (3.32)-(3.35), we obtain ζ˙ σ i (t) = ζ σ+1 i (t) +T σ i Biui(t) +T σ i Didi(t) +T σ i Eifi(t) +T dσ i Biui(t−τii)
Forσ=ni ζ and from (3.42), we have ζ˙ n i iζ(t) = T n i iζAiixi(t) + (T n i iζAd ii +T dn i iζAii)xi(t−τii) +
Now, by using (3.25)-(3.28), equation (3.46) can be expressed as ζ˙ n i iζ(t) = h
Finally, (3.40)-(3.42), (3.46) and (3.47) can now be expressed in the form (3.36)-(3.37) This completes the proof of Theorem3.1.1.
Fault detection observer
Provided that systems (3.1)-(3.3) are transformed into the form (3.36)-(3.37), let us now consider a functional observer of orderq i for thei-th subsystem as follows ˙ ωi(t) = Niωi(t) +Hiui(t) +H i 1 ui(t−τii) +
The equation \( G¯ ` si ys(t−τs`−τ`i), t⩾2τmax \) describes a system where \( \omega_i(t) \) belongs to \( R^{q_i} \) with constraints \( 1≤q_i≤n_i \) The parameters \( N_i, G_i, G^1_i, G^2_i, G^1_{`i}, G^2_{`i}, ¯G_{`si}, H_i, H_{i1}, H_{i2}, \) and \( H_{`i} \) are observer parameters that need to be determined The goal is to ensure that \( \omega_i(t) \) serves as an asymptotic estimate of \( L_iζ_i(t) \) when \( f_i(t) = 0 \) Here, \( L_i \) is a matrix in \( R^{q_i \times n_i} \) that is also to be determined for effective fault detection.
We introduce a residual function, \( r_i(t) \), designed to activate the i-th system faults, expressed as \( r_i(t) = T_i \omega_i(t) + F_i y_i(t) \) For effective fault detection, the matrices \( T_i \) and \( F_i \) must be configured such that as time approaches infinity, \( \lim_{t \to \infty} r_i(t) \) equals 0 when \( f_i(t) = 0 \), and either \( c_i \) (where \( c_i \neq 0 \)) or remains undefined when \( f_i(t) \neq 0 \) Here, \( f_i(t) = 0 \) indicates a faultless state, while \( f_i(t) \neq 0 \) signifies a faulty condition Additionally, we define the error vector as \( e_i(t) = \omega_i(t) - L_i \zeta_i(t) \).
Theorem 3.2.1 Under no fault conditions, i.e.,fi(t) = 0,ωi(t)is an asymptotic estimate ofLiζi(t)(i.e., e i (t)→0 asymptotically) and the residual generator function r i (t) as in (3.48)for any initial condition φi(θ),ζi(0), ωi(0) and anyui(t)if
L i D¯ i = 0, (3.54) and matricesGi,G 1 i ,G 2 i ,G 3 i ,G 1 `i ,G 2 `i ,G¯ ` si ,Hi,H i 1 andH`i are chosen according to the following
Under fault conditions, residualri(t)satisfies (3.49) if and only if all the parameters satisfy the conditions (3.51)-(3.57) and
Proof If f i (t) = 0 and (3.55), (3.57), (3.58), (3.52), (3.53), (3.54) satisfied then according to (3.47), (3.36), (3.37) we have ˙ e i (t) =N i e i (t)
Therefore ifNi is Hurwitz thenei(t)→0 asymptotically According to (3.48) we haveri(t)→0 asymp- totically.
If fi(t) 6= 0 and also (3.55), (3.57), (3.58), (3.52), (3.53), (3.54) are satisfied then according to (3.47), (3.36), (3.37) we have ˙ ei(t) =Niei(t)−LiE¯ifi(t).
Therefore ifL i E¯ i 6= 0 thene i (t)90 According to (3.48) we haver i (t)90.
A numerical example
To demonstrate the effectiveness of the obtained results in this chapter, we now consider the moti- vated example in this chapter.
Subsystem 1: A state transformationζ 1 (t) of the form (3.17) is obtained as ζ1(t) = h
, and a transformed system of the form (3.36)-(3.37), where
The transformed system's matrix pair ( ¯A1,C¯1) is observable, allowing us to determine the unknown matrices N1, L1, ¯G1, T1, and F1 to meet the specified conditions Consequently, we have developed a third-order functional observer-based fault detection observer for the 1-subsystem.
. Subsystem 2: A state transformationζ 2 (t) of the form (3.17) is obtained as ζ2(t) = h
, and a transformed system of the form (3.36)-(3.37), where
The transformed system's matrix pair ( ¯A 2 ,C¯ 2 ) is observable, allowing us to determine the unknown matrices N2, L2, ¯G2, T2, and F2 to meet the specified conditions Consequently, we have developed a third-order functional observer-based fault detection observer for the 2-st subsystem.
Figures 3.1-3.2 demonstrate that the residual generators effectively identify faults in both the first and second subsystems Faults \( f_1(t) \) and \( f_2(t) \) occur at \( t = 10s \) and \( t = 25s \), respectively, and are resolved by \( t = 15s \) and \( t = 35s \) During the fault occurrences, the residual generators successfully trigger alerts, while in the absence of faults, they converge to zero as anticipated Additionally, the residuals remain unaffected by the inputs \( u_1(t) \) and \( u_2(t) \), as expected.
Figure 3.1: Residual generator using third-order observer effectively triggers fault in the 1-st subsystem.
Figure 3.2: Residual generator using third-order observer effectively triggers fault in the 2-st subsystem.
In Chapter 3, we introduced a design for distributed functional observers aimed at detecting actuator faults in interconnected time-delay systems with unknown inputs We built upon state transformations from previous works to convert each subsystem into an observable canonical form, providing conditions for the existence of these transformations and a novel approach for identifying unknown parameters Within this new coordinate system, we developed a functional observer to create a residual function capable of identifying subsystem faults A numerical example with simulations demonstrates the effectiveness of our proposed design method However, this chapter does not address the uncertainty of system matrices in interconnected time-delay systems, and extending our method to incorporate time delays in outputs presents an intriguing avenue for future research.
Distributed functional interval observer design for large-scale networks impulsive systems
Impulsive dynamical systems are a significant subclass of hybrid systems that integrate both continuous and discontinuous behaviors, typically represented by ordinary differential equations and instantaneous state jumps at specific time instants These systems are crucial for modeling physical phenomena that experience abrupt changes and serve as effective tools for stabilizing and synchronizing chaotic systems They find applications across various fields, including epidemiology, sampled-data and networked control systems, and power electronics.
Despite numerous findings on the stability of impulsive systems, as noted in references [5], [34], and [35], there has been limited focus on the design of observers for this category of systems.
[21], [64], [77] In [21], the construction of observers for linear impulsive systems with the objective of stabilizing the state of the system about the origin thanks to a dynamic output feedback was reported.
In recent studies, researchers have developed innovative observer designs for various classes of linear impulsive systems and switched linear systems One approach involves constructing an impulsive observer that generates general cubic spline signals, with the observer gain explicitly defined by the impulse times Additionally, a method for designing asymptotic observers for switched linear systems has been introduced, leveraging the property of determinability to combine partial information from different modes for state vector estimation This observer accounts for the time required to process information, and under persistent switching conditions, error analysis confirms that the state estimate converges exponentially to the actual state Furthermore, sufficient conditions for designing a straightforward class of interval observers for linear systems have also been established.
Designing distributed linear functional interval observers
We consider an interconnection ofnimpulsive subsystems with inputs ˙ xi(t) = Aiixi(t) +Biui(t) +di(t) +
The equations describe the dynamics of a composite system composed of multiple subsystems, where the local state, remote state, control input, and measured output vectors are represented as \(x_i(t)\), \(x_j(t)\), \(u_i(t)\), and \(y_i(t)\), respectively Each subsystem is indexed by \(i\) (ranging from 1 to \(N\)), with \(N\) indicating the total number of subsystems The relationship between the local state at time \(t_{i k+1}\) and its previous state is defined by the equation \(x_i(t_{i k+1}) = G_ix_i(\bar{t}_{i k+1}) + D_iu_i(t_{i k+1}) + g_i(t_{i k+1})\) Additionally, the impulse events \(t_{i k}\) must satisfy the condition \(θ_{i k} = t_{i k+1} - t_{i k} \in [T_{i \text{min}}, T_{i \text{max}}]\), where \(0 < T_{i \text{min}} \leq T_{i \text{max}} < \infty\) are constants The matrices \(A_{ii}\), \(G_i\), \(A_{ij}\), \(B_i\), \(D_i\), and \(C_i\) are real known system matrices that define the interactions and dynamics within the subsystems.
This work aims to design N stand-alone distributed interval observers for N subsystems It is recognized that for subsystems with a high number of state variables, designing interval observers for state functions can be advantageous To accomplish this, we define \( z_i(t) = F_i x_i(t) \) for \( 1 \leq r_i \leq n_i - p_i \) and \( i = 1, \ldots, N \), where \( F_i \in \mathbb{R}^{r_i \times n_i} \) is a specified matrix To derive an interval observer for the linear function \( z_i(t) \), we examine the following distributed linear functional observers: \[\dot{z}_i(t) = N_i z_i(t) - |J_i|^{1/p_i} V_i + F_i B_i u_i(t) + J_i y_i(t) + F_i d_i(t)\]
|Hij|1p jVj, t∈[ti k , ti k+1 ), (4.5) z i − (ti k+1 ) = Liz − i (¯ti k+1 ) +FiDiui(ti k+1 )− |Ki|1p i Vi+Kiyi(ti k+1 ) +Fig i − (ti k+1 ),
(4.6) ˙ z i + (t) = Niz i + (t) +Jiyi(t) +|Ji|1p i Vi+FiBiui(t) +Fid + i (t)
|Hij|1p jV j , t∈[t i k , t i k+1 ), (4.7) z i + (ti k+1 ) = Liz + i (¯ti k+1 ) +Kiyi(ti k+1 ) +|Ki|1p i Vi+Fig i + (ti k+1 ) +FiDiui(ti k+1 ),
(4.8) where z i + (t) ∈ R r i , z i − (t) ∈ R r i Matrices N i , J i , L i , K i and H ij are unknown observer parameters. z − i (¯ti k+1 ) andz + i (¯ti k+1 ) are the left-sided limit ofz i − (t) andz i + (t) fort→ti k+1 , respectively.
We first introduce the following assumptions.
(H1)zi(t)∈ L r ∞ i withz i − (0)⩽zi(0)⩽z i + (0) for some knownz i − (0)∈R r i ,z i + (0)∈R r i
(H3) d − i (t)⩽di(t)⩽d + i (t), t⩾0,g − i (t)⩽gi(t)⩽g i + (t), t⩾0, for some knowns d − i (t), d + i (t)∈
Now, we define a functional interval observer of system (4.1)-(4.3) as follows.
Definition 4.1.1 Assume that (H 1 )-(H 3 ) are satisfied Then systems (4.5)-(4.6) and (4.7)-(4.8) are called a linear functional interval observer for the i- subsystem of system (4.1)-(4.3) if for any initial conditions, z i + (0), z i − (0) ∈ R r i , the solutions of equations (4.1)-(4.2), (4.5)-(4.6) and (4.7)-(4.8) exist, z − i (t), z i + (t)∈ L r ∞ i and z i − (t)⩽z i (t)⩽z i + (t) (4.9) for allt⩾0.
Existence conditions of distributed linear functional interval observers
The following theorem provides conditions for the existence of distributed functional interval ob- servers for large-scale networks impulsive system (4.1)-(4.3).
Theorem 4.2.1 Systems (4.5)-(4.6)and (4.7)-(4.8)are an interval observer of linear functionz i (t)for the i- subsystem of system (4.1)-(4.3)if the following conditions are satisfied
L i F i +K i C i −F i G i = 0, (4.14) There exist matricesP i , Q i ∈ S >0 n i such that
L T i e N i T θ P i e N i θ L i −P i =−Q i (4.15) hold for allθ∈[T i min , T i max ].
Proof We first define the upper errore + i (t) and the lower errore − i (t) as e + i (t) =z + i (t)−z i (t), t∈[t i k , t i k+1 ), (4.16) e + i (t i k+1 ) =z + i (t i k+1 )−z i (t i k+1 ), (4.17) e − i (t) =z i (t)−z − i (t), t∈[t i k , t i k+1 ), (4.18) e − i (t i k+1 ) =z i (t i k+1 )−z i − (t i k+1 ) (4.19)
In regard to (4.1)-(4.2), (4.5)-(4.6) and (4.7)-(4.8), the derivatives ofe − i (t) ande + i (t) are given by ˙ e − i (t) = z˙i(t)−z˙ − i (t)
−Jivi(t) +|Ji|1p i Vi−FiBiui(t)−Fid − i (t)−
Thus, if conditions (4.10)-(4.14) of Theorem4.2.1are satisfied then we obtain ˙ e − i (t) = Nie − i (t) +ϕ − i
1(t), t∈[ti k , ti k+1 ), (4.25) e − i (ti+1) = Lie − i (¯ti k+1 ) +ϕ − i
2(ti k+1 ) = |Ki|1p i Vi+Fi(gi(ti k+1 )−g i − (ti k+1 )), ϕ + i
By using (H3) and (H2), we obtain ϕ − i
2(t i k+1 )⩾0,∀k⩾1 Therefore, it follows from conditions (4.10)-(4.11), Lemma 1.1.1 and Lemma 1.1.2 that e − i (t) ⩾ 0 and e + i (t) ⩾ 0 for all t ⩾ 0 Hence, we obtain (4.9).
We demonstrate that the errors \$e^{-i}(t)\$ and \$e^{+i}(t)\$ are bounded According to condition (4.15) and Lemma 1.1.3, the systems defined by equations (4.25)-(4.26) and (4.27)-(4.28) exhibit bounded state variables when the ranged dwell-time \$\theta_{ik} \in [T_{i}^{min}, T_{i}^{max}]\$ is applied, provided that \$\phi^{-i}\$ remains bounded.
2 Finally, the boundedness ofz − i (t),z i + (t) is implied by boundedness ofzi(t),e − i (t) ande + i (t) The proof is completed.
Corollary 4.2.1 Assume that T i max =∞, conditions (4.10)-(4.14) of Theorem 4.2.1 are satisfied and the condition (4.15) is replaced by the following condition:
There exist matrixP i ∈ S >0 n i such that
Then systems (4.5)-(4.6)and (4.7)-(4.8)are an interval observer of linear functionzi(t)for thei- subsystem of system (4.1)-(4.3).
Solving unknown matrices
The next phase in designing stand-alone functional interval observers involves identifying the unknown observer parameters \(N_i\), \(J_i\), \(L_i\), \(K_i\), \(H_{ij}\), \(P_i\), and \(Q\) to ensure compliance with the conditions outlined in Theorem 4.2.1 The process for achieving this is elaborated on in the following sections.
F i A ij 1 F i A ij 2 F i A ij N i and eN i ∈R σ i ×r i , eJ i ∈R σ i ×p i , eT i ∈R σ i ×r i , eK i ∈R σ i ×p i , eH ijk ∈R σ i ×p jk are as below: eN i
By using the above notations, we can express equations (4.12)-(4.14) into the following form: χ i X i =Y i (4.38)
SinceXi andYi are two known constant matrices, a solution forχi always exists if and only if rank
Under condition (4.39), a general solution forχi is given by χi=YiX i ‡ +Zi
In equation (4.40), the matrix \(X_i^{\dagger} \in \mathbb{R}^{\tau_i \times \sigma_i}\) represents the Moore-Penrose inverse of \(X_i\), while \(Z_i \in \mathbb{R}^{r_i \times \sigma_i}\) is an arbitrary matrix that needs to be determined Additionally, the matrices \(N_i\), \(J_i\), \(L_i\), \(K_i\), and \(H_{ij}^k\) for \(k = 1, 2, \ldots, N\) and \(k \neq i\) can be derived from equation (4.40).
To implement functional interval observers, we will consolidate conditions to create a Linear Programming (LP) problem for verifying design parameters Specifically, conditions can be expressed using equations that involve the terms \( e^T N_i \Phi^T_i \lambda_i \) and \( e^T L_i \Phi^T_i \lambda_i \), ensuring they meet the necessary criteria for the observer design.
Based on the above discussion we obtain the following theorem which provides a computational approach which is based on LP for the determination of unknown observer matrices.
Theorem 4.3.1 Conditions (4.10)-(4.11)and (4.39)are feasible if and only if the following LP problem in the variablesλi∈R r i andZi∈R σ i ×r i is feasible:
Moreover, the observer gainsNi,Ji,Li,Ki,Hij k , k= 1,2, , N, k6=iare obtained as in (4.41)-(4.45) whereZ i = (diag(λ i )) −1 Z i T
We now propose an effective algorithm to obtain the interval observer parameters for each subsys- temi- (i= 1, , N).
Step 1: For given matrices A ii ∈R n i ×n i ,G i ∈R n i ×n i , A ij ∈R n i ×n j andC i ∈R p i ×n i , obtain matrices
Xi andYi from (4.35)-(4.36) Check the existence condition (4.39).
Step 2: Compute the matrices Φ i and Ψ i from (4.46).
Step 3: Solve the LP (4.49) with respect to Ziand λi.
Step 4: Compute the matrixZ i = (diag(λ i )) −1 Z i T where (λ i ,Zi) is a solution obtained in Step 3. Step 5: SubstituteZiinto (4.41)-(4.45) to obtain observer gainsNi,Ji,Li,Ki,Hij k , k= 1,2, , N, k6=i. Step 6: WithN i ,L i from Step 5, solve (4.15) to obtain matricesP i ,Q i
Remark 4.3.1 For the case where Aij = 0, j = 1, , N, j 6=i, we consider the following functional interval observer ˙ z i − (t) = Niz − i (t)− |Ji|1p i Vi+FiBiui(t) +Jiyi(t) +Fid − i (t), t∈[ti k , ti k+1 ),
(4.53) wherez i + (t)∈R r i ,z − i (t)∈R r i MatricesNi, Ji,Li andKi are unknown observer parameters z i − (¯ti k+1 ) andz + i (¯t i k+1 ) are the left-sided limit ofz i − (t) andz i + (t) fort→t i k+1 , respectively.
By following the proof of Theorem4.2.1, 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 functionz i (t) of system (4.1)-(4.3) (in caseAij = 0, j= 1, , N, j6=i) if the following conditions are satisfied
L i F i +K i C i −F i G i = 0, (4.57) There exist matricesP i , Q i ∈ S >0 n i such that
L T i e N i T θ P i e N i θ L i −P i =−Qi (4.58) hold for allθ∈[T i min , T i max ].
Remark 4.3.2 MatricesNi, Ji, Li andKi satisfying conditions of Corollary4.3.1 can be obtained by using Theorem4.3.1, where matricesX i andY i in (4.35) and (4.36) are replaced by
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 ast i k = 10 + 20k,i= 1,2,k∈Z+, where x T 1 (t) =h x 11 (t) x 12 (t) x 13 (t) i
In this example, the robust observer referenced in [46] and [17] is not applicable Instead, we will utilize the method outlined in this chapter to design distributed functional interval observers for the impulsive interconnected system We begin by focusing on the 1-subsystem, where we apply Step 1 of Algorithm 4.1 to derive the matrices X₁ and Y₁ from equations (4.35) and (4.36), taking into account the rank conditions.
, condition (4.39) is satisfied By solving the LP problem (4.49) with the constraint 0.1≤λ 1 ≤1.2 we obtain a solution λ1= 0.1, Z1= 01,8 (4.59)
Now, taking (4.59) into account for Step 4 and Step 5 of Algorithm 4.1, the observer gains are obtained as N 1 =−5,J 1 =h
1 0 i Next, we found that the LMI conditions (4.29)-(4.30) hold for allθ∈[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 z 1 (t) = F 1 x 1 (t) x13(t).
Next, we consider the 2- subsystem: According to Step 1 of Algorithm 4.1, we obtain matricesX 2 andY2from equations (4.35)-(4.36) Since rank
By solving the LP problem (4.49) with the constraint 0.15≤λ 1 ≤2.5 we obtain a solution λ1= 0.15, Z1= 01,8 (4.60)
Now, taking (4.60) into account for Step 4 and Step 5 of Algorithm 4.1, the observer gains are obtained as N 2 = −6, J 2 = h
1 0 i 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) =F2x2(t) =x23(t).
We consider the input u1(t) =u2(t) = sint, 0 ⩽t ⩽100, the signalsd1(t), g1(t) and v1(t), d2(t), g2(t) andv 2 (t) are d1(t)=d2(t) "
# for 0⩽t⩽100 and the initial conditions arex11(0) = 1,x12(0) = 2,x13(0) = 3,x21(0) = 4,x22(0) = 5, x23(0) = 6,z − 1 (0) = 0,z 1 + (0) = 1,z 2 − (0) = 1,z 2 + (0) = 5 Figure4.1shows the responses ofx11(t),x12(t), x13(t), x21(t), x22(t) and x23(t) With the above data, Figure 4.2shows the response of z1(t) =x13(t), z − 1 (t),z + 1 (t), while Figure 4.3shows the response ofz2(t) =x23(t),z − 2 (t),z 2 + (t), respectively.
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, j6=i), where x T i (t) " x i1 (t) xi2(t)
In [17], the authors introduced a method for deriving a full-order interval observer for the state vectors of a single impulsive system However, they overlooked the design of functional interval observers and did not demonstrate how to select the matrices L and M to ensure that \( A - LC \) meets the necessary criteria.
Metzler andG−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 z i (t) =F i x i (t) =h
By following steps (Step 1-Step 4) of Algorithm 4.1we obtain
The LMI conditions (4.29)-(4.30) are satisfied for all values of \(\theta\) in the range \([0.41, \infty)\) Consequently, all requirements of Corollary 4.3.1 are met, allowing us to establish a first-order interval observer for estimating the function \(z_i(t)\) with \(F_i(t) = x_{i1}(t)\).
We consider boundsd − i (t), d + i (t), g − i (t), g i + (t) and Vi ared − i (t) "
# , for 0⩽t⩽100 andV i = 0.1 The initial conditions are chosen asx i1 (0) = 1, x i2 (0) = 2, z i − (0) = 0, z i + (0) = 2 Figure 4.4 shows the responses of x i1 (t) andx i2 (t) while Figure4.5 shows the response ofzi(t) =xi1(t),z i − (t),z i + (t).
In this chapter, we introduced a novel method for deriving interval observers for linear functions within large-scale network impulsive systems characterized by bounded uncertainties We outlined the conditions necessary for the existence of these interval observers and provided an effective algorithm for determining the unknown observer matrices To demonstrate the simplicity of our design method, we included two examples with simulation results However, it is important to note that this chapter does not address the impact of time delays in the state and output vectors when the system matrices of large-scale impulsive network systems are uncertain.
Figure 4.5: Responses ofzi(t) =xi1(t),zi −(t) andzi +(t)
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.
Theorem 2.1.1 plays a crucial role by converting a nonlinear time-delay model into a new observable format, facilitating the straightforward design of a third-order delayed state observer.
- Simulation results have been given to illustrate the effectiveness of our results (see Subsection 2.2.3).
The matrix pair (A, C) is unobservable, prompting the use of a transformation with a delayed state variable to revert to a system characterized by the observable matrix pair ( ¯A, C) Consequently, the designed state observer is capable of estimating a delayed version of the state vector rather than the instantaneous state vector.
• We have received the design of distributed functional observers to detect actuator faults of inter- connected 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 thei-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 algo- rithm 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).
In the future, we intend to continue the investigation in the following directions:
• Extend the results of Chapter 2 to design dynamic event-triggered state observers to estimate the state vectors of nonlinear time-delay systems.
This article extends the findings from Chapter 3 to address fault detection issues in interconnected time-delay systems It focuses on large-scale impulsive network systems characterized by uncertain system matrices and unknown time-varying delays present in the output vectors.
This article expands upon the findings of Chapter 4 by addressing large-scale impulsive network systems characterized by uncertain system matrices and the presence of unknown time-varying delays in both state and output vectors.
List of Author’s Related Publication
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3 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.
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