Observer canonical form based robust fault detection and estimation for hyperbolic spatiotemporal dynamic systems

In this study, the authors propose a novel state and a fault estimation scheme for a class of hyperbolic spatiotemporal dynamic systems in the presence of unknown external disturbance. They consider the occurrence of multiplicative actuator and sensor faults. In detail, they consider two cases of fault occurrence: (i) only one type (actuator or sensor) of fault happens; (ii) two types of faults occur simultaneously. This study discusses the fault detectability conditions by proposing a fault detection observer. To complete the estimation problem, three difficulties arise: (i) no prior information shows the type of faults; (ii) the observer design is non-linear due to multiplication between plant signals (state or input) and unknown fault parameters; (iii) only one boundary measurement is available. They convert the original faulty plant into its observer canonical form. By proposing two filters based on the resulting observer canonical form, they develop novel parameter update laws for fault parameter estimation. With the proposed update laws, the true state of the faulty plant can be estimated by the proposed observers. By selecting appropriate Lyapunov functions, they prove that estimation error of state and fault parameters exponentially decays to an arbitrarily small neighbourhood of zero despite unknown external disturbance.


Introduction
Process monitoring, including fault detection and estimation, plays an important role in aspects of process safety. Existing fault detection techniques can be classified into three categories: a mathematical model based, data-driven empirical model-based, and combination of empirical models with prior information of plants. Data-driven methods refer to the identification filed [1][2][3][4]. The advantages of mathematical models and empirical models are the former has a better extrapolation and the latter is more convenient to design. Significant contributions to fault detection have been made by a lot of papers and books [5][6][7][8][9]. These contributions may include sliding-mode observer (SMO) [10], adaptive observer [11,12] and high-gain observer [13]. Recent works [14,15] proposed full-order and reduced-order descriptor by SMO to solve the robust fault estimation problem for Markov jump systems in the presence of unknown bounded external disturbances. Similarly, in [16], simultaneous actuator and sensor fault estimations were achieved for Markovian jump systems using linear matrix inequality (LMI) theory. Moreover, the stochastic process model was constructed to approximate the fault, and time to failure (TTF) was estimated in [17,18].
In practice, robust fault estimation is crucial to real-time monitoring, diagnosis and fault-tolerance control. To address the relevant problems, many contributions have made. In [19], an augmented system was first configured for the augmented state that includes the plant state and faults of interest, and then an unknown input observer (UIO) was proposed based LMI optimisation technique to the stability of estimation error dynamics and minimise the influence of disturbances. In [20], a novel robust estimator was developed based on the LMI technique and descriptor system theory to address the simultaneous state and fault estimation problem in discrete-time systems. Liu et al. [21] proposed an observer design for stochastic Takagi-Sugeno fuzzy systems suffering from process uncertainties, Brownian parameter perturbations, as well as unexpected faults. In particular, UIO techniques, augmented system theory and sliding mode control approaches were employed, and a robustly simultaneous full-plant state estimation and fault estimation were achieved. Recently, Yang and Yin [14,15] developed full-order and reduced-order descriptor SMO to solve the robustly simultaneous full state estimation and fault estimation problem for Markov jump systems in the presence of unexpected sensor and actuator faults and unknown bounded external disturbances.
Various industrial systems including fluid flows, thermal convection and chemical reaction processes exhibit spatiotemporal dynamics, namely states of systems are space varying and timevarying. Therefore, common temporal dynamical representation is unsuitable to demonstrate their behaviour [22] and the corresponding observer design is challenging [23]. Furthermore, it is more challenging to explore fault detection and estimation of spatiotemporal systems. Existing fault detection methods are mostly based on modal analysis technique. For example, Armaou and Demetriou [24] employed the modal analysis technique to approximate the original spatiotemporal system by a largedimensional temporal system, and developed a fault detection observer for the resulting approximated temporal system. Along the same line, Ghantasala and El-Farra [25] designed a robust observer to resolve fault detection and state estimation issues.
The techniques mentioned above utilised model approximation which leads to false and missed alarms caused by model reduction. In [26], adaptive spatiotemporal observers were designed for positive systems to achieve fault detection. In [27], the design of functional observers for fault detection was extended to Rieszspectral systems. In [28], conventional Luenberger observer was divided into generating two subsystems based on superposition principle and then adaptive parameter update laws were developed and embedded into the subsystems to complete state and boundary fault estimation for hyperbolic systems without exogenous disturbance.
This paper proposes the development of simultaneous state and fault estimation schemes for linear hyperbolic spatiotemporal systems in the presence of unknown disturbances based on the observer canonical form of the plants. Different from [29], where the additive fault was considered and some real positive conditions need to be met, Multiplicative faults are taken into account in this paper and there is no any special restriction. A non-linear problem arises due to the multiplication between unknown fault parameters IET Cyber-syst. Robot and plant input/state. In particular, the estimation problem becomes more challenging and intractable when both actuator and sensor faults happen to the plant simultaneously.
To overcome these difficulties, our contributions are made as follows: • Simultaneous state and fault estimation problems are tackled with respect to two cases: (i) only one type (sensor or actuator) of fault occurrence; (ii) simultaneous sensor and actuator fault occurrence. • The plant is converted into its observer canonical form and two filters are proposed to filter actuator input and sensor output and to account for solutions to observer canonical form. By designing appropriate Lyapunov candidates, we develop novel adaptive update laws (AULs) to estimate fault parameters that will be incorporated with the proposed filters. • By constructing different Lyapunov functions, it is proved that estimates of state and fault converge to arbitrarily small neighbourhoods of their true values exponentially despite unknown external disturbances.
The paper is organised as follows: In Section 2, the hyperbolic spatiotemporal plant and multiplicative actuator/sensor faults are introduced. Section 3 explores the fault detectability and provides time-varying fault detection threshold. More importance is the observer canonical form-based design of fault parameter update laws and state observer corresponding to different fault occurrences. In Section 4, illustrative numerical simulation is carried output to present the effectiveness of theoretical results.

Notation
For the vector signals v(z) for all z ∈ [0, 1], the following integral operator is introduced: with the derived norm The operator (1) is linear and has the following property: where v z denotes the derivative of v(z) w.r.t the spatial variable z. In the subsequent sections, we will write ∂ t v(z, t) (or v t (z, t)) and ∂ z v(z, t) (or v z (z, t)) to represent ∂v(z, t)/∂t and ∂v(z, t)/∂z, respectively. The norm ∥ v ∥ α is equivalent to the standard L 2 -norm, in the sense that there exist positive constants k 1 , k 2 so that For brevity, we will in subsequent sections often omit writing the argument in time and space, so that, e.g. v(z) = v(z, t) or v = v(z, t), and ∥ u ∥ = ∥ u(t) ∥.

Problem formulation
We consider the following hyperbolic spatiotemporal dynamical systems: for all time variable and space variable t ∈ ℝ + , z ∈ 0, 1 , where g, f and h stand for system coefficient functions that are realvalued continuous functions, v(z, t) ∈ L 2 (0, 1), ∀(z, t) ∈ [0, 1] × ℝ + stands for the distributed state of the plant, L 2 (0, 1) is a real Hilbert space equipped with the inner product v 1 , v 2 = ∫ 0 1 v 1 (z)v 2 (z) dz, ∀v 1 , v 2 ∈ L 2 (0, 1), v(0, t) is system output and sensor input, v(1, t) is system input and actuator output, u(t) denotes actuator input, and y(t) is the sensor output (measurement), as shown in Fig. 1. ϕ(t) represents an external unknown disturbance which satisfies the following assumption: Assumption 1: The disturbance ϕ(t) is bounded by a known positive constant φ: ϕ(t) ≤ φ, ∀t ∈ ℝ + (6) As depicted in Fig. 1, the relation between system boundary conditions and sensor output (actuator input) before and after fault occurrences (at t = T o ) can be expressed as follows: Sensor fault where θ s denotes a positive sensor fault parameter and θ a represents an actuator fault parameter, i.e. θ s ∈ ℝ + and θ a ∈ ℝ. In practice, when designing a state observer for the faulty plant, the knowledge of boundary conditions v (1, t) and v(0, t) is not available due to the prevention of unknown fault parameters θ a and θ s . Only the actuator input u(t) and sensor output y(t) can be used.
In this work, the investigation of fault detectability is one objective and simultaneous state and fault estimation is another objective in the presence of the following fault occurrences (also see Fig. 1): [p1] Only the actuator fault (7) occurs at time t = T o .
[p2] Only the sensor fault (8) happens at time t = T o .
[p3] Both sensor and actuator faults simultaneously occur at time Before dealing with the problems, for the plant (5), we make the following assumptions [30]: Assumption 2: : Define the triangles T L = z, ζ ∈ 0, 1 × 0, 1 , z ≥ ζ T U = z, ζ ∈ 0, 1 × 0, 1 , z ≤ ζ and the spaces V L = C T L ; ℝ and V U = C T U ; ℝ . Accordingly, the norms are defined by then the coefficient functions in (5) satisfy: g ∈ C([0, 1]; ℝ), f ∈ V L and h ∈ V U . Additionally, functions g, f and h satisfy: 25. In particular, if g(z) ≡ 0, then the coefficients f and h satisfy: We make Assumption 2 to separate the influence of faults from the instability of the plant. For fault detection and estimation, the closed-loop system should be stable.
Assumption 3: We assume that fault parameters are timeinvariant and bounded by where θ¯a m > 0, θ sl > 0 and θ¯s m > 0 are known constants.

Fault detection and estimation
In practice, only the sensor output y(t) is considered measurable. We first convert the plant (5) into its observer canonical form by applying the following backstepping transformation and invertible bounded transformation: and with k o , p o ∈ V L and l o , q o ∈ V U , where I L 2 is the identity operator on L 2 (0, 1) and the operator ℚ ψ 1 , ψ 2 is given by (14) for all ζ ∈ L 2 (0, 1), and z ∈ [0, 1] provided ψ 1 ∈ V L and ψ 2 ∈ V U . In (11), , and Φ(z) and Ψ(z) provide an additional degree of freedom required to make the transformation successful and will be calculated in the following.
Here, kernels k o , l o have to satisfy the following set of kernel equations: with boundary conditions From (13) and (17), it follows that: As a consequence, during the transformation procedure from the original plant to the observer canonical form, the auxiliary terms Φ(z) and Ψ(z) are computed as Due to the boundedness of transformations in (12) and (13), Under normal operating conditions where θ a = θ s = 1, we consider u(t), v o (0, t) and v o (1, t) as external inputs of a linear spatiotemporal system (11). According to the superposition principle, the solution to (11) can be solved by summing the response of the subsystems driven by each external input. Consequently, the solution v o (z, t) to (11) in the absence of exogenous disturbance ϕ(t) is computed by combining the solutions to the following subsystems: and One can easily obtain the explicit solutions to the subsystems (20)- Inspirited by and based on the above results on the observer canonical form, we first briefly introduce different observer configurations using Λ(z, t) and Γ(z, t) according to different fault occurrence analysis.
Here v^o(z, t) and ŷ(t) represent estimates of v o (z, t) and y(t), respectively. e y (t) denotes the detection residual used to indicate the fault occurrence. c2 (Actuator fault estimation): To address the problem [p1], the observer is designed as where θ^a(t) is the estimate of θ a and is generated by an AUL that will be designed latter.

c3 (Sensor fault estimation):
To solve the problem [p2], the observer is constructed as where θ^s(t) denotes the estimate of θ s and is generated by an AUL that will be proposed.

c4 (Simultaneous fault estimation):
To tackle the problem [p3], the proposed observer is configured

Fault detection
During normal operating situations with θ a = θ s = 1, if we employ (26) as the fault detection observer and define the state residual: , then ṽ o (z, t) satisfies the following system: and the corresponding detection residual is given by Let us choose a Lyapunov function candidate with a constant α > 1 (to be discussed). The first derivative of V(t) with respect to time becomes By inserting (30) into the above V˙(t) and taking integration by parts, we get with T¯ϕ defined in Remark 1. Based on Young and Cauchy-Schwarz inequality, we have Therefore, we finally have where Therefore, V(t) is exponentially decreasing and there exists a finite time Therefore, from the definition in (2), we have e y (t) = ṽ o (0, t) ≤ κ l T¯ϕ with κ l = 2/(α − 1). In conclusion, when under healthy condition θ a = θ s = 1, the detection residual e y (t) stays within [0, κ l T¯ϕ] if we employ the observer (26) as the fault detection observer. Then, it is reasonable to choose ∥ e y (t) ∥ ≤ ϑ → no faults happen ∥ e y (t) ∥ > ϑ → at least one fault happens where ϑ is a threshold which is set artificially for fault detection, one can determine whether the system is affected by a fault. A fault is detected once the detection residual e y (t) exceeds the predefined threshold ϑ > 0. The fault detectability condition is discussed in the next theorem.
Proof: From (35), it is straightforward to obtain the boundedness of the detection residual e y (t). In the following, we mainly discuss the conditions for fault detectability.
Sensor fault: Now we turn to the case of a sensor fault (θ s ≠ 1). (z, t), ∀z ∈ [0, 1] and thus the detection residual is e y (t) = δv o (0, t). The dynamics of δv o (z, t) is given by

Let us define a new state variable
Then, we consider a new system Let η(z, τ) = δv o (z, τ) at time τ > T o . Consequently, we have δv o (z, t) ≥ η(z, t) for z ∈ [0, 1] and t ≥ τ. Moreover, η(0, t) − ℋ < κ l T¯ϕθ¯s m is ultimately bounded. In other words, for any 0 < Δh < ℋ, there exists a time τ d > τ so that Then, when Δh < ℋ − ϑ − κ l T¯ϕθ¯s m , there exists a time τ d one has e y (t) = δv o (0, τ d ) > ϑ, ensuring the detection of a sensor fault. □ During normal operating situations with θ a = θ s = 1, the estimation error ṽ o (z, t) always satisfies (30) and (31). By utilising the method of characteristics, the analytical solution to (30) and (31) is given by and hence the detection residual e y (t) = ṽ o (0, t) can be written as Apparently, the detection residual e y (t) will not be zero unless ṽ o (z, 0) and ϕ(t) are zero. When the initial state ṽ o (z, 0) is unknown, then e y (t) ≠ 0 may exceed the predefined threshold ϑ defined in (36) and hence the large value of the residual e y (t) will yield a false alarm in the absence of faults. In the following, a new time varying threshold will be defined to avoid false alarms. Lemma 1: Consider the plant (5): v − system, whose equivalent system is v o −system defined in (11), with an exogenous disturbance ϕ(t) with a known bound φ. Then, the detection observer defined in (26) has the following properties: and a fault will be reported if the detection residual e y (t) exceeds the threshold ϑ(t), where T¯ϕ is defined in Remark 1.
The information required to estimate the plant state in the presence of faults is the knowledge of fault parameters. In the following, we turn to the estimation of time-invariant parameters θ a and θ s corresponding to Problems

Single type of fault estimation
In this subsection, we consider estimation Problems [p1] and [p2]. AULs will be developed to generate fault parameter estimates θ^a(t) and θ^s(t) that will be incorporated into the observers defined in (27) and (28). The output residual e y (t) will be different: for actuator fault estimation [p1] e y (t) = v o (0, t) − v^o(0, t); for sensor fault estimation [p2] e y (t) = θ s v o (0, t) − θ^s(t)v^o(0, t). As shown in Assumption 3, θ a and θ s are bounded by (9)  boundedness must be ensured when designing the parameter update laws for θ^a(t) and θ^s(t).

Theorem 2: (State and fault estimation):
Suppose that a single fault occurs and the fault type is known in advance. For actuator fault estimation problem [p1], we apply the observer (27) and the following parameter update law:

as well as v(z, t) in the original plant) and output y(t).
For sensor fault estimation problem [p2], we use the observer (28) and the updated law to estimate the plant state. The projection operators are defined in (95) and (103). Then, the estimates θ^a(t) and θ^s(t) satisfy (9) and (10), namely The parameter estimation errors θ a (t) = θ a − θ^a(t) and θ s (t) = θ s − θ^s(t) converge to arbitrarily small values in the neighbourhood of zero. The state estimation error 1] converges to an arbitrarily small neighbourhood of zero, and therefore ṽ(z, t) = v(z, t) − v^(z, t) in z ∈ [0, 1] converges to an arbitrarily small neighbourhood of zero.
For an actuator fault, a new error signal is defined as and for a sensor fault, the variable is defined as The error signal in both cases satisfies μ t (z, t) = μ z (z, t) + T (ϕ(t)), μ(1, t) = 0 (55) (a) Actuator fault: Define a Lyapunov function candidate where the design parameters α, ε and η are positive constants. By differentiating (56) and substituting the dynamics (55), we calculate where T (ϕ(t)) ≤ T¯ϕ. Using property (3) and Young's inequality, inserting the parameter update law (49), and applying the property (98) in the Appendix yield where θ a = − θ^a is used, since θ a = θ a − θ^a and θ a is assumed to be time invariant. Moreover, the inequality was applied. By combining (27) and (53), in the case of an actuator fault, we have Using (59) to replace θ a G Γ(t) (0) in the above inequality, inserting θ^a = θ a − θ a and applying Cauchy-Schwartz inequality yield with the positive constant c 1 = (1/2) min 2α − 1 , γ and where we choose the design parameters satisfying In (61), if we choose a large η, a small γ and a large ε, then K 1Δ becomes arbitrarily small in a neighbourhood of the origin. From the inequality (60), if the function V 1 (t 0 ) > K 1Δ /c 1 and V 1 (t) > K 1Δ /c 1 for all t > t 0 , then V˙1(t) ≤ − c 1 V 1 (t), which implies V 1 (t) ≤ V 1 (t 0 )e −c 1 t . Therefore, V 1 (t) is exponentially decreasing. Then, after a finite time t f = (1/c 1 )ln c 1 V 1 (t 0 )/K 1Δ , we get V 1 (t) ≤ K 1Δ /c 1 , for all t > t f . As a conclusion, by choosing appropriately the values of positive design parameters γ, β, η, α and ε, one can arbitrarily reduce the value of K 1Δ , μ(z, t) and θ a (t) will exponentially converge to the neighbourhood N r θ a (0) with the value of r θ a > 0 proportionally depending on K 1Δ . Therefore, from (27) and (53), the estimate v^o(z, t) converges to the neighbourhood N r v v o (z, t) along the convergence of μ(z, t) and θ a (t). From the bounded transformation from v^o(z, t) to v^(z, t) defined in (12) and (13), the estimate v^(z, t) converges to the neighbourhood N r v v(z, t) .
(b) Sensor fault: Consider the Lyapunov candidate with positive design parameters α, ε and η. By taking the derivative of V 2 w.r.t time and plugging in the dynamics (55), we have Using property (3) and Young's inequality, inserting the parameter update law (50), and using the property (106) in the Appendix yield where θ s = − θ^s is used, since θ s = θ s − θ^s and the sensor fault parameter θ s is assumed to be time invariant. By combining (28) and (54), in the case of a sensor fault, we have Inserting (66) to replace θ s (t)G Γ(t) (0) and applying Cauchy-Schwartz inequality yield with the positive constant c 2 = (1/2) min γ, 2α − 1 and where the design parameters are chosen such that Based on the inequality (67) and for V 2 (t 0 ) > K 2Δ /c 2 and V 2 (t) > K 2Δ /c 2 , ∀t > t 0 , we have V˙2(t) ≤ V 2 (t 0 )e −c 2 t , which implies that V 2 (t) is decreasing exponentially and there exists a finite time Furthermore, we can reduce K 2Δ to an arbitrarily small value by selecting appropriate values of design parameters α, η, β, γ and ε. μ(z, t) in z ∈ [0, 1] and θ s (t) decay to the neighbourhoods N r μ (0) and N r θ (0), respectively. The values of r μ and r θ can be reduced arbitrarily by reducing K 2Δ . From (28) and (54), we conclude that the state estimate v^o(z, t) converges to a neighbourhood N r v (v o (z, t)) with very small r v > 0. From the bounded transformation from v^o(z, t) to v^(z, t), the estimate v^(z, t) converges to the neighbourhood N r v v(z, t) . □ Remark 2: In order to prevent θ^a(t) and θ^s(t) to increase unboundedly, we have used the so-called γ −modification technique, which consists in adding the term −γθ^a(t) in (49) and −γθ^s(t) in (50). The projection operator denoted by 'proj' defined in (95) or (103) is applied to ensure that the fault parameter estimate θ^a(t) or θ^s(t) stays within the specified bound, i.e. θ^a(t) ∈ (0, θ¯a m ] and θ^s(t) ∈ [θ sl , θ¯s m ].

Simultaneous actuator and sensor faults estimation
In this subsection, we focus on the estimation problem [p3] introduced in Section 2. , it requires the development of coupled parameter update laws to simultaneously estimate θ a and θ s . We utilise the observer (3) to estimate the state in the presence of actuator and sensor faults, where the update laws for θ^a(t) and θ^s(t) are given in the following theorem.
Theorem 3: (Simultaneous fault estimation): Suppose that both actuator and sensor faults simultaneously happen to the plant (5) as sensor fault parameter θ s ; for sensor fault estimation problem [p2], θ^s(t) will converge around 'θ s = 1', and θ^a(t) converges around the true actuator fault parameter θ a .

Simulation results
In this section, we will study the performance of the proposed methods via simulation corresponding to different fault occurrences [p1], [p2] and [p3] introduced in Section 2.
As an example, we utilise the system in the form of (5) given in [30] as a representative. The functions in the system are defined by with parameters a = 1.25, b = c = 0.1 and d = 10. Suppose that the boundary control input is given as where U s is a constant input U s = 50, and the second term as full state feedback is employed to stabilise the closed-loop system. p s (z, ζ) is the solution to the kernel (14) in [31]. Moreover, the initial condition is v(z, 0) = 18sin(2πz) for z ∈ [0, 1]. The faults are defined as follows:  Table 1. In Table 1, '✓' indicates that the fault happens and '-' means that the fault does not happen.

Fault detection
Based   In addition, fault occurrence time and fault detection time are listed in Table 2. In Table 2, by comparing actuator fault detection and sensor fault detection, we can find that the fault detection observer was more sensitive to a sensor fault. This is because more time was required to transport the influence of actuator fault to the measurement, but the influence of sensor fault can be measured by the output immediately.  Table 3.  Table 1. By applying the proposed observer (27) and update laws (49), the output estimate y^(t) = v^(0, t) (blue dash line) follows the measurement y(t) (solid line) as shown in Fig. 5 and the parameter estimate θ^a(t) converges around the true value of θ a shown in Fig. 2. We also apply the fault detection observer (26), but the output estimate ŷ x (t) (black dash line) diverges away from the measurement in Fig. 5. In particular, the mean square error (MSE) of the output estimate is calculated as a performance metric to show the effectiveness of the proposed methods.

State estimation with sensor fault:
In this subsection, we study the performance of the proposed observer (28) and the AUL (50). We simulate the same system in the presence of fault    Table 3. Since ϕ 2 is different from ϕ 1 , the profile output y(t) in Fig. 6 is different from the output in Fig. 5. The comparison between two estimates ŷ x (t) and ŷ(t) = θ^s(t)v^(0, t) is shown in Figs. 6 and 3, where ŷ x (t) is generated by fault detection observer (26) and ŷ(t) is generated by the proposed observer (28). Apparently, ŷ(t) (blue dash line) follows the measurement and the corresponding residual e y (t) in Fig. 3 converges to a small value around zero. However, the fault detection observer (26) does not provide satisfactory results.

State estimation with simultaneous faults:
In this subsection, we carry out the system simulation in the presence of fault occurrence [p3]: both actuator and sensor faults occurred at T o = 5 s and maintained constant after t = 8 s. The exogenous disturbance ϕ(t) = ϕ 3 defined in Table 1. We employ the proposed observer (29) and coupled parameter update laws (70) and (71). It is expected that the state estimate v^(z, t), parameter estimate θ^a(t) and θ^s(t) converge to arbitrarily small neighbourhoods of v(z, t), θ a and θ s simultaneously. Results shown in Figs. 4, 7 and 8 directly reflect the effectiveness of the proposed methods. The observation error ṽ(0, t) = v(z, t) − v^(z, t) and output residual e y (t) generated by the proposed observer (29) decay to very small values in a neighbourhood of zero, see the blue dash line in Fig. 4 and the left subfigure in Fig. 7. However, the fault detection observer (26) fails to provide acceptable estimation results shown by the black dash line in Fig. 4 and the right subfigure in Fig. 7.

Conclusions
We presented a robust observer design for a class of linear hyperbolic spatotemporal systems with separate and simultaneous faults and unknown persistent disturbances. The proposed observer using the only available boundary measurement achieves both state estimation and fault parameter estimation based on the observer canonical form. The main result in Section 3 was stated to show the detailed observer design, and we also associated parameter update laws to estimate faults and reject the influence of the disturbance. Through numerical simulations, we showed that despite the unknown exogenous disturbances, the proposed observer always ensures arbitrarily small state and fault parameter estimation errors. The advantages of the proposed methods include convenient design and strong portability. It has the potential to extend the results developed in this work to more complex spatiotemporal systems as long as their observer canonical forms can be obtained. In this paper, we have achieved that the fault parameter and state estimate errors are ultimately bounded. However, it is more challenging and interesting to achieve the exponential convergence of these estimation errors to zero. Future works may include the observer design for coupled systems with the faulty measurement directly affected by the feedthrough faulty input.