FAULT ESTIMATION AND OPTIMIZATION FOR UNCERTAIN DISTURBED SINGULARLY PERTURBED SYSTEMS WITH TIME-DELAY

. This paper presents a observer-based fault estimation method for a class of singularly perturbed systems subjected to parameter uncertainties and time-delay in state and disturbance signal with ﬁnite energy. To solve the estimation problem involving actuator fault and sensor fault for the uncertain disturbed singularly perturbed systems with time-delay, the problem we studied is ﬁrstly transformed into a standard H ∞ control problem, in which the performance index γ represents the attenuation of ﬁnite energy disturbance. By adopting Lyapunov function with the ε -dependence, a suﬃcient condition can be derived which enables the designed observer to estimate diﬀerent kinds of fault signals stably and accurately, and the result obtained by dealing with small perturbation parameter in this way is less conservative. A novel multiobjective optimization scheme is then proposed to optimal disturbance atten- uation index γ and system stable upper bound ε ∗ , in this case, the designed observer can estimate the fault signals better in the presence of interference when the systems guarantee maximum stability bound. In the end, the validity and correctness of proposed scheme is veriﬁed by comparing the error between the estimated faults and the actual faults.

certain performance constraints. In addition, the stability upper bound represented by ε * of singularly perturbed systems has attracted wide attention [10]- [5].
In the process of actual engineering system control, the occurrence of actuator faults and sensor faults and control components faults in dynamic systems could greatly reduce the control performance and make the system inoperable, such that the fault diagnosis problem of singular perturbation systems is widely concerned. In singularly perturbed systems, faults may also cross time scales due to the existence of small perturbed parameter, which increases the difficulty of fault diagnosis. At the beginning of the study, it can only detect whether the fault occurs or not without obtaining the information about the shape and magnititude [17,18], and then the conservative fault diagnosis can be realized along with the further research [19,20]. Recently, with the gradual improvement of the theoretical system, the research on fault estimation has yielded fruitful results [5], [9]- [27]. In observer-based fault diagnosis methods, a fault diagnosis method based on Proportional-Integral(PI) observer is proposed for systems with sensor and actuator faults in [9], and then the faulttolerant controller is designed based on optimal control theory. An observer-based fault estimation method is proposed for the Lipschitz nonlinear singular perturbation systems with sensor fault in [5], in which a optimal scheme is presented to estimate the maximum stability bound and the best uncertainty decay ability. A feasible scheme for estimating sensor fault vector by observer based on descriptor approach is adopted in paper [24]. Different from the above, the transient behavior of the system is improved to a satisfactory level by designing fault detection filters and pole assignment based on the finite frequency method in [1]. Generally speaking, the literature research systems discussed above are relatively single. However, for the fault estimation problem of uncertain time-delay singularly perturbed systems with faults and external disturbances, although it is significant, has not been researched yet.
In view of above, the fault estimation problem for a class of singularly perturbed systems with time-delay and uncertainties in the presence of disturbance is addressed in the paper. Inspired by the literature [5], a residual observer to exactly estimate actuator and sensor faults by taking into account the diversity of actual industrial system faults is designed. A multi-objective optimization scheme is then proposed and the stability upper bound and the H ∞ performance index of the system are optimized simultaneously, in this way, the designed observer can estimate the fault signals better in the presence of interference when the systems guarantee maximum stability bound. The final numerical example will demonstrated the feasibility and correctness of this method.
2. Preliminary knowledge and problem description. Consider a class of disturbed singularly perturbed systems with uncertainty and time-delay as follows where x(t) ∈ R n is the state vector of the system and satisfies n 1 + n 2 = n. u(t) ∈ R m , y(t) ∈ R p are the control input vector and measure output vector, respectively. w(t) ∈ R l represents finite energy disturbance signal. h ∈ 0,h is the system constant time-delay, and 0 < ε 1 is the perturbation parameter of singular perturbation system. ∆A and ∆A h are time-invariant matrices that describe the bounded uncertainty of the parameter where M, N, N h are known constant real matrices with appropriate dimensions. The uncertain matrix F (σ) is satisfied σ ∈ Θ, Θ is a dense set of R. Suppose given arbitrary matrix F : F F T ≤ I, It exists σ ∈ Θ such that F = F (σ). If 2 and 3 are both established, ∆A and ∆A h are said to be tolerable. The fault f (t) studied in this paper includes actuator fault f a (t) and sensor fault f s (t) Assuming the fault signals can be described by a dynamic system as the expression 4 is an outer system that describes the fault signals, ϕ(t) ∈ R (na+ns) is the system fault state vector, and The outer system 4 is a common expression for continuous faults, such as step faults, sinusoidal faults, attenuation faults, exponential faults and so on.
Set z(t) = x(t) ϕ(t) , an augmented system can be obtained wherē Some lemmas for subsequent proof references are given.

Lemma 2.3. [26]
If the fast subsystem and slow subsystem of system 1 are controllable and observable when the perturbation parameter is satisfied 0 < ε 1 , the system is said to be strongly controllable and observable.

Lemma 2.4. [30]
If the symmetric matrices P 1 , P 2 , P 3 , P 4 and the appropriate dimension matrix P 5 are satisfied then, the following formula holds. where Construct a residual observer as follows whereẑ(t),x(t),φ(t) represent the estimated value of the augmented system state vector z(t), the original system state vector x(t) and the fault state vector ϕ(t), respectively.L = L 1 L 2 ∈ R (n+na+ns)×p is the observer gain matrix to be solved.
Define the estimation error e z (t) = z(t) −ẑ(t). ∆Ā, ∆Ā h and w(t) are consistent with the uncertainty of the original system (1), and w(t) is also the anti-interference ability of the observer. The expression about estimation error can be got In order to realize the design goals of fault estimation and optimization, an observer is first designed and the input u(t) and output y(t) of the original systems are taken as the input vectors of the observer. The observer can accurately output the state vectors of the original system and outer system representing the fault as well as possesses the ability of anti-disturbance. Furthermore, the stability upper bound and the attenuation ability of finite energy disturbance are optimized jointly based on multi-objective optimization algorithm. Detailed description is as follows.
(1) Fault estimation 1) When the disturbance does not exist(w(t) = 0), the estimation error e z (t) gradually tends to 0, it is shown that the observer can estimate the state vectors of the original system and the external system representing the fault.
2) When the disturbance exist(w(t) = 0), the observer can accurately output the state vectors of the original system and the external system representing the fault as well as possesses the ability of anti-disturbance, the estimation error e z (t) satisfies H ∞ performance under zero initial conditions which can be converted to the following form where γ is the attenuation performance index of finite energy disturbance.
(2) Optimization A multi-objective optimization algorithm is proposed to optimize H ∞ performance index γ and system perturbation upper bound ε * , in which a compromise scheme can be found to ensure both larger stable upper bound and stronger antidisturbance capability.
Remark 2. In engineering practice, finding an accurate non-conservatively stable upper bound can ensure that the controller proposed in the singular perturbation systems (including the residual estimator) exists, and the increase in the stability upper bound indicates that there is greater opportunity to obtain an effective controller [5].
3. Residual observer design and optimization.

Design of Residual Observer.
Theorem 3.1. Assuming that prerequisites of Lemmas 2.1-2.2 are satisfied in system 1, given parameters γ > 0, 0 < ε * < 1, h ∈ (0,h] and Q = diag{Q 1 , Q 2 } > 0. The residual observer can be constructed by equation 8 if it exists symmetric matrices P, P 1 , P 2 , P 3 , P 4 and the appropriate dimension matrices P 5 , Y satisfying the linear matrix inequalities 6 and 12-13. And the estimation error 9 is robust stable and satisfies the H ∞ performance index, the gain matrix can be obtained bȳ Proof of Theorem 3.1. The system is strongly observable when the prerequisites in Lemmas 2.1-2.2 are satisfied, so the residual observer constructed by equation 8 exists. Then we demonstrate that the residual observer satisfies the design requirements of fault estimation. Consider the Lyapunov function as follows The following formula can be deduced from Lemma 2.4 obviously, V is positive definite when Q > 0 holds.
Derivative of the Lyapunov function, we can geṫ where Π = (Ā + ∆Ā −LC) TP T (ε) +P T (ε)(Ā + ∆Ā −LC) + Q. From Theorem 3.1 we know that the linear matrix inequality 17 holds By substitutingL =P −T (ε)Y into equation 17 and using Schur complement, linear matrix inequality 18 can be derived as which can be derived to so, the inequalityV < 0 holds when w(t) = 0, and the state estimation error e z gradually tends to 0. when w(t) = 0, the performance index function 9 is converted to the following form The above equation can be converted into The equations 19 and 17 are equivalent by substitutingL =P −T (ε)Y , which means that Ψ < 0 holds for every ε in the range of 0 to ε * , so J < 0.
To sum up, system 9 is robust stable and satisfies the H ∞ performance index. Proof is completed.

Multi-objective optimization.
A multi-objective optimization algorithm is proposed to optimize both the stability upper bound ε * and the H ∞ performance index γ of the system simultaneously. The system maximum stability upper bound can be expressed by the maximum perturbation upper bound ε * , while γ represents the strength of the damping capacity of the finite energy disturbance. Consequently, the optimization problem can be transformed into the following mathematical form min γ, max ε * s.t. 6,12,13 Optimization algorithm is as follows: Step1: Select the initial value of the stability bound ε * as ε 0 (0 < ε 0 1) and step length as d(0 < d 1). Step2: Set i = 1, ε 1 * =ε 0 , calculate the result at this time and write it as γ 1 : min γ s.t. 6, 12, 13 Step3: Step 2 loop and record the result as γ i . Then, define the function to evaluate the optimization performance where δ 1 , δ 2 are the weight of the evaluation function and satisfy the equation δ 1 + δ 2 = 1. And the values of weight could be got from practical requirements.
Step4: Go through all the combination solutions from 0 to 1, and find the value of i that corresponds to the smallest evaluation function f . ie i op = arg min f i Step5: The optimized combination is and the optimal observer gain matrixL can be obtained after optimization value is substituted into the system. 4. Numerical example. The parameters of an uncertain disturbed singularly perturbed system with time-delay are as follows: We select the initial value of the perturbation parameter and the step length as ε 0 = 0.01, d = 0.01. The weight coefficients data from paper [5] can be chosen as δ 1 = 0.2, δ 2 = 0.8 according to practical requirements.
The optimized perturbation parameter ε * opt = 0.10. Then the perturbation parameter are selected as ε = ε ver1 = 0.11(ε ver1 / ∈ (0, ε * ])and ε = ε ver2 = 0.09(ε ver2 ∈ (0, ε * ]) in order to verify the superiority of multi-objective optimization algorithms. The gain matrix of residual observer   Fig. 1 when the perturbation parameters are 0.11 and 0.10, respectively. The simulation curve shows that when the perturbation parameters ε = ε * = 0.10, the estimated fault basically coincides with the actual fault trajectory. However, when the value of the observer is not in the upper bound of maximum stability, ie ε = ε ver1 = 0.11(ε ver1 / ∈ (0, ε * ]), the deviation between the     Fig. 3, it is concluded that the output of the observer is difficult to describe the actual fault signal and does not meet the design requirements. Fig. 2 depicts the comparison between the estimated fault and the actual fault signals when the perturbation parameters are 0.09 and 0.10. It can be seen that the three curves basically coincide at this time. It shows that the observer designed by selecting the parameters within the maximum stability upper bound meets the requirements of fault estimation. The estimated error under the two values described in Fig. 4 shows that the order of magnitude of error is kept at 10 −3 and the difference between them is very small. It shows that the system has strong anti-disturbance ability at this time and the anti-disturbance ability of the two systems is almost same. Therefore, when the perturbation parameter is 0.10, the system has the maximum stability upper bound and the best anti-disturbance characteristic, which proves the correctness of this method. In addition, the state vector estimation error waveforms of the outer fault system and original system are respectively described in Figs. 9-10 when ε = ε * = 0.10. From the estimated error trajectories, it can be seen that the state vector curves tend to zero quickly and the fluctuation range is very small and the magnitude of the curve is even in the order of 10 −4 . It implies that the observer can accurately estimate the state vectors of the outer fault system and original system with better anti-disturbance characteristics.

5.
Conclusion. In this paper, the problem of optimal fault estimation for perturbed singular perturbation systems with actuator and sensor faults is considered. Firstly, H ∞ performance index is used to characterize the attenuation ability of finite energy disturbance by reasonably constructing augmented system. And then the sufficient condition is given base on Lyapunov stability theory, such that the observer can accurately estimate the state and fault signals of the system. Furthermore, the maximum stability upper bound and the best anti-disturbance capability are guaranteed based on the multi-objective optimization algorithm. Last but not the least, a numerical example is given to compare the fault output trajectories in different perturbation parameter values. When the perturbation parameter takes the maximum stable upper bound, the stability and anti-disturbance performance of the system are both optimal, which proves the correctness of the scheme.