# Investigation of Unmeasured Parameters Estimation for Distributed Control Systems

Authors: Hao Wang; Shousheng Xie; Weixuan Wang; Lei Wang; Jingbo Peng
Journal: Complexity
 (2020)

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## Abstract

The problem of unmeasured parameters estimation for distributed control systems is studied in this paper. The Takagi–Sugeno fuzzy model which can appropriate any nonlinear systems is employed, and based on the model, an observer-based fuzzyH∞ filter which has robustness against time-delay, external noise, and system uncertainties is designed. The sufficient condition for the existence of the desired filter is derived in terms of linear matrix inequalities (LMIs) solutions. Moreover, the underdetermined estimation problem in which the number of sensors available is typically less than the number of state variables to be estimated is specifically addressed. A systematic method is proposed to produce a model tuning parameter vector of appropriate dimension for the estimation of the filter, and the optimal transformation matrix is selected via iterative solution to minimize the estimated error. Finally, a simulation example for turbofan aeroengine is given to illustrate the effectiveness of the proposed method, and the estimated error is less than 2.5%.


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## Body

## 1. Introduction

Along with the continuous development of computer and network technology, distributed control systems (DCSs) have gradually become a new trend and attract much attention [1, 2]. Compared with traditional point-to-point control systems, DCSs have advantages such as high reliability, reduced weight, low cost, and ease of maintenance [3, 4]. The DCSs provide the low-level processing function via intelligent unit and are more conductive to the implementation of complex control algorithms [5, 6]. However, the signal of DCSs is transmitted via bus network, which brings new challenges, such as networked-induced time-delay, packet dropout, and packet disordering.In practice, many parameters are difficult to obtain due to the limitation of the system itself (for example, the sensors cannot be installed in poor working conditions) or cost constraints. Thus, the estimation of unmeasured system parameters is significantly important, since the estimated parameters can not only be used in the design of controller but also for online monitoring [7, 8]. For the estimation problem of unmeasured system parameters, there are two main methods: (1) high-precision mathematical model [9]; (2) filtering technique and its extension methods [10]. The first method is to establish the mathematical model based on the structure of system to solve the unmeasured parameters. However, the modeling process is often cumbersome and complicated. And for some systems, the structure is extremely complicated and cannot be accurately modeled. Moreover, since this method generally requires iterative calculation, the amount of calculation is large.For the second method, the estimation of unmeasured parameters is based on the value of measurable parameters using the filtering technique. Because this method uses a recursive algorithm, it is easy to implement on computer and meet the requirements of accuracy and real-time. Among the filters, theH∞ filter stands out because it has robust stability against external noise without priori knowledge of noise and precise mathematical model [11–13].However, in some cases, the number of sensors available is typically less than the number of state variables to be estimated, which is an underdetermined estimation problem and cannot be solved directly. A common approach to address this shortcoming is to estimate a subset of the unknown parameters and assuming that others remain unchanged [14–16]. Although this approach enables online filter-based estimation, it will introduce error in the accuracy of overall estimation application. If any of the parameters which are assumed unchanged moves away from their nominal values, the estimates can no longer represent the true parameters. Litt [17] presented a novel approach based on singular value decomposition (SVD) that selects a model tuning parameter vector of low enough dimension. The model tuning parameter vector is constructed as a linear combination of all unmeasured parameters and the transformation matrix is generated by selecting the k most significant terms of singular values, where k is the number of available sensors. Similarly, the SVD is also used in [18] to determine which parts of the systems are observable if the whole system is unobservable. However, the unaccounted terms of singular values may also have great importance on the accuracy of overall estimation application even if they are small under some circumstances.Furthermore, in [19], the transformation matrix is selected to minimize the theoretical mean squared estimation error at a steady-state open-loop linear design point. But the estimation results are affected by the parameter perturbation, and there are many limitations in application. Moreover, all the approaches above do not consider the impact of time-delay and uncertainties of the system. Thus, it has great significance and practical value to propose a novel approach for the underdetermined estimation problem.Besides, due to the strong nonlinear fitting characteristics of complex nonlinear systems, the Takagi–Sugeno (T-S) fuzzy model has been widely used in the study of nonlinear systems since it was proposed [20–22]. Thus in this paper, an observer-based fuzzy H∞ filter is constructed, and the unmeasured parameters estimation problem for T-S fuzzy distributed control systems with time-delay and parameters perturbation is studied.The contributions of this paper can be concluded as follows: (i) a model tuning parameter vector of appropriate dimension is produced for the estimation of the filter; (ii) a systematic method is proposed for the selection of optimal transformation matrix to minimize the estimated error via iterative solution; (iii) the parameter perturbation is considered, so that the designed filter has a certain range of margins of disturbance.The remainder of this paper is organized as follows. Section2 introduces the modeling method of DCSs and the observer-based fuzzy H∞ filter. The main results are presented in Section 3, including the analysis and synthesis of the filtering error system.Section 4 presents the approach of optimal transformation matrix selection. Numerical simulation results are shown in Section 5 and finally conclusion and a discussion of further application of the method are presented.


## 2. Problem Formulation

Consider the DCSs with time-delay, which can be described by a class of T-S fuzzy model as follows:Plant rulei: If f1t is Φ1i and ⋯ fgt is Φgi, then(1)x˙t=Ai+ΔAixt+Bi+ΔBiut−τt+Liht+E1i+ΔE1iwt,yt=Ci+ΔCixt+Di+ΔDiut−τt+Miht+E2i+ΔE2iwt,zt=Fi+ΔFixt+Gi+ΔGiut−τt+Nihtxt=ϕt,t∈−τm,0,where Ωjii=1,…,r;j=1,…,g denotes the fuzzy set, r denotes the number of IF-THEN rules, and fjt denotes the premise variable.xt∈ℝn is the vector of state variables; ut∈ℝm is the vector of control inputs; yt∈ℝp is the vector of measured outputs; zt∈ℝl is the vector of unmeasured outputs and wt is noise signal which belongs to L20,∞. Ai,Bi,Ci,Di,E1i,E2i,Fi,Gi,Li,Mi,Ni are matrices with appropriate dimension. ΔAi,ΔBi,ΔCi,ΔDi,ΔE1i,ΔE2i,ΔFi,ΔGi are unknown matrices which represent the time-varying uncertainties of the system. ht∈ℝq is the health parameters of system, which represents the physical characteristics of each component.Remark 1.
The health parameters are considered in this paper, because the performance degradation of each component in the system is inevitable during the working process. And the system’s performance is affected by the level of degradation, which is generally described in terms of unmeasured health parameters such as efficiencies and capacities related to each major module in most cases. If the health parameters move away from their nominal values, the shift in other performance variables will be induced. They may be treated as a set of biases and can be augmented to the system states.
It is assumed that both sensors and actuators are time-driven. The data has timestamp and is transmitted in a single-packet, and incorrect order of the data packet does not exist. Since time-delay is related to the bus load at a certain moment, it is assumed that the time-delay has an upper bound, which isτt≤τm (under the above assumption, the upper bound of time-delay can be estimated; see [23] in detail). As a consequence, τt can be modeled as a finite state Markov stochastic process on a finite set Λ=1,2,…,τm. The transition probability from τt=i at time t to τt=jj≠i at time t+Δt is(2)Prτt+Δt=jτt=i=πijΔt+οΔt,i≠j,1+πijΔt+οΔt,i=j,where Δt>0 and limΔt⟶0οΔt/Δt=0. πij≥0 is the transition probability rates from τt=i at time t to τt=jj≠i at time t+Δt, and there is ∑j=1,j≠iτmπij=−πii.
The state feedback controller is applied in this paper, which isut=Kcxt. Then, by fuzzy blending, the global fuzzy augmented model can be obtained as follows:(3)x˙ht=∑i=1rhiftAhi+ΔAhixht+Bhi+ΔBhixht−τt+Eh1i+ΔEh1iwt,yt=∑i=1rhiftChi+ΔChixht+Dhi+ΔDhixht−τt+E2i+ΔE2iwt,zt=∑i=1rhiftFhi+ΔFhixht+Ghi+ΔGhixht−τt,xht=ϕt0,t∈−τm,0,where(4)xt=xtht,Ahi=AiLi0I,Bhi=BiKc000,ΔAhi=ΔAi000,ΔBhi=ΔBiKc000,Eh1i=E1i0,Chi=CiMi,ΔChi=ΔCi0,ΔEh1i=ΔE1i0,Dhi=DiKc0,ΔDhi=ΔDiKc0,Eh2i=E2i,ΔEh2i=ΔE2i,Fhi=FiNi,Ghi=GiKc0,ΔFhi=ΔFi0,ΔGhi=ΔGiKc0,hift=ϑift∑i=1rϑift,ϑift=∏j=12Ωjift.
For the underdetermined estimation problem, a low-dimensional model tuning parameter vectorqt is produced to represent the information of high-dimensional health parameter vector ht. The model tuning parameter vector qt is constructed as a linear combination of all health parameters, given by(5)qt=V∗ht,where ht∈ℝq,qt∈ℝk,k<q, and V∗ is a k×q transformation matrix, which is applied to construct the tuning parameter vector. The selection of optimal V∗ is obtained in Section 4.
Thus, the estimation of the health parametersh^ can be obtained as(6)h^=V∗†q,where V∗† is the pseudoinverse of V∗.
The final augmented dynamic fuzzy model can be rewritten as(7)x˙qt=∑i=1rhiftAqi+ΔAqixqt+Bqi+ΔBqixqt−τt+Eq1i+ΔEq1iwt,yt=∑i=1rhiftCqi+ΔCqixqt+Dqi+ΔDqixqt−τt+E2i+ΔE2iwt,zt=∑i=1rhiftFqi+ΔFqixqt+Gqi+ΔGqixqt−τt,xqt=ϕt0,t∈−τm,0,where(8)xqt=xtqt,Aqi=AiLiV∗†0I,Eq1i=E1i0,ΔAqi=ΔAi000,Bqi=BiKc000,ΔBqi=ΔBiKc000,ΔEq1i=ΔE1i0,Cqi=CiMiV∗†,ΔCqi=ΔCi0,Dqi=DiKc0,ΔDqi=ΔDiKc0,Eq2i=E2i,ΔEq2i=ΔE2i,Fqi=FiNiV∗†,Gqi=GiKc0,ΔFqi=ΔFi0,ΔGqi=ΔGiKc0.Remark 2.
If the model tuning parameter vector is of appropriate dimension, the augmented system (7) is observable and the observer-based filter can be used to estimate the unknown health and performance parameters. And the key of the estimation is to find the optimal transformation matrix so that the low dimension tuning vector can represent as much of the information as possible. Moreover, different from the method in [17–19], the solution of the optimal transformation matrix is transformed into an optimization problem.
It is assumed that the uncertainties of the system can be described in the following form:(9)ΔAqiΔBqiΔEq1iΔFqiTΔCqiΔDqiΔEq2iΔGqiT=U1iU2iSiV1iV2iV3iV4i,where U1i,U2i,V1i,V2i,V3i,V4i are known matrices and Si is a time-varying unknown matrix which satisfies(10)SiTSi≤I.
Then the following observer-based fuzzy filter is constructed:(11)x^˙t=∑i=1rhiftAqi+ΔAqix^t+Bqi+ΔBqix^t−τt+Kiyt−y^t,y^t=∑i=1rhiftCqi+ΔCqix^t+Dqi+ΔDqix^t−τt,z^t=∑i=1rhiftFqi+ΔFqix^t+Gqi+ΔGqix^t−τt,where x^t is the filter state, and y^t and z^t are the filter outputs. Ki is the filter parameter to be determined.
The estimation error of state is(12)x˜t=xqt−x^t=Aqi−KiCqi+ΔAqi−KiΔCqix˜t+Bqi−KiDqi+ΔBqi−KiΔDqix˜t−τt+Eq1i−KiE2i+ΔEq1i−KiΔE2iwt.
Defineξt=xqTx˜TtT and et=zt−z^t, and the filtering error system is given by(13)ξ˙t=∑i=1rhiftA¯i+ΔA¯iξt+B¯i+ΔB¯iξt−τt+E¯i+ΔE¯iwt,e˙t=∑i=1rhiftF¯i+ΔF¯iξt+G¯i+ΔG¯iξt−τt,where(14)A¯i=Aqi00Aqi−KiCqi,B¯i=Bqi00Bqi−KiDqi,ΔA¯i=ΔAqi00ΔAqi−KiΔCqi,E¯i=Eq1iEq1i−KiE2i,ΔB¯i=ΔBqi00ΔBqi−KiΔDqi,ΔE¯i=ΔEq1iΔEq1i−KiΔE2i,F¯i=0Fqi,ΔF¯i=0ΔFqi,G¯i=0Gqi,ΔG¯i=0ΔGqi.
Define the following parameters:(15)A¯=∑i=1rhiftA¯i,ΔA¯=∑i=1rhiftΔA¯i,B¯=∑i=1rhiftB¯i,ΔB¯=∑i=1rhiftΔB¯i,E¯=∑i=1rhiftE¯i,ΔE¯=∑i=1rhiftΔE¯i,F¯=∑i=1rhiftF¯i,ΔF¯=∑i=1rhiftΔF¯i,G¯=∑i=1rhiftG¯i,ΔG¯=∑i=1rhiftΔG¯i.
The goal is to design a filter in form of (11) so that the filtering error system can meet the following requirements simultaneously:(1)
The filtering error system is asymptotically stable whenwt=0(2)
Under the zero initial condition, the filtering error system satisfies(16)et2≤γwt2,for any nonzero wt∈L20,∞.


## 3. Main Results

In this section, the sufficient condition for the existence of the desired filter is derived in terms of LMIs solutions. Before proceeding with the study, the following Lemma is needed.Lemma 1.
(see [24]). D,E,F are real matrices with appropriate dimensions, and F is a time-varying unknown matrix which satisfies FTF≤I. Then for a scalar ε>0, the following inequality(17)DFE+ETFTDT≤ε−1DDT+εETE,always holds.Theorem 1.
If there exist positive matricesP>0,Q>0,R>0, and matrix W such that the following inequality holds:(18)ψ¯=ΓPB¯+ΔB¯+WPE¯+ΔE¯τmWτmA¯+ΔA¯TRF¯+ΔF¯T∗−Q00τmB¯+ΔB¯TRG¯+ΔG¯T∗∗−γ2I0τmE¯+ΔE¯TR0∗∗∗−τmR00∗∗∗∗−τmR0∗∗∗∗∗−I<0,where(19)Γ=PA¯+ΔA¯+A¯+ΔA¯TP+Q−W−WT,then the filtering error system is asymptotically stable and the prescribed H∞ performance γ can be guaranteed.Proof.
Select the Lyapunov function as(20)Vt=V1t+V2t+V3t,where(21)V1t=ξTtPξt,V2t=∫t−τttξTαQξαdα,V3t=∫−τm0∫t+βtξ˙TαRξ˙αdαdβ.
Whenwt=0, the derivation of Vt is(22)V˙1t=2ξTtPA¯+ΔA¯ξt+B¯+ΔB¯ξt−τt,V˙2t≤ξTtQξt−ξTt−τtQξt−τt,V˙3t≤τmξ˙TtRξ˙t−∫t−τttξ˙TαRξ˙αdα.
It is obvious that there is(23)ξt−τt=ξt−∫t−τttξ˙Tαdα.
Thus,Vt can be further rewritten as(24)V˙t≤2ξTtPA¯+ΔA¯ξt+B¯+ΔB¯ξt−τt+ξTtQξt−ξTt−τtQξt−τt+τmξ˙TtRξ˙t−∫t−τttξ˙TαRξ˙αdα+2ξTtWξTt−τt−ξTt+∫t−τttξ˙Tαdα≤1τt∫t−τttξtξt−τtξ˙αTΨ0ξtξt−τtξ˙α,where(25)Ψ0=ΓPB¯+ΔB¯+WτmW∗−Q0∗∗−τmR+τmA¯+ΔA¯TB¯+ΔB¯T0RA¯+ΔA¯TB¯+ΔB¯T0T.
By the Schur complement, if the inequality (24) holds, it can be obtained that there is Ψ0<0. Thus, there is V˙t<0 and the filtering error system is asymptotically stable.
Secondly, define a new function(26)J=∫0TeTtet−γ2wTtwtdt,where T is a scalar which satisfies T>0.
Thus, under the zero initial condition, there is(27)J=∫0TeTtet−γ2wTtwt+V˙tdt−VT≤∫0TeTtet−γ2wTtwt+V˙tdt≤ξtξt−τtwtξ˙αTψ1ξtξt−τtwtξ˙α,where(28)ψ1=ΓPB¯+ΔB¯+WPE¯+ΔE¯τmW∗−Q00∗∗−γ2I0∗∗∗−τmR+τmA¯+ΔA¯TB¯+ΔB¯TE¯+ΔE¯T0RA¯+ΔA¯TB¯+ΔB¯TE¯+ΔE¯T0T+F¯+ΔF¯TG¯+ΔG¯T00F¯+ΔF¯TG¯+ΔG¯T00T.
According to the Schur complement, it follows that(29)ψ1<0⇔ψ¯<0.
Thus, if the inequalityψ¯<0 holds, there is J<0 and equation (16) holds for any nonzero wt∈L20,∞. The proof is completed.Theorem 2.
For a given scalarγ>0, the filtering problem is solvable under the conditions above with a H∞ performance level γ, if there exist positive matrices P1,P2,Q1,Q2,R1,R2, matrices W1,W2,Zi and scalars ε1i,ε2i,ε3ij,ε4i,ε5i>01≤i≤j≤r such that the following inequalities hold:(30)Ωii<0,i=1,2,…,r,(31)Ωij+Ωji<0,i<j,i,j=1,2,…,r,where(32)Θ1i=P1Ui1TP2Ui1T00000τmR1U1iTτmR2U1iT0T,Ξ1i=V1i0V2i0V3i00000,Θ2i=0P2U1iT000000τmR2U1iT0T,Ξ2i=0V1i0V2i000000,Θ3ij=0P2KiU2iT000000τmR2KiU2iT0T,Ξ3i=0V1i0V2iV3i00000,Θ4i=000000000U1iTT,Ξ4i=0V4i00000000,Θ5i=000000000U2iTT,Ξ5i=000V4i000000,Γ11=P1Aqi+AqiTP1+Q1−W1−W1T,Γ22=P2Aqi−Z1iCqi+AqiTP2−CqiTZ1iT+Q2−W2−W2T,Γ13=P1Bqi+W1,Γ24=P2Bqi−Z1iDqi+W2,Γ25=P2Eq1i−Z1iE2i,Ωij=Ω0ij+ε1iΞ1iTΞ1i+ε2iΞ2iTΞ2i−ε3ijΞ3iTΞ3i+ε4iΞ4iTΞ4i+ε5iΞ5iTΞ5iΘ1iΘ2iΘ3ijΘ4iΘ5i∗−ε1i0000∗∗−ε1i2000∗∗∗ε3ij00∗∗∗∗−ε4i0∗∗∗∗∗−ε5i,Ω0ij=Γ110Γ130P1Eq1iτmW10τmAqiTR100∗Γ220Γ24Γ250τmW20τmAqiTR2−CqiTZ2iFqiT∗∗−Q10000τmBqiTR00∗∗∗−Q20000τmBqiTR2−DqiTZ2iGqiT∗∗∗∗−γ2I00τmEq1iTR1τmEq1iTR2−E2iTZ2i0∗∗∗∗∗−τmR10000∗∗∗∗∗∗−τmR2000∗∗∗∗∗∗∗−τmR100∗∗∗∗∗∗∗∗−τmR20∗∗∗∗∗∗∗∗∗−I,
Moreover, the filter parameter can be solved by(33)Ki=P2−1Z1i.Proof.
Let(34)P=diagP1,P2,Q=diagQ1,Q2,R=diagR1,R2,W=diagW1,W2.
Thenψ¯ can be rewritten as(35)ψ¯=ϒ11ϒ12ϒ13τmWτmϒ15F¯+ΔF¯T∗−Q00τmϒ25G¯+ΔG¯T∗∗−γ2I0τmϒ350∗∗∗−τmR00∗∗∗∗−τmR0∗∗∗∗∗−I,where(36)ϒ11=diagδ1,δ2,δ1=P1Aqi+ΔAqi+Aqi+ΔAqiTP1+Q1−W1−W1T,δ2=P2Aqi+ΔAqi−P2KiCqi+ΔCqi+Aqi+ΔAqiTP2−Cqi+ΔCqiTKiTP2+Q2−W2−W2T,ϒ12=diagP1Bqi+ΔBqi+W1,P2Bqi+ΔBqi−P2KiDqi+ΔDqi+W2,ϒ13=P1Eq1i+ΔEq1iP2Eq1i+ΔEq1i−P2KiE2i+ΔE2i,ϒ15=diagAqi+ΔAqiTR1,Aqi+ΔAqiTR2−Cqi+ΔCqiTKiTR2,ϒ25=diagBqi+ΔBqiTR1Bqi+ΔBqiTR2−Dqi+ΔDqiTKiTR2,ϒ35=Eq1i+ΔEq1iTR1Eq1i+ΔEq1iTR2−E2i+ΔE2iTKiTR2.
Then, for further analysis, based on equation (9) and Lemma 1, equation (35) can be decomposed into(37)ψ¯=∑i=1r∑j=1rhifthjftΩ0ij+Θ1iSiΞ1i+Ξ1iTSiTΘ1iT+Θ2iSiΞ2i+Ξ2iTSiTΘ2iT−Θ3ijSiΞ3i−Ξ3iTSiTΘ3ijT+Θ4iSiΞ4i+Ξ4iTSiTΘ4iT+Θ5iSiΞ5i+Ξ5iTSiTΘ5iT≤∑i=1r∑j=1rhifthjftΩ0ij+1ε1iΘ1iΘ1iT+ε1iΞ1iTΞ1i+1ε2iΘ2iΘ2iT+ε2iΞ2iTΞ2i−1ε3ijΘ3ijΘ3ijT−ε3ijΞ3iTΞ3i+1ε4iΘ4iΘ4iT+ε4iΞ4iTΞ4i+1ε5iΘ5iΘ5iT+ε5iΞ5iTΞ5i.Moreover, define(38)Z1i=P2Ki,Z2i=KiTR2.By applying the Schur complement, equation (37) is equivalent to(39)∑i=1rhi2ftΩii+∑i<jrhifthjftΩij+Ωji<0.Thus, the inequalities (30) and (31) hold. The proof is completed.


## 4. Optimal Transformation Matrix Selection

It can readily be obtained that the estimation precision is directly affected by the estimation of the unknown states (health parameters). Thus, the key of the parameters estimation is to find an optimal transformation matrixV∗, which can ensure that the low-dimensional tuning vector q can represent as much of the information of the health parameters as possible.In this paper, the optimization objective is to find an optimal transformation matrixV∗, which can ensure that the estimation error of the measured outputs ρ=y−y^Ty−y^ is minimum.Thus, the optimization problem can be described as(40)minρs.t.ρ=y−y^Ty−y^.And the method based on iterative solution is used in this paper, and the process of optimal transformation matrix selection is shown in Figure1:Figure 1
Flowchart ofV∗ iterative search.(1)
To begin with, the initial value ofV∗ is given randomly. And during the estimation process, the optimal V∗ of the last moment is used as the initial value to reduce the quantity of calculation. In order to avoid the calculation converging to a poorly scaled result, the Frobenius norm of V∗ must satisfy V∗F=1.
(2)
With reference to equation (7), construct the reduced-order state-space model.
(3)
Solve the parameters of theH∞ filter.
(4)
Calculate estimated error using equation (40).
(5)
Determine whether the estimated errorρ achieves convergence within a tolerance (user-specified):(1)
If converged, skip step 6 and proceed directly to step 7.(2)
If not converged, proceed to step 6.(3)
If the number of iterations exceeds 100, skip step 6 and proceed directly to step 7 to meet the real-time requirements.
(6)
Use the MATLABlsqnonlin function to update V∗, and the new value still needs to satisfy V∗F=1.
(7)
Return the optimal value ofV∗, and ends.


## 5. Simulation Example

Before proceeding with the simulation, the high-precision aero-thermodynamic mathematical model of aeroengine in full envelope is the basis of the simulation. It can not only build the small deviation dynamic state space models, but can also replace the real engine in simulation. The detailed modeling process is carried out by reference to [25, 26]. This paper takes a type of double-rotor turbofan engine as research object and uses modeling method based on component characteristics. The afterburner is not considered in this engine model. For greater adaptability, faster calculation speed, and stronger convergence, the self-tuning Broyden quasi-Newton method [27] is used to solve equilibrium equations in the component-level mathematical model. And the small deviation dynamic model is solved by fitting method [28–30].Assuming that the aeroengine works in the maximum state and then based on the methods of flight envelop division in [31–33], a T-S fuzzy model is constructed and the initial membership function is shown in Figure 2. In this paper, the T-S fuzzy model has two state variables, four health parameters, two control inputs, five measured outputs, and three unmeasured outputs, all shown in Table 1.Figure 2
Initial membership function.Table 1
Variables of aeroengine T-S fuzzy model.
State variables
Health parameters
Control inputs
Measured outputs
Unmeasured outputs
n
l
—low-pressure spool speed
Low-pressure compressor (LPC) efficiency
w
f
—fuel flow
n
l
—low-pressure spool speed
T
5
—LPT exit total temperature
n
h
—high-pressure spool speed
LPC flow capacity
A
8
—nozzle area
n
h
—high-pressure spool speed
F
n
—net thrust
Low-pressure turbine (LPT) efficiency
P
2b
—LPC exit total pressure
ΠT—turbine pressure ratio
LPT flow capacity
T
2b
—LPC exit total temperature
T
3
—HPC exit total temperatureThen the four T-S fuzzy rules with the selection ofHf and Ma can be obtained as follows:Rule 1. IfHf is about 0 km and Ma is about 0, then(41)A1=−1.261.02−0.14−1.24,B1=0.350.540.360.32,L1=−1.020.90−0.620.51−0.26−0.110.280.23,E11=−0.51.0,C1=10010.55−0.290.51−0.070.360.54,D1=00000.19−0.820.05−0.220.04−0.16,M1=000000000.94−0.180.02−0.070.24−0.390.01−0.020.13−0.23−0.09−0.08,E21=000.20.10.15,F1=0.25−0.54−0.141.270.400.04,G1=0.450.09−0.100.390.240.40,N1=−0.23−0.290.07−0.18−0.090.26−0.33−0.190.81−0.080.04−0.08.Rule 2. IfHf is about 4.8 km and Ma is about 1.24, then(42)A2=−2.661.10−0.61−1.39,B2=0.390.420.360.33,L2=−0.860.87−0.640.52−0.28−0.100.270.23,E12=−0.30.1,C2=10012.03−0.340.66−0.060.420.58,D2=00000.14−0.670.02−0.110.02−0.10,M2=000000000.92−0.160.02−0.050.15−0.300−0.010.08−0.18−0.09−0.07,E22=000.30.250.1,F2=−0.34−0.52−0.061.262.100.12,G2=0.450.09−0.100.430.54−0.57,N2=−0.31−0.220.08−0.1700.19−0.34−0.190.94−0.180.09−0.19.Rule 3. IfHf is about 12.8 km and Ma is about 0.7, then(43)A3=−0.310.19−0.05−0.28,B3=0.110.110.090.09,L3=−0.240.21−0.160.13−0.07−0.020.070.06,E13=0.20.3,C3=10010.53−0.180.49−0.040.350.55,D3=00000.19−0.840.05−0.210.04−0.15,M3=000000000.94−0.170.02−0.080.23−0.410.01−0.020.13−0.25−0.10−0.09,E23=000.10.20.1,F3=0.22−0.31−0.101.040.590.12,G3=0.450.12−0.080.360.45−0.58,N3=−0.25−0.250.07−0.20−0.070.21−0.34−0.191.05−0.150.07−0.17.Rule 4. IfHf is about 18.8 km and Ma is about 1.72, then(44)A4=−0.630.24−0.13−0.29,B4=0.080.100.080.07,L4=−0.200.19−0.140.11−0.06−0.020.060.05,E14=0.30.5,C4=10012.29−0.360.73−0.070.460.58,D4=00000.14−0.660.03−0.120.03−0.11,M4=000000000.95−0.160.02−0.050.18−0.300−0.010.09−0.17−0.09−0.08,E24=000.20.10.1,F4=−0.37−0.50−0.071.252.000.14,G4=0.460.05−0.110.470.61−0.43,N4=−0.28−0.210.07−0.18−0.020.18−0.33−0.180.79−0.200.09−0.22.Moreover, the uncertainties of the system are(45)U1i=0.2−0.3000T,V4i=−0.10.20.15,U2i=0.10.3−0.10.20.05T,V3i=0.1,V1i=0.10.5000,V2i=0.2−0.1000,i=1,2,3,4.The system sampling period is set asT=20ms, and the external disturbance wt is(46)wt=0.3sin0.8t5sin6t+3cos8t.The state space of time-delay isτt∈1,2,3, and the state transition matrix of τt is(47)Π=−0.60.40.20.3−0.50.20.10.4−0.5.The working condition is chosen asH = 5 km and Ma = 0.3. The given value of γ is γ=2.Two types of performance degradation are selected to simulate, which are slow varying and sudden change of health parameters. For the sudden change type (Type I), there is a step change for health parameters, and this type is used to represent the sudden damage, such as foreign objects damage. And the slow varying type (Type II) is defined as the linear variation of health parameters to represent the gradual performance degradation during using process. These two types can cover almost all the health parameters changes in practice, and they are simulated respectively to illustrate the effectiveness of the method proposed in this paper.The results are shown in Figures3–6. It can be seen that the method proposed in this paper can effectively simulate the health and performance parameters of the aeroengine. The estimated values can accurately track the changes as the health parameters shift.Figure 3
The estimation of health parameters for Type I.Figure 4
The estimation of health parameters for Type II.Figure 5
The estimation of performance parameter for Type I.Figure 6
The estimation of performance parameter for Type II.By comparing the results of the two types of health parameters change, it can be concluded that the estimated values for slow varying type are more accurate. For the type of sudden change, it takes a while for the estimated values to stabilize and the overshoot is slightly larger.Table2 shows the average estimated error of the health and performance parameters. It can be seen that the estimated error is within 2.5% for all types. At the same time, the results are more accurate except for the beginning of simulation.Table 2
Estimated errors of health and performance parameters.
Estimated error
Type I (%)
Type II (%)
h
1
0.46
0.40
h
2
0.38
0.31
T
5
1.27
0.98
F
n
1.95
1.57
ΠT
2.46
2.03Furthermore, the results show that the method proposed in this paper can accurately estimate most system parameters of interest. On this basis, the method can also be used in the investigations of direct control, fault diagnosis, health management, and online monitoring. And for aeroengine, there are already some researches which indicate that the changes of health parameters can characterize specific faults, allowing for online monitoring and alerting [34–36].


## 6. Conclusion

In this paper, a method is proposed for the underdetermined estimation problem of the distributed control systems. First, the T-S fuzzy model for DCSs is constructed and a model tuning parameter vector of appropriate dimension is produced. Then an observer-based fuzzy filter is designed and the sufficient condition for the existence of the designed filter is derived in terms of LMIs solutions. Besides, the method based on iterative solution is used to select the optimal transformation matrix to minimize the estimated error. Finally, the results of the simulation show that the proposed method can effectively estimate unmeasured parameters of the aeroengine and the estimated error is less than 2.5%.



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*Source: 7518039-2020-03-30.xml*