Optimal robust fault detection of discrete‐time LPV systems with measurement error‐affected scheduling variables combining ZKF and pQP

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systems with measurement error-affected scheduling variables combining ZKF and pQP

Junbo Tan, Sorin Olaru, Feng Xu, Xueqian Wang, Bin Liang

To cite this version:

Junbo Tan, Sorin Olaru, Feng Xu, Xueqian Wang, Bin Liang. Optimal robust fault detection of discrete-time LPV systems with measurement error-affected scheduling variables combining ZKF and pQP. International Journal of Robust and Nonlinear Control, Wiley, 2020, 30 (16), pp.6782-6802.

�10.1002/rnc.5138�. �hal-02960783�

Systems with Measurement Error-affected Scheduling Variables Combining ZKF and pQP

Junbo Tan^1,2,3, Sorin Olaru², Feng Xu^3∗, Xueqian Wang^3∗ and Bin Liang¹

1Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, 100084 Beijing, P.R.China.

2Laboratory of Signals and Systems, Univ. Paris-Sud-CentraleSupelec-CNRS, Universit´e Paris Saclay.

3Center for Artificial Intelligence and Robotics, Tsinghua Shenzhen International Graduate School, Tsinghua University, 518055 Shenzhen, P.R.China.

SUMMARY

Optimal robuststate estimation(SE) and fault detection(FD) methods of discrete-timelinear parameter varying (LPV) systems with measurement error-affected scheduling variables are proposed under the boundedness assumption of system uncertainties. By using the weighted Frobenius norm of the generator matrix of SE zonotope to characterize the set size, the optimal observer gain can be computed by aZonotopic Kalman Filter(ZKF) procedure for the purpose of observation. Meanwhile, by minimizing the influence of system uncertainties while maximizing that of faults on SE to enhance the sensitivity of FD, the optimal FD criterion is characterized based on an on-line fractional programming problem, which can be equivalently transformed into aparametric quadratic programming(pQP) problem. The pQP problem can be efficiently solved by searching the root of its nonlinear characteristic equation using secant method. In general, as long as sensors with sufficiently high precision are equipped to measure the scheduling variables, the bounds of measurement errors of scheduling variables can be less conservative than those direct bounds of scheduling variables, which can reduce the conservatism of FD or SE in this way. At the end of this paper, a case study based on a practical circuit model is used to illustrate the effectiveness of the proposed method. Copyright

c

2016 John Wiley & Sons, Ltd.

Received . . .

KEY WORDS: Optimal Robust Faut Detection, State Estimation, LPV systems, Inexact Scheduling Variables.

1. INTRODUCTION

As industrial systems become more and more complex, there always exist faults occurrence during the operation of system. State estimation (SE) and fault detection (FD) are often key steps for the aim of advanced monitory and control requirements in many engineering applications [1][2].

Fault diagnosis techniques have attracted a great number of attentions since they play an important role in real-time detecting, isolating and estimating (identifying) the occurred faults in sensors, actuators or system plant itself [3] [4]. The model-based SE and FD rely on establishing an effective mathematical model for real physical systems by using differential and difference equations.

∗Correspondenceto:FengXuandXueqianWang

Email:[email protected]and{xu.feng,wang.xq}@sz.tsinghua.edu.cn

R3-1

R3-1 Inrecentyears,thereisalargevolumeofpublishedstudiesdealingwithLPVmodelingmethods

sinceLPV systemscan beconsideredas arepresentationof nonlinearsystemsbut owna model structurally-similarwith linearsystemsateach equilibriumpoint[5]. Sometechniquesforlinear systemscanbefurtherusedtohandlesub-classesofnonlinearsystemsbymeansofLPVsystemsas a bridge collectinglinear and nonlinear systems [6][7][8][9]. It is worthy to mention that most of works on SE and FD for LPV systems only consider perfectly known scheduling variables [10][11][12][13],whileamoregeneralsituationisthattheschedulingvariablesareoftenaffected bythemeasurementsnoisesandlimitedtotheaccuracyofequippedsensors[14].

In general, model-based SE and FD can be divided into two categories according to the different approaches dealing with system uncertainties. In particular, the stochastic approaches characterizesystemuncertaintiesusingrandomvariablestoimplementSEandFD[15][16],whose classicalrepresentationisKalmanFilteringtechnology.Thedeterministicapproachesconsiderthat system uncertaintiesare unknownand bounded to generatebounding sets for completion ofSE and FD,such as intervalobservers,invariant-set methods, set-membership estimationand so on [2][17][18][19][20]. Since these set-theoretic methods consider the sets of system uncertainties and all possible situations of system states, the robustness of SE and FD can be guaranteed.

[21]proposed arobust FDmethod forLPV systemsusing intervalobserverandzonotopes. The recentstudy in[22]combinedthestandardKalmanFiltering withzonotopicgeometry torealize the optimal SE for linear time varying (LTV) systems based on the so-called ZKF procedure with a proof of robust convergence, which results in an explicit bridge between the zonotopic set-membership and the stochasticparadigms for KalmanFiltering by introducing thenotion of covariation.Basedon[22],[20]furtherconsideredtheZKFoptimizingFDratherthanSEforLTV systemsbyusing generalizedeigenvalues/eigenvectorstechnique,inwhichthemainoptimization problem is approximately transformed into maximizingthe general Rayleighquotient ableto be solvedbyusinggeneralizedeigenvalues/eigenvectors.[23]usedzonotopicunknowninputobserver (UIO)toimplementrobustfaultdetectionandisolationfordiscrete-timeLTVdescriptorsystems.

Forarealphysicalsystem,systemdisturbancesandmeasurementnoiseshaveanimportanteffect ontheresultsofSEandFD.ThemainideabehindmostofrobustSEandFDmethodsistoattenuate theimpactofuncertaintiesasmuchaspossiblewithoutprovidinganyinformationoftheestimation accuracyanddetectioneffectiveness.FDfiltersshouldbedesignedsuchthatthegeneratedresidual signalissensitivetothefaultswhileinsensitivetosystemuncertainties[24].Basedontheworkin [22],wefurtherconsidertheoptimalFDbymaximizingtheeffectoffaultsonthesizeoftheSE zonotope whileminimizingthatofsystemuncertaintiestodramaticallyimprovetheperformance ofFDatthepriceofslightlysacrificingtheaccuracyofSE.Notethat,comparedwith [20]using thegeneralizedeigenvalues/eigenvectorstomaximizethegeneralizedRayleighquotienttoobtain the approximately optimalgain ofFD observer,our proposed method can obtain a strict global optimalsolutionforthegaindesignofFDobserver.Meanwhile,theproposedmethodmainlydeals withtheoptimalrobustFDforLPVsystemswithmeasurement-erroraffectedschedulingvariables representingawiderpracticalapplicationprocesscomparedtotheLTVsystemsusedin[20]and the ordinary LPVsystems in [21]. Certainly, thecomputational burden of ourproposed method is relativelyhigherthanthegeneralizedeigenvalues/eigenvectors proposedin [20].However,the optimizationproblemoftheproposedmethodtobesolvedateachstepisjustaconvexquadratic programming problem, which hasbeen able to be solved by many efficientt oolboxes,s uchas YALMIP[25],CVX[26]andsoon.Inthissense,thecomputationburdenofourproposedmethod isalsoacceptableandtolerable.

For the clarity of this paper, the main contributions of this paper are summarized as follows:

• An optimal observer gain for observation purpose is computed by using the ZKF procedure for discrete-time LPV systems with measurement error-affected scheduling variables.

• Establish a fractional programming to maximize the effect of faults while minimize that of system uncertainties on SE to enhance the sensitivity of FD.

• Transform the fractional programming to the pQP problem to obtain optimal observer gain for FD purpose, which can be efficiently solved by searching a root for its characteristic equation using secant method.

The remainder of this paper is organized as follows. Section 2 mainly introduces a general model of discrete-time LPV systems affected by additive actuator and sensor faults and the corresponding design of FD observer considering measurement error-affected scheduling variables. The system behaviors and stability are analyzed and the ZKF-based optimal SE is proposed in Section 3.

The optimal robust FD based on the pQP approach is further illustrated in Section 4. Simulation results are provided to show the effectiveness of the proposed method in Section 5. Finally, some conclusions are drawn in Section 6.

2. PLANT MODEL

This section introduces a general model of discrete-time LPV systems affected by additive actuator and sensor faults and the corresponding FD observer for this kind of LPV systems.

2.1. System Plant

The LPV plant under the effect of actuator and sensor faults is modeled as

xk+1=A(θk)xk+B(θk)uk+G(θk)fk+E(θk)wk, (1a) y_k =C(θ_k)x_k+D(θ_k)u_k+H(θ_k)s_k+P(θ_k)v_k, (1b) whereA(θk)∈R^nx×nx,B(θk)∈R^nx×nu,G(θk)∈R^nx×nf,E(θk)∈R^nx×nw,C(θk)∈R^ny×nx, D(θk)∈R^ny×nu, H(θk)∈R^ny×ns and P(θk)∈R^ny×nv are parametric matrices dependent on a scheduling vector θk∈R^nθ. k denotes the k-th discrete time instant.xk∈R^nx and yk∈R^ny denote the system state and output vectors, respectively.uk ∈R^nu,wk ∈R^nwandvk ∈R^nv denote the known inputs, the unknown inputs and the measurement noises, respectively. fk ∈R^nf and sk ∈R^ns represent the additive actuator and sensor faults, respectively. It is assumed that the unknown inputs wk (including process disturbances, modeling errors, etc.) and the measurement noises vk are bounded by zonotopes W=w^c⊕MwB^{nw=hwc, Mwi} and V=v^c⊕MvB^{nv =} hv^c, Mvi, respectively. The additive actuator faultsfkand sensor faultsskare contained in zonotopes F=f^c⊕M_fB^{nf =hfc, Mfi}andS=s^c⊕M_sB^{ns=hsc, Msi}, respectively.

Assumption 2.1

The original system (1) is bounded-input, bounded-output (BIBO) stable. That is, given bounded input vectoruk, the system outputykis also bounded.

It is assumed that then_θ-dimensional scheduling vectorθ_kis bounded by a polytopic hypercube Θ, i.e.,θk ∈Θ, andθ_kⁱ is thei-th component of vectorθk= [θ¹_k, ..., θ_kⁱ, ..., θ_k^nθ]^T. Therefore, a linear affine functionΞ(θk)ofθkis also bounded by a polytopic set and can be written as the sum of vertex matrices of this set:

Ξ(θ_k) =

N

X

i=1

λ_i(θ_k)Ξ_i, (2)

whereΞi is thei-th vertex matrix of the setΞ(Θ),N = 2^nθ is the number of vertex matrices and the weighting coefficients satisfyPN

i=1λi(θk) = 1,0≤λi(θk)≤1. HereΞcan represent matrices A,B,G,E,C,D,H andP in (1). In some existing works on SE and FD of LPV systems, it is often considered that the scheduling vectorθkis perfectly measured. However, this is an unrealistic assumption because there always exists a measurement error between the actual valueθk and the measured valueθ^ˆk, which is denoted as

θ˜k =θk−θˆk, (3)

whereθ˜

k denotesthemeasurementerrorvectorofsensorsequippedintherealsystemtomeasure the scheduling variables. In general, the measurement errors of sensors are derived from their measurementprecisionanditisconsideredthatthemeasurementprecisionofsensorsusedtoobtain

the bounds ofmeasurement errors can beread fromtheir performance specifications, whichare R2-1

denoted asθ^˜k∈Θ˜.

In general, the size of measurement error setΘ^˜ of scheduling vectorθkis much smaller than that of the polytopic hypercubeΘ, based on which we are able to obtain less conservative results of SE or FD.

2.2. Design of FD Observer

We consider designing a Luenberger-structure FD observer for the discrete-time LPV system (1) as ˆ

xk+1=A(ˆθk)ˆxk+B(ˆθk)uk+L(ˆθk)(yk−yˆk) (4a) ˆ

yk=C(ˆθk)ˆxk+D(ˆθk)uk, (4b) wherexˆk∈R^nxandyˆk∈R^ny are the estimated state vector and output vector of (1), respectively.

θˆk =_ˆ

θ¹_k,θˆ²_k, ...,θˆ_k^nθˆT

denotes an actual measurement ofθkwith θˆk ∈Θ,

whereθ^ˆkdenotes the measurement values of scheduling vectorθkandθ^ˆi_kdenotes thei-th component of θ^ˆk. Note that, by projecting θ^ˆk into the bounding set Θ, we can obtain the measurement θ^ˆk

confined insideΘ( see [27] for more detailed explanations on this point). Therefore, the following polytopic decomposition

Ξ(ˆθ_k) =

N

X

i=1

λ_i(ˆθ_k)Ξ_i

can be obtained. Here, it is considered thatΞ(θ_k)(orΞ(ˆθ_k)) is the affine function ofθ_k(orθ^ˆ_k). That is

Ξ(θ_k) = Ξ⁰+

n_θ

X

i=1

Ξⁱθ_kⁱ,Ξ(ˆθ_k) = Ξ⁰+

n_θ

X

i=1

Ξⁱθˆⁱ_k (5)

whereΞ⁰,Ξ¹,· · ·, andΞ^nθare constant matrices.

By combining (5) and (3), we have the following results:

Ξ(θ_k) = Ξ(ˆθ_k) + Ξ(˜θ_k). (6) whereΞ(˜θ_k) =Pnθ

i=1Ξⁱθ˜_kⁱ. Remark 2.1

It must be emphasized that the notationsΞ_iin (2) andΞⁱin (5) have different meanings. The former denotes thei-th vertex matrix of the hypercube setΞ(Θ), while the latter denotes the coefficient matrix of thei-th component ofθk(orθ^ˆk)of linear form ofΞ(θk)(orΞ(ˆθk)).

Fortheconvenienceofillustration,wesimplifythesymbolΞ(θk)(Ξ(θˆ_k)orΞ(θ˜_k))asΞ^θ_k (Ξ^θˆ kor R2-2 Ξ^θ_k^˜).Foraclearexplanationoftheproposedmethods,boththeaffineandpolytopicformsofLPV

systemareusedinthemathematicalderivationsofthispaper.

3. SYSTEMBEHAVIORS

Thissectionmainlyderivesthestate-estimation-errordynamicsandoptimalSEsetinthehealthy situationbyusingaZKFprocedure.

3.1. Analysis of System Behaviors

With (1) and (4), the state-estimation error is defined as

ek=xk−xˆk, (7)

whose dynamics can be further derived as

ek+1=(A^θ_k−L^θ_k^ˆC_k^θ)ek+ (A^θ_k^˜−L_k^θˆC_k^θ˜)ˆxk+ (B_k^θ˜−L^θ_k^ˆD^θ_k^˜)uk

+G^θ_kf_k+E_k^θw_k−L^θ_k^ˆH_k^θs_k−L^θ_k^ˆP_k^θv_k. (8) Moreover, the output estimation error (i.e., the residual signal) is defined as

r_k =y_k−yˆ_k=C_k^θe_k+C_k^θ˜xˆ_k+D^θ_k^˜u_k+H_k^θs_k+P_k^θv_k. (9) We consider the dynamics (8) and (9) in the healthy situation without the effect of the fault signals fkandsk:

¯

ek+1=(A^θ_k−L^θ_k^ˆC_k^θ)¯ek+ (A^θ_k^˜−L^θ_k^ˆC_k^θ˜)ˆxk+ (B^θ_k^˜−L^θ_k^ˆD_k^θ˜)uk+E_k^θwk−L^θ_k^ˆP_k^θvk, (10a)

¯

rk=C_k^θe¯k+C_k^θ˜xˆk+D^θ_k^˜uk+P_k^θvk. (10b) The set version of the dynamics (10) can be derived as

E¯k+1=(A^θ_k^ˆ+A^Θ˜ −L_k^θˆC_k^θˆ−L^ˆθ_kC^Θ˜) ¯Ek⊕(A^Θ˜ −L^θ_k^ˆC^Θ˜)ˆxk⊕(B^Θ˜ −L^θ_k^ˆD^Θ˜)uk

⊕(E_k^θˆ+E^Θ˜)W⊕(−L_k^θˆP_k^θˆ−L^θ_k^ˆP^Θ˜)V, (11a) R¯_k=(C_k^θˆ+C^Θ˜) ¯E_k⊕C^Θ˜xˆ_k⊕D^Θ˜u_k⊕(P_k^θˆ+P^Θ˜)V. (11b) whereA^Θ˜,B^Θ˜,C^Θ˜,D^Θ˜,E^Θ˜ andP^Θ˜ are the interval matrices containingA^θ_k^˜,B_k^θ˜,C_k^θ˜,D^θ_k^˜,E_k^θ˜and P_k^θ˜respectively. More detailed introduction about interval analysis and operation can be referred to[28]. Furthermore, the center-generator matrix form of (11) can be obtained as

M_k+1^e¯ =h

(A^θ_k^ˆ−L^θ_k^ˆC_k^θˆ) ¯M_k^e¯ seg((A^Θ˜M¯_k^e¯)) rad(A^Θ˜¯e^c_k) −L_k^θˆseg((C^Θ˜M¯_k^¯e))

−L^θ_k^ˆrad(C^Θ˜¯e^c_k) −L^θ_k^ˆrad(C^Θ˜xˆ_k) −L^θ_k^ˆrad(D^Θ˜u_k) rad(A^Θ˜xˆ_k) rad(B^Θ˜uk) −L_k^θˆseg(((P_k^θˆ+P^Θ˜)Mv)) −L^θ_k^ˆrad((P_k^θˆ+P^Θ˜)v^c) seg(((E_k^θˆ+E^Θ˜)M_w)) rad((E_k^θˆ+E^Θ˜)w^c)

i

, (12a)

¯

e^c_k+1= (A^θ_k^ˆ+mid(A^Θ˜)−L^θ_k^ˆC_k^θˆ−L_k^θˆmid(C^Θ˜))¯e^c_k+ (mid(A^Θ˜)−L^θ_k^ˆmid(C^Θ˜))ˆx_k + (mid(B^Θ˜)−L^θ_k^ˆmid(D^Θ˜))uk−L^θ_k^ˆ(P_k^θˆ+mid(P^Θ˜))v^c+ (E_k^θˆ+mid(E^Θ˜))w^c,

(12b) M_k^r¯=h

seg(((C_k^θˆ+C^Θ˜) ¯M_k^e¯)) rad((C_k^θˆ+C^Θ˜)¯e^c_k) rad(C^Θ˜xˆk) rad(D^Θ˜uk) seg(((P_k^θˆ+P^Θ˜)Mv)) rad((P_k^θˆ+P^Θ˜)v^c)

i

, (12c)

¯

r_k^c = (C_k^θˆ+mid(C^Θ˜))¯e^c_k+mid(C^Θ˜)ˆxk+mid(D^Θ˜)uk+ (P_k^θˆ+mid(P^Θ˜))v^c (12d) where M_k^e¯, ¯e^c_k, M_k^r¯ and ¯r^c_k are the generator matrices and centers of E^¯k and R^¯k, respectively.

M¯_k^e¯ is theorder-reduction matrix ofM_k^e¯,i.e., M¯_k^¯e=↓λ,W (M_k^e¯) ( Zonotope-orderreduction is a ^R2-3 contributionfrom[22].PleasereferAppendixBformoredetails.)

The FD criterion lies in real-time checking whether

¯

r_k∈R¯_k (13)

holds or not. (13) is equivalent to the following relationship:

0∈ {−¯rk} ⊕R¯k=Rk=hcR_k, MR_ki. (14) If there is a violation of (14), i.e.,06∈Rk, it is considered that faults have occurred in system (1).

Otherwise, we still consider that the system (1) operates in the healthy situation.

3.2. Stability analysis

According to (10a), it can be found that the dynamics (10a) of the SE errore¯k involves in the state estimationxˆk. Thus, in order to analyze the stability of the dynamics (10a), we integrate (4a) and (10a) to construct the following augmented dynamics:

ζk+1= Φ0ζk+ Φ1µk+ Φ2wk+ Φ3vk, (15) where

Φ₀=

"

A^θ_k^ˆ−L^θ_k^ˆC_k^θˆ 0 A^θ_k^˜−L^θ_k^ˆC_k^θ˜ A^θ_k−L^θ_k^ˆC_k^θ

# ,Φ₁=

"

L^θ_k^ˆ B^θ_k^ˆ−L^θ_k^ˆD_k^θˆ 0 B^θ_k^˜−L^θ_k^ˆD_k^θ˜

# , Φ2=

0 E_k^θ

,Φ3=

₀

−L^θ_k^ˆP_k^θ

, ζk= xˆk

¯ ek

, µk =

yk

uk

.

As we can see, the inputµkof the augmented dynamics (15) is composed of the known system input u_k and the known system outputy_k. Under Assumption2.1, the augmented input µ_k is bounded.

Meanwhile, considering that the uncertain factorsw_k andv_k are also bounded, i.e.,w_k ∈Wand v_k∈V, they will not affect the BIBO stability of the augmented dynamics (15). Thus, we directly consider the stability of the following nominal system:

ζ¯_k+1= Φ₀ζ¯_k, (16)

Here, stability results in Lemmas 3.1 and 3.2 taken from [29] and [30] are first recalled for the stability analysis of the dynamics (16).

Lemma 3.1

Considering the discrete LPV system:

xk+1=

"

A^θ_11,k^ˆ 0 A^θ_21,k^ˆ A^θ_22,k^ˆ

#

xk (17)

and assume thatA^θ_11,k^ˆ andA^θ_22,k^ˆ are poly-quadratically stable^†, respectively, then the LPV system (17) is poly-quadratically stable.

Lemma 3.2

The dynamics x_k+1=A^θ_k^ˆx_k is poly-quadratically stable if and only if there exists symmetric definite matricesS_i,S_jand matricesG_isuch that

h_G

i+G^T_i−Si ∗ AiGi Sj

i0, ∀i, j= 1,2, , N, (18) where the symbol ∗ denotes the transpose of AiGi. In this case, the time-varying parameter dependent Lyapunov function for the stability is given as V(x_k, λ^ˆθ_i,k) =x^T_kP_k^θˆx_k with P_k^θˆ= PN

i=1λ^θ_i,k^ˆ S_i⁻¹,PN

i=1λ^θ_i,k^ˆ = 1and0≤λ^θ_i,k^ˆ ≤1.

†The expression on the stability of a matrixA^θ_k^ˆmeans the stability of the systemxk+1=A^ˆθ_kxk.

Theorem 3.1

The dynamics (16) is poly-quadratically stable if there exist symmetric definite matricesSi,Sjand matricesGisuch that

Gi+G^T_i −Si ∗ (A_i−L_tC_i)G_i S_j

0 (19)

∀i, j, t= 1, 2,· · ·, N with the indices defined asA^θ_k=PN

i=1λ^θ_i,kA_i,A^θ_k^ˆ=PN

i=1λ^θ_i,k^ˆ A_i,C_k^θ= PN

i=1λ^θ_i,kCi,C_k^θˆ=PN

i=1λ^θ_i,k^ˆ Ci, andL^θ_k^ˆ=PN

t=1λ^θ_t,k^ˆ Lt, where the symbol∗denotes the transpose of(A_i−L_tC_i)G_i.

Proof

From Lemma3.1, the stability condition ofΦ0 can be formulated into the simultaneous stability problems ofA^θ_k^ˆ−L^θ_k^ˆC_k^θˆandA^θ_k−L_k^θˆC_k^θ. For the stability conditions ofA^θ_k^ˆ−L^θ_k^ˆC_k^θˆ, by using the results in Lemma3.2, we can first derive the following stability conditions:

Gi+G^T_i −Si ∗ ΛiGi Sj

0 with Λi=Ai−L_k^θˆCi by decomposing A^θ_k^ˆ=PN

i=1λ^θ_i,k^ˆ Ai and C_k^θˆ=PN

i=1λ^θ_i,k^ˆ Ci for i, j= 1,2, ...,N, where Si and Sj are symmetric definite matrices with proper dimensions, and Gi

are matrices with proper dimensions. Furthermore, with the polytopic decomposition of L^θ_k^ˆ= PN

t=1λ^θ_t,k^ˆ Lt, we can haveΛi =PN

t=1λ^θ_t,k^ˆ (Ai−LtCi). Thus, we are able to further have G_i+G^T_i −S_i ∗

Λ_iG_i S_j

=

N

X

t=1

λ^θ_t,k^ˆ

G_i+G^T_i −S_i ∗ (A_i−L_tC_i)G_i S_j

which implies that as long as (19) holds for ∀t= 1,2, ...,N, the matrix

Gi+G^T_i −Si ∗ Λ_iG_i S_j

is also positive definite. Similarly, we can also derive the same stability conditions forA^θ_k−L^θ_k^ˆC_k^θ. Therefore, according to Lemmas3.1and3.2, the dynamics (16) is quadratically stable.

3.3. Optimal observer gain for robust SE

Regarding the design of the gain for SE purposes, the main goal is to reduce the effect of system uncertainties on SE. In the healthy situation, the estimated state set X_k at time instant k can be computed byX_k = ˆx_k⊕E¯_k. It can be found that the precision of SE is determined by the size of the setE^¯k. That means, if we can compute the gain matrixL^θ_k^ˆto minimize the size ofE^¯k, we can obtain the optimal SE. It is known that the shape and size of a zonotope are completely determined by its generator matrix. Because the weighted Frobenius norm of a matrix sufficiently considers its effects, we use the weighted Frobenius norm of the generator matrix of a zonotope to describe its size here (the details on the weighted Frobenius norm based size of zonotope is referred to [31] and [22]). As presented in (11a) and (12a), the size of zonotopeE^¯_k+1 can be directly assessed by the M_k+1^e¯ . Moreover, for brevity, we use the square of the weighted Frobenius norm of the generator matrixM_k+1^e¯ to describe the size ofE^¯_k+1:

k

¯

e^c_k+1, M_k+1^e¯

k²_F,W =kM_k+1^¯e k²_F,W =tr(W M_k+1^¯e M_k+1^¯eT ).

Theorem 3.2

(Optimal Gain for Robust SE) Considering ¯ek∈E¯k=h¯e^c_k,M¯_k^¯ei at time instant k, the optimal observer gainL^∗_kminimizing the weighted Frobenius-norm based size of bounding zonotopeE^¯k+1

at time instantk+ 1is computed as

L^∗_k =A^θ_k^ˆK^∗, (20)

with

K^∗=LS⁻¹,L=QMC_k^θˆT,S=C_k^θˆQMC_k^θˆT +QΘ˜ +Qv,QM = ¯M_k^¯eM¯_k^e¯T, Qv =seg(((P_k^θˆ+P^Θ˜)Mv))seg(((P_k^θˆ+P^Θ˜)Mv))^T,

QΘ˜ =seg((C^Θ˜M¯_k^¯e))seg((C^Θ˜M¯_k^¯e))^T+rad(C^Θ˜e¯^c_k)rad(C^Θ˜e¯^c_k)^T +rad(C^Θ˜xˆ_k)rad(C^Θ˜xˆ_k)^T +rad(D^Θ˜uk)rad(D^Θ˜uk)^T+rad((P_k^θˆ+P^Θ˜)v^c)rad((P_k^θˆ+P^Θ˜)v^c)^T.

Proof

According to (12a), we have

kM_k+1^e¯ k²_F,W =tr(W M_k+1^e¯ M_k+1^e¯T )

=tr

W(A^θ_k^ˆ−L^θ_k^ˆC_k^θˆ)QM(A^θ_k^ˆ−L_k^θˆC_k^θˆ)^T+W L_k^θˆ(Qv+QΘ˜)L^θ_k^ˆT

+Wseg((A^Θ˜M¯_k^e¯))seg((A^Θ˜M¯_k^¯e))^T+Wrad(A^Θ˜¯e^c_k)rad(A^Θ˜e¯^c_k)^T +Wrad(A^Θ˜ˆx_k)rad(A^Θ˜xˆ_k)^T +Wrad(B^Θ˜u_k)rad(B^Θ˜u_k)^T +Wseg(((E_k^θˆ+E^Θ˜)Mw))seg(((E_k^θˆ+E^Θ˜)Mw))^T +Wrad((E^θ_k^ˆ+E^Θ˜)w^c)rad((E_k^θˆ+E^Θ˜)w^c)^T

.

It can be found thatkM_k+1^¯e k²_F,W is convex with respect toL^θ_k^ˆ. Moreover,L^θ_k^ˆ is a free parametric matrix and there are no constraints onL^θ_k^ˆ. Therefore, we can directly compute the optimal valueL^∗_k to obtain the minimal value ofkM_k+1^e¯ k²_F,W by solving the partial differential equation

∂kM_k+1^¯e k²_F,W

∂L^θ_k^ˆ

= 0. (21)

We solve (21) and obtainA^θ_k^ˆQMC_k^θˆT =L^θ_k^ˆ(C_k^θˆQMC_k^θˆT +QΘ˜ +Qv). By settingK^∗,L andS as in Theorem3.2, sinceS =C_k^θˆQMC_k^θˆT +QΘ˜ +Qv is positive definite, an explicit solution can be obtained in Theorem3.2.

Remark 3.1

Solving the optimal parametric matrixL^∗_k in Theorem3.2is actually equivalent to a Kalman filter procedure searching for the optimal Kalman gain matrix but with bounded disturbances and noises instead of stochastic disturbances and noises. Since here the zonotopic framework and Kalman filter are combined to solve the problem of optimal parametric matrix, we denote this optimization procedure as zonotopic Kalman filter (ZKF). More information on this point can be referred to [22], which shows that there is a strong analogy between the Kalman filter and ZKF where the usually Gaussian probability density functions are replaced by zonotopic sets.

4. OPTIMAL OBSERVER GAIN FOR ROBUST FD

In this section, we aim to design the optimal observer gain for the purposes of robust FD. Such a gain is computed to maximize the effect of faults while minimize the effect of disturbances to enhance the sensitivity of FD.

4.1. Structure of dynamics

Let us first consider the state-estimation-error dynamics (8) in the faulty situation, which can split into two independent dynamical systems:

¯

e_k+1=(A^θ_k−L^θ_k^ˆC_k^θ)¯e_k+ (A^θ_k^˜−L^θ_k^ˆC_k^θ˜)ˆx_k+ (B^θ_k^˜−L^θ_k^ˆD_k^θ˜)u_k+E_k^θw_k−L^θ_k^ˆP_k^θv_k, (22a) e_k+1^{f s} =(A^θ_k−L_k^θˆC_k^θ)e^{f s}_k +G^θ_kfk−L_k^θˆH_k^θsk, (22b)

withek = ¯ek+e^{f s}_k . It can be found that (22) allows the seperation of the effects of the disturbances (i.e.,wkandvk) and the faults (i.e.,fkandsk). Furthermore, (22a) is considered for the computation of the observer gain for SE purposes, which has been clearly illustrated in Section 3. Next, tuning the observer gain for FD purposes is done by combining (22a) and (22b). As a result, the advantage of the proposed FD scheme lies in the fact that not only the effect of the system disturbances is considered, but also the relative influence of the disturbances and the faults are characterized to formulate an optimization criterion to satisfy the optimal FD goal.

For the robustness of FD, we consider the set version of (22b)

E^{f s}_k+1=(A^θ_k^ˆ+A^Θ˜ −L_k^θˆC_k^θˆ−L^θ_k^ˆC^Θ˜)E^{f s}_k ⊕(G^θ_k^ˆ+G^Θ˜)F⊕(−L^θ_k^ˆ)(H_k^θˆ+H^Θ˜)S. (23)

Furthermore, the center-generator matrix form of (23) can be obtained as

M_k+1^{f s} =h

(A_k^θˆ−L^θ_k^ˆC_k^θˆ) ¯M_k^{f s} seg((A^Θ˜M¯_k^{f s})) rad(A^Θ˜e^{f s,c}_k ) −L_k^θˆseg((C^Θ˜M¯_k^{f s}))

−L^θ_k^ˆrad(C^Θ˜e^{f s,c}_k ) seg(((G^θ_k^ˆ+G^Θ˜)M_f)) rad((G^θ_k^ˆ+G^Θ˜)f^c)

−L^θ_k^ˆseg(((H_k^θˆ+H^Θ˜)Ms)) −L^θ_k^ˆrad((H_k^θˆ+H^Θ˜)s^c) i

, (24a)

e^{f s,c}_k+1= (A^θ_k^ˆ+mid(A^Θ˜)−L^θ_k^ˆC_k^θˆ−L^θ_k^ˆmid(C^θ˜))e^{f s,c}_k + (G^θ_k^ˆ+mid(G^Θ˜))f^c

−L^θ_k^ˆ(H_k^θˆ+mid(H^Θ˜))s^c, (24b)

whereM_k^{f s,c}ande^{f s,c}_k are the generator matrix and center ofE^{f s}_k .M^¯_k^{f s}is the order-reduction matrix ofM_k^{f s}, i.e.,M^¯_k^{f s}=↓λ,W (M_k^{f s}).

4.2. Optimal observer gain for FD purposes

Now, we concentrate on the design of optimal observer gain to improve the sensitivity of robust FD with respect to the system disturbances. We consider the following optimization criterion minimizing the influences of disturbances while maximizing that of faults:

J= kE¯k+1k²_F,W

kE^{f s}_k+1k²_F,W. (25)

For the convenience of handling the optimization criterion (25), Theorem4.1gives an equivalent form of the FD optimization criterion (25).

Theorem 4.1

Consideringξ=vec(L^θ_k^ˆ), the FD optimization criterion (25) can be parameterized withξas

J(ξ) = ξ^T(Z1⊗W)ξ+vec(Z2)^Tξ+tr(Z3)

ξ^T(Z4⊗W)ξ+vec(Z5)^Tξ+tr(Z6), (26)