Simultaneous input and state set-valued \mathcal{H}_{\infty}-observers for linear parameter-varying systems
aa r X i v : . [ ee ss . S Y ] J a n Simultaneous Input and State Set-Valued H ∞ -Observers For LinearParameter-Varying Systems Mohammad Khajenejad Sze Zheng Yong
Abstract — A fixed-order set-valued observer is presented forlinear parameter-varying systems with bounded-norm noise andunder completely unknown attack signals, which simultaneouslyfinds bounded sets of states and unknown inputs that include thetrue state and inputs. The proposed observer can be designedusing semidefinite programming with LMI constraints and isoptimal in the minimum H ∞ -norm sense. We show that thestrong detectability of each constituent linear time-invariantsystem is a necessary condition for the existence of such anobserver, as well as the boundedness of set-valued estimates.Furthermore, sufficient conditions are provided for the upperbounds of the estimation errors to converge to steady statevalues and finally, the results of such a set-valued observer areexhibited through an illustrative example. I. I
NTRODUCTION
The security of Cyber-Physical Systems (CPS) is emergingas an extremely critical and important issue. Since physicaland software components are deeply intertwined in CPS,such systems are potentially vulnerable to adversarial attacks,which could be harmful for both the physical systems andtheir operators. Given that adversarial attackers may behavestrategically, there are many potential avenues through whichthey can cause harm, steal information/power, etc.Misleading the system operator by inserting counterfeitdata into sensor and actuator signals (false data injection)is among the most common and extensive attacks on CPS.Hence, a significant amount of effort has been invested innew designs for estimation and control against false datainjection attacks. Due to the nature of the attack signals, itis not justifiable to impose any kind of restrictive assump-tions on them (e.g., stochastic with normal distribution ordeterministic with bounded norm) i.e., they can be anything ,so none of the classical Kalman filtering based methods areapplicable. Moreover, considering the complicated structureof CPS (several physical, computing and communicationcomponents), they can be modeled and represented morerealistically by time-varying and nonlinear dynamic systemsrather than linear time-invariant ones. Taking all these factsinto account, this paper attempts to design a resilient observerfor a particular class of linear time-varying systems knownas linear parameter-varying systems, which to the best of ourknowledge is a novel approach.
Literature review.
There are several different frameworksfor simultaneous input and state estimation of linear time-varying stochastic systems with unknown inputs, assuming
M. Khajenejad and S.Z. Yong are with the School for Engineering ofMatter, Transport and Energy, Arizona State University, Tempe, AZ, USA(e-mail: [email protected], [email protected] ). that the noise signals are Gaussian and white. The authorsin [1]–[4] apply Kalman filtering inspired recursive filterdesign approaches (modified versions of unbiased minimum-variance estimation methods), where some additional as-sumptions are needed to guarantee the stability of the filters,while [5] uses a modified double-model adaptive estimationmethod. However, these Kalman filtering inspired approachesare not applicable for set-membership estimation problems(cf. [6] for a comprehensive discussion), as is considered inthis paper.In the context of attack-resilient estimation, where ad-versarial signals can be malicious and strategic and thus,bounds on the unknown inputs/attack signals are completelyunknown, there have been a number of proposed approachesin the literature for systems with bounded errors (e.g., [7]–[9]), but all of them only consider point estimates, i.e, themost likely or best single estimate as opposed to a set-valuedestimate. Specifically, the work in [7] only computes errorbounds for the initial state and [8] assumes zero initial statesand does not consider any optimality criteria. The authorin [6] and references therein discussed the advantages ofset-valued observers (when compared to point estimators)in terms of providing hard accuracy bounds, which areimportant to guarantee safety [10]. In addition, the useof fixed-order set-valued methods can help decrease thecomplexity of optimal observers [11], which grows withtime. Hence, the work in [6] presents a fixed-order set-valuedobserver for linear time-invariant discrete time systems withbounded errors, that simultaneously finds bounded sets ofcompatible states and unknown inputs that are optimal inthe minimum H ∞ -norm sense, i.e., with minimum averagepower amplification, which we aim to generalize in this paperfor linear parameter-varying systems. Contributions.
We propose a novel fixed-order set-valuedobserver for linear parameter-varying systems with unknowninput and bounded noise signals that simultaneously findsbounded sets of states and unknown inputs that contain thetrue state and unknown input and are compatible/consistentwith the measurement outputs. Specifically, we consider lin-ear parameter-varying system dynamics that can be presentedas a convex combination of linear time-invariant constituent dynamics. In addition, we provide necessary conditions forthe boundedness of the set-valued estimates. We furtherprove the optimality of the filter in the minimum H ∞ -norm sense, i.e., minimum average power amplification,by converting the corresponding problem into a tractableformulation using semi-definite programming with LMI con-traints that is readily implementable using off-the-shelfoptimization solvers. We also show that strong detectabilityof each constituent system is a necessary condition for theexistence of such an H ∞ -observer. Then, we provide somesufficient conditions for the convergence of upper bounds ofthe state and input estimation errors to steady state and forobtaining these steady state bounds. Finally, we demonstratethe effectiveness of our proposed set-valued observer throughan illustrative example. Notation. R n denotes the n -dimensional Euclidean spaceand N nonnegative integers. For a vector v ∈ R n and amatrix M ∈ R p × q , k v k , √ v ⊤ v and k M k denote their(induced) 2-norm. Moreover, the transpose, inverse, Moore-Penrose pseudoinverse and rank of M are given by M ⊤ , M − , M † and rk( M ) . For a symmetric matrix S , S ≻ ( S (cid:23) ) is positive (semi-)definite.II. P ROBLEM S TATEMENT
System Assumptions.
Consider the following linearparameter-varying discrete-time bounded-error system: x k +1 = P Ni =1 λ i,k ( A i x k + B i u k + w ik ) + Gd k ,y k = Cx k + P Ni =1 λ i,k ( D i u k + v ik ) + Hd k , (1)where λ i,k is known and satisfies ≤ λ i,k ≤ , P Ni =1 λ i,k =1 , ∀ k . x k ∈ R n is the state vector at time k ∈ N , u k ∈ R m isa known input vector, d k ∈ R p is an unknown input vector,and y k ∈ R l is the measurement vector. The process noise w ik ∈ R n and the measurement noise v ik ∈ R l are assumedto be bounded and ℓ ∞ sequences, with k w ik k ≤ η w and k v ik k ≤ η v .We also assume an estimate ˆ x of the initial state x is available, where k ˆ x − x k ≤ δ x . The matrices A i , B i , C , D i , G and H are known for i ∈ { , , . . . , N } andof appropriate dimensions, where G and H are matrices thatencode the locations through which the unknown input orattack signal can affect the system dynamics and measure-ments and N is the number of constituent systems. Note thatno assumption is made on H to be either the zero matrix (nodirect feedthrough), or to have full column rank when thereis direct feedthrough. Without loss of generality, we assumethat rk[ G ⊤ H ⊤ ] = p , n ≥ l ≥ , l ≥ p ≥ , m ≥ and each ( A i , B i , C, D i , G, H ) , i ∈ { , , . . . , N } represents a lineartime-invariant constituent system: x ik +1 = A i x ik + B i u k + Gd k + w ik ,y ik = Cx k + D i u k + Hd k + v ik . (2) Unknown Input (or Attack) Signal Assumptions.
The un-known inputs d k are not constrained to be a signal of anytype (random or strategic) nor to follow any model, thus noprior ‘useful’ knowledge of the dynamics of d k is available(independent of { d ℓ } ∀ k = ℓ , { w ℓ } and { v ℓ } ∀ ℓ ). We alsodo not assume that d k is bounded or has known bounds andthus, d k is suitable for representing adversarial attack signals.The simultaneous input and state set-valued observer de-sign problem can be stated as follows: Problem 1.
Given a linear parameter-varying discrete-timebounded-error system with unknown inputs (1) , design an optimal and stable filter that simultaneously finds boundedsets of compatible states and unknown inputs in the min-imum H ∞ -norm sense, i.e., with minimum average poweramplification. III. P
RELIMINARY M ATERIAL
A. System Transformation
In order to decouple the output equation into two com-ponents, first a transformation is carried out for each ofthe constituent subsystems, one with a full rank directfeedthrough matrix and the other without direct feedthrough.Note that this similarity transformation is similar to the onein [6] and is not the same as the one in [4], which is no longerapplicable as it was based on the noise error covariance.Let p H , rk( H ) . Using singular value decomposition,we rewrite the direct feedthrough matrix H as H = (cid:2) U U (cid:3) (cid:20) Σ 00 0 (cid:21) (cid:20) V ⊤ V ⊤ (cid:21) , where Σ ∈ R p H × p H is a diagonalmatrix of full rank, U ∈ R l × p H , U ∈ R l × ( l − p H ) , V ∈ R p × p H and V ∈ R p × ( p − p H ) , while U , (cid:2) U U (cid:3) and V , (cid:2) V V (cid:3) are unitary matrices. When there is no directfeedthrough, Σ , U and V are empty matrices a , and U and V are arbitrary unitary matrices.Then, we decouple the unknown input into two orthogonalcomponents: d ,k = V ⊤ d k , d ,k = V ⊤ d k . (3)Considering that V is unitary, d k = V d ,k + V d ,k and wecan represent the system (1) as: x k +1 = P Ni =1 λ i,k ( A i x k + B i u k + w ik ) + G d ,k + G d ,k ,y k = Cx k + P Ni =1 λ i,k ( D i u k + v ik ) + H d ,k (4)where G , GV , G , GV and H , HV = U Σ . Next,the output y k is decoupled using a nonsingular transforma-tion T = (cid:2) T ⊤ T ⊤ (cid:3) ⊤ , U ⊤ = (cid:2) U U (cid:3) ⊤ to get z ,k ∈ R p H and z ,k ∈ R l − p H given by z ,k , T y k = U ⊤ y k = C x k + Σ d ,k + P Ni =1 λ i,k D i u k + P Ni =1 λ i,k v i ,k z ,k , T y k = U ⊤ y k = C x k + P Ni =1 λ i,k D i u k + P Ni =1 λ i,k v i ,k (5)where C , U ⊤ C , C , U ⊤ C , D i , U ⊤ D i , D i , U ⊤ D i , v i ,k , U ⊤ v ik and v i ,k , U ⊤ v ik . This transform is alsochosen such that k h v i ,k ⊤ v i ,k ⊤ i ⊤ k = k U ⊤ v ik k = k v ik k .IV. F IXED -O RDER S IMULTANEOUS I NPUT AND S TATE S ET -V ALUED O BSERVERS
A. Set-Valued Observer Design
We consider a recursive three-step set-valued observerdesign. This design utilizes a similar framework as in [6]and contains an unknown input estimation step that usesthe current measurement and the set of compatible statesto estimate the set of compatible unknown inputs, a time a Based on the convention that the inverse of an empty matrix is an emptymatrix and the assumption that operations with empty matrices are possible. pdate step which propagates the compatible set of statesbased on the system dynamics, and a measurement update step that uses the current measurement to update the set ofcompatible states. To sum up, our target is to design a three-step recursive set-valued observer of the form:
Unknown Input Estimation: ˆ D k − = F d ( ˆ X k − , u k ) , Time Update: ˆ X ⋆k = F ⋆x ( ˆ X k − , ˆ D k − , u k ) , Measurement Update: ˆ X k = F x ( ˆ X ⋆k , u k , y k ) , where F d , F ⋆x and F x are to-be-designed set mappings, while ˆ D k − , ˆ X ⋆k and ˆ X k are the sets of compatible unknown inputsat time k − , propagated, and updated states at time k ,correspondingly. It is important to note that d ,k cannot beestimated from y k since it does not affect z ,k and z ,k . Thus,the only estimate we can obtain in light of (5) is a (one-step)delayed estimate of ˆ D k − . The reader may refer to a previouswork [3] for a complete discussion on when a delay is absentor when we can expect further delays. Similar to [6], [10],[12], as the complexity of optimal observers increases withtime, only the fixed-order recursive filters will be considered.In particular, we choose set-valued estimates of the form: ˆ D k − = { d ∈ R p : k d k − − ˆ d k − k ≤ δ dk − } , ˆ X ⋆k = { x ∈ R n : k x k − ˆ x ⋆k | k k ≤ δ x,⋆k } , ˆ X k = { x ∈ R n : k x k − ˆ x k | k k ≤ δ xk } . In other words, we restrict the estimation errors to ballsof norm δ . In this setting, the observer design problem isequivalent to finding the centroids ˆ d k − , ˆ x ⋆k | k and ˆ x k | k aswell as the radii δ dk − , δ x,⋆k and δ xk of the sets ˆ D k − , ˆ X ⋆k and ˆ X k , respectively. In addition, we limit our attention toobservers for the centroids ˆ d k − , ˆ x ⋆k | k and ˆ x k | k that belongto the class of three-step recursive filters given in [2] and[4], defined as follows for each time k (with ˆ x | = ˆ x ): Unknown Input Estimation : ˆ d ,k = M ( z ,k − C ˆ x k | k − P Ni =1 λ i,k D i u k ) , (6) ˆ d ,k − = M ( z ,k − C ˆ x k | k − − P Ni =1 λ i,k D i u k ) , (7) ˆ d k − = V ˆ d ,k − + V ˆ d ,k − . (8) Time Update : ˆ x k | k − = P Ni =1 λ i,k − ( A i ˆ x k − | k − + B i u k − )+ G ˆ d ,k − , (9) ˆ x ⋆k | k = ˆ x k | k − + G ˆ d ,k − . (10) Measurement Update : ˆ x k | k = ˆ x ⋆k | k + L ( y k − C ˆ x ⋆k | k − P Ni =1 λ i,k D i u k )= ˆ x ⋆k | k + ˜ L ( z ,k − C ˆ x ⋆k | k − P Ni =1 λ i,k D i u k ) , (11)where L ∈ R n × l , ˜ L , LU ∈ R n × ( l − p H ) , M ∈ R p H × p H and M ∈ R ( p − p H ) × ( l − p H ) are observer gain matrices thatare designed according to Theorem 1. The main result inTheorem 1 is derived by minimizing the “volume” of theset of compatible states and unknown inputs, quantified bythe radii δ dk − , δ x,⋆k and δ xk . Note also that we applied L = LU U ⊤ = ˜ LU ⊤ from Lemma 1 into (11). The state and in-put estimation errors are defined as ˜ x k | k , x k − ˆ x k | k , ˜ d k − , d k − − ˆ d k − , ˜ d ,k − , d ,k − − ˆ d ,k − , ˜ d ,k − , d ,k − − ˆ d ,k − respectively. In Lemmas 1 and 2, we will providenecessary conditions for boundedness of estimation errorsand sufficient conditions for stability of the observer. All theproofs are provided in the Appendix. Lemma 1 (Necessary Conditions for Boundedness of Set–Valued Estimates [6, Lemma 1]) . The input and state estima-tion errors, ( ˜ d k − and ˜ x k | k ), are bounded for all k (i.e., theset-valued estimates are bounded with radii δ dk − , δ x,⋆k , δ xk < ∞ ), only if M Σ = I , p ≤ l , M C G = I and LU = 0 . Consequently, rk( C G ) = p − p H , M = Σ − , M =( C G ) † and L = LU U ⊤ = ˜ LU ⊤ . Lemma 2 (Sufficient Conditions for Observer Stability) . Asufficient condition for the stability of the set-valued observeris that ( A k , C ) is uniformly detectable b for each k , where A k , ( I − G M C ) ˆ A k and ˆ A k , P Ni =1 λ i,k A i − G M C .B. Optimal H ∞ -Observer In this section, we provide sufficient conditions for the existence of a set-valued observer for system (1) with anysequence { λ i,k } ∞ k =0 for all i ∈ { , , . . . , N } that satisfies ≤ λ i,k ≤ , P Ni =1 λ i,k = 1 , ∀ k in the sense of H ∞ (i.e.,minimizing the sum of squares of the state estimation errorsequence). Furthermore, we introduce a relatively simpleapproach to find such an observer, which involves solving asemi-definite program with Linear Matrix Inequalities (LMI)as constraints. We will also show that given some structuralconditions for the system, the upper bounds of the estimationerrors for both states and unknown inputs are guaranteed toconverge to steady state. Theorem 1 ( H ∞ -Observer Design) . Suppose Lemma 1 holdsand there exist matrices Y and S ≻ with appropriatedimensions such that S ( A i ) ⊤ ( S − C ⊤ Y ⊤ ) 0 I ∗ S (cid:2) S − Y C − Y (cid:3) ∗ ∗ ηI ∗ ∗ ∗ ηI ≻ for all i ∈ { , , . . . , N } . Then, there exists an η per-formance bounded H ∞ -observer for system (1) with anysequence { λ i,k } ∞ k =0 for all i ∈ { , , . . . , N } that satisfies ≤ λ i,k ≤ , P Ni =1 λ i,k = 1 , ∀ k when using ˜ L = S − Y ,i.e., k T ˜ x,w,v k ≤ η , where T ˜ x,w,v is the transfer functionmatrix that maps the noise signals P Ni =1 λ i,k (cid:2) w i ⊤ k v i ⊤ k (cid:3) T to the updated state estimation error ˜ x k | k , x k − ˆ x k | k .Furthermore, the optimal filter gain ˜ L = S ⋆ − ˜ Y ⋆ with η ⋆ H ∞ -performance can be obtained from the following semi-definite programming with LMI constraints: ( η ⋆ , S ⋆ , Y ⋆ ) ∈ arg min η,S,Y ηs.t S ( A i ) ⊤ ( S − C ⊤ Y ⊤ ) 0 I ∗ S (cid:2) S − Y C − Y (cid:3) ∗ ∗ ηI ∗ ∗ ∗ ηI ≻ , ∀ i ∈ { , , .., N } . (12) b For conciseness, the readers are referred to [13, Section 2] for thedefinition of uniform detectability. A spectral test can be found in [14]. lthough Theorem 1 equips us with an approach fordesigning an H ∞ -observer for the linear parameter-varyingsystem in (1) when one exists, it would still be valuableto find a structural and conveniently testable property forthe constituent linear time-invariant systems in (2) that is necessary for the existence of such an observer. Knowingsuch conditions would be beneficial in the sense that if theyare not satisfied, the designer knows a priori that there doesnot exist any H ∞ -observer for such an attacked system. Thiswill be the goal of Theorem 2. Theorem 2 (Necessary Conditions for the Existence of an H ∞ -observer) . There exists a simultaneous state and un-known input H ∞ -observer for system (1) with any sequence { λ i,k } ∞ k =0 for all i ∈ { , , . . . , N } that satisfies ≤ λ i,k ≤ , P Ni =1 λ i,k = 1 , ∀ k , only if each ( A i , G, C, H ) is stronglydetectable c for all i ∈ { , , . . . , N } . Next, we characterize the resulting radii δ xk and δ dk − whenusing the proposed H ∞ -observer. Theorem 3 (Radii of Set-Valued Estimates) . The radii δ xk and δ dk − can be obtained as: δ xk = δ x θ k + η P ki =1 θ i − ,δ dk − = βδ xk − + k V M C k η w + (cid:2) k ( V M C G − V ) M T k + k V M T k (cid:3) η v , where β , max i ∈{ , ,...,N } k V M C + V M C A e,i k , Ψ , I − ˜ LC , Φ , I − G M C , A e,i , ΨΦ( A i − G M C ) ,θ , max i ∈{ , ,...,N } k A e,i k . The resulting fixed-order set-valued observer is summa-rized in Algorithm 1.So far, we have designed an H ∞ -observer for our linearparameter-varying system and provided necessary conditionsfor the boundedness of the set-valued estimates. It is worthmentioning that for the linear time-invariant case in [6],strong detectability of the system is also a sufficient conditionfor the convergence of the radii δ xk and δ dk − to steady state.In our parameter-varying case, even if all constituent lineartime-invariant systems are strongly detectable, there is noguarantee that the radii converge. The reason is that theconvergence hinges on the stability of the product of time-varying matrices (cf. proof of Theorem 4), which is notguaranteed even if all the multiplicands are stable. In thenext theorem, we discuss some sufficient conditions for theconvergence of the radii to steady state. Theorem 4 (Convergence) . Suppose the conditions of The-orem 1 hold. Then, the radii δ xk and δ dk − are convergent if k A e,i k < for all i ∈ { , , . . . , N } , where A e,i is definedin Theorem (3) . Moreover, the steady state radii is given by: lim k →∞ δ xk = η − θ , lim k →∞ δ dk = ηβ − θ + η w k V M C k + η v ( k V M T k + k R k ) , where η , ( k Γ k η v + k ΨΦ k η w ) , R , V M C G M T − V M T , Γ , − (ΨΦ G M T + Ψ G M T + ˜ LT ) . c For brevity, the readers may refer to [6] for the definition of strongdetectability.
Algorithm 1
Fixed-Order Input & State Set-Valued Observer Initialize: M = Σ − ; M = ( C G ) † ; Φ = I − G M C ;Compute ˜ L via Theorem 1; Ψ = I − ˜ LC ; θ , max i ∈{ , ,...,N } k ΨΦ( A i − G M C ) k ; ˆ x | = ˆ x = centroid ( ˆ X ) ; δ x = min δ {k x − ˆ x | k ≤ δ, ∀ x ∈ ˆ X } ; ˆ d , = M ( z , − C ˆ x | − D u ) ; for k = 1 to K do ⊲ Estimation of d ,k − and d k − ˆ x k | k − = P Ni =1 λ i,k A i ˆ x k − | k − + P Ni =1 λ i,k B i u k − + G ˆ d ,k − ; ˆ d ,k − = M ( z ,k − C ˆ x k | k − − P Ni =1 λ i,k D i u k ); ˆ d k − = V ˆ d ,k − + V ˆ d ,k − ; δ dk − = δ xk − k V M C + V M C ˆ A k k + η v ( k ( V M C G − V ) M T k + k V M T k )+ η w k V M C k ; ˆ D k − = { d ∈ R l : k d − ˆ d k − k ≤ δ dk − } ; ⊲ Time update ˆ x ⋆k | k = ˆ x k | k − + G ˆ d ,k − ; ⊲ Measurement update ˆ x k | k = ˆ x ⋆k | k + ˜ L ( z ,k − C ˆ x ⋆k | k − P Ni =1 λ i,k D i u k ) ; δ xk = δ x θ k + η P ki =1 θ i − ; ˆ X k = { x ∈ R n : k x − ˆ x k | k k ≤ δ xk } ; ⊲ Estimation of d ,k ˆ d ,k = M ( z ,k − C ˆ x k | k − P Ni =1 D i u k ) ; end for Remark 1.
Alternatively, we can trade off between “opti-mality” of the observer and “convergence” of the radii. Wecan iteratively find η (e.g., by line search) that satisfies thefollowing feasibility problem: Find (
S, Y ) s.t S ∗ I ( S − Y C ) A i S (cid:2) S − Y C − Y (cid:3) ∗ ∗ η I ∗ ∗ ∗ η I ≻ , ∀ i ∈ { , , .., N } , as well as the sufficient condition in Theorem 4, i.e., k A e,i k < for all i ∈ { , , . . . , N } . Although the designedobserver may not be optimum in minimum H ∞ sense whenusing this alternative method, we can guarantee the steadystate convergence of the radii instead. V. S
IMULATION E XAMPLE
In this section, we consider a convex combination of twoconstituent linear time-invariant strongly detectable subsys-tems that have been used in the literature as a benchmarkfor some state and input filters (e.g., [12]): A = (cid:20) . . − . (cid:21) ; A = (cid:20) . . − .
35 1 (cid:21) ; C = (cid:20) . . . (cid:21) ; G = (cid:20) − .
02 0 . . − . (cid:21) ; H = (cid:20) . . (cid:21) ; B = B = I × ; D = 0 × . The unknown inputs used in this example are as given inFigure 1, while the initial state estimate and noise signals(drawn uniformly) have bounds δ x = 0 . , η w = 0 . and η v = 10 − . We also picked uniformly random coefficients, λ i,k , that satisfies ≤ λ i,k ≤ , P Ni =1 λ i,k = 1 , ∀ k . Basedon the results of Theorem 1 and by solving the correspondingsemi-definite programming problem using YALMIP [15] andOSEK [16] as the solver, we find S ⋆ = (cid:20) . . . . (cid:21) , Y ⋆ = (cid:20) . . (cid:21) and the H ∞ -observer gain as ˜ L = S ⋆ − Y ⋆ = (cid:20) − . . (cid:21) . Then, applying Algorithm 1, wesummarized the set-valued state and unknown input resultsin Figures 1 and 2. The radii are observed to be convergentto steady state in Figure 2.Fig. 1: Actual states x , x and their estimates, as well asunknown inputs d and d and their estimates.Fig. 2: Actual estimation errors and radii of set-valuedestimates of states, k ˜ x k | k k , δ xk , and unknown inputs, k ˜ d k k , δ dk .VI. C ONCLUSION
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ROOFS
A. Proof of Lemma 2 (5)-(9) and plugging M = Σ − into (13) imply that ˆ d ,k = M ( C ˜ x k | k + Σ d ,k + P Ni =1 λ i,k v i ,k ) , (13) ˆ d ,k − = M ( C ( P Ni =1 λ i,k − A i ˜ x k − | k − + G ˜ d ,k − (14) + G d ,k − + P Ni =1 λ i,k − w ik − )+ P Ni =1 λ i,k v i ,k ) . ˜ d ,k = d ,k − ˆ d ,k = − M ( C ˜ x k | k + P Ni =1 λ i,k v i ,k ) . (15) (15) and setting M = ( C G ) † (Lemma 1) in (14), return ˜ d ,k − = − M ( C ˆ A k − ˜ x k − | k − − C G M P Ni =1 λ i,k − v i ,k − + C P Ni =1 λ i,k − w ik − + P Ni =1 λ i,k v i ,k ) . (16) Defining ˜ x ⋆k | k , x k − ˆ x ⋆k | k , from (1), (9) and (10) we obtain ˜ x ⋆k | k = P Ni =1 λ i,k − ( A i ˜ x k − | k − + w ik − )+ G ˜ d ,k − + G ˜ d ,k − (17) In addition, from (5) and (11) and (15)-(17) we conclude: ˜ x k | k = ( I − ˜ LC )˜ x ⋆k | k − ˜ L P Ni =1 λ i,k v i ,k . (18) ˜ x ⋆k | k = A k − ˜ x k − | k − − ( I − G M C )( G M (19) P Ni =1 λ i,k − ( v i ,k − − w ik − )) − G M P Ni =1 λ i,k v i ,k . Now, we define w k − , − G M P Ni =1 λ i,k v i ,k − ( I − G M C )( G M P Ni =1 λ i,k − ( v i ,k − − w ik − )) , v k − , P Ni =1 λ i,k v i ,k . Then, (18)-(19) imply that ˜ x ⋆k | k = A k − ˜ x k − | k − + w k − , ˜ x k | k = ( I − ˜ LC ) A k − ˜ x k − | k − +( I − ˜ LC ) w k − − ˜ Lv k − . (20) ow, consider the following linear time-varying system: x k +1 = A k x k + w k , y k = C x k + v k . (21)Systems (20) and (21) are equivalent from the viewpointof estimation, since the estimation error equations for bothproblems are the same, hence they both have the sameobjective. Therefore, the pair ( A k , C ) needs to be uniformlydetectable such that the observer is stable [13, Section 5]. (cid:4) B. Proof of Theorem 1
Starting from (20), we have ˜ x k | k = ( I − ˜ LC ) A k − ˜ x k − | k − +( I − ˜ LC ) w k − − ˜ Lv k − , from which we can define a system with ˜ x k | k as its state and ˜ z k | k = ˜ x k | k as the output: ˜ x k | k = ( I − ˜ LC ) A k − ˜ x k − | k − + (cid:2) I − ˜ LC − ˜ L (cid:3)(cid:20) w k − v k − (cid:21) , ˜ z k | k = ˜ x k | k . By [17, Lemma 3], this system has an H ∞ performancebounded by η , if there exists a symmetric positive definitematrix P with rank n such that: P ( I − ˜ LC ) A i P (cid:2) I − ˜ LC − ˜ L (cid:3) ∗ P P ∗ ∗ ηI ∗ ∗ ∗ ηI ≻ , ∀ i ∈{ , , . . . , N } . (22) Notice that the referenced lemma requires the existence of a bounded matrix sequence , which in our case is a sequenceof time-invariant matrices ( P is the same for each k ), that isobviously bounded. By plugging S = P − ≻ and applyingsome similarity transformations, we obtain S ∗ ∗ ∗ I ∗ ∗ ∗ I P ( I − ˜ LC ) A i P (cid:2) I − ˜ LC − ˜ L (cid:3) ∗ P P ∗ ∗ ηI ∗ ∗ ∗ ηI S ∗ ∗ ∗ I ∗ ∗ ∗ I = S A i ⊤ ( I − C ⊤ ˜ L ⊤ ) S I ∗ S (cid:2) I − ˜ LC − ˜ L (cid:3) ∗ ∗ I ∗ ∗ ∗ ηI ≻ ∀ i ∈{ , , . . . , N } . Setting Y , S ˜ L completes the proof. (cid:4) C. Proof of Theorem 2
Suppose, for contradiction, that there exists an H ∞ -observer for system (1) with any sequence { λ i,k } ∞ k =0 for all i ∈ { , , . . . , N } that satisfies ≤ λ i,k ≤ , P Ni =1 λ i,k =1 , ∀ k , but one of the constituent linear time-invariant systems(e.g., ( A j , G, C, H ) ) is not strongly detectable. Since the H ∞ -observer exists for any sequence of λ i,k , particularlyit exists when λ j,k = 1 and λ ij,k = 0 , ∀ i = j for all k . However, we know from [6] that strong detectability isnecessary for the stability of the linear time-invariant system ( A j , G, C, H ) , which is a contradiction. (cid:4) D. Proof of Theorem 3
To prove Theorem 3, we first find closed form expressionsfor the state and input estimation errors in the following:
Lemma 3.
The state and input estimation errors are ˜ x k | k =( Q k − j =0 A e,k − j )˜ x | + P ki =1 ( Q i − j =0 A e,k − j )(Ψ w k − i − ˜ Lv k − i ) , ˜ d k − = P Ni =1 λ i,k − ( − V M C − V M C A e,i )˜ x k − | k − + ( − V M + V M C G M ) T P Ni =1 λ i,k − v ik − − V M C P Ni =1 λ i,k − w ik − − V M T P Ni =1 λ i,k v ik . Proof.
From (20), we have ˜ x k | k = Ψ A k ˜ x k − | k − + Ψ w k − − ˜ Lv k − . (23)We use induction and (23) to obtain ˜ x | = Ψ A ˜ x | + Ψ w − ˜ Lv = A e, x | + Ψ w − − ˜ Lv − = ( Q − j =0 A e, − )˜ x | + P i =1 ( Q i − j =0 A e, − j )(Ψ w k − − ˜ Lv − i )˜ x k +1 | k +1 = Ψ A k +1 ˜ x k | k + Ψ w k − ˜ Lv k =Ψ A k +1 (cid:2) ( Q k − j =0 A e,k − j )˜ x | + P ki =1 ( Q i − j =0 A e,k − j )(Ψ w k − i − ˜ Lv k − i ) (cid:3) +Ψ w k − ˜ Lv k = ( A e,k +1 A e,k ...A e, )˜ x | + Ψ w k − ˜ Lv k + P ki =1 ( A e,k +1 A e,k ...A e,k − ( i − )(Ψ w k − i − ˜ Lv k − i ))=( Q k +1 j =0 A e,k +1 − j )˜ x | + P ki =0 ( Q i − j =0 A e,k − j )(Ψ w k − i − ˜ Lv k − i )=( Q k +1 j =0 A e,k +1 − j )˜ x | + P k +1 i =1 ( Q i − j =0 A e,k +1 − j )(Ψ w k +1 − i − ˜ Lv k +1 − i ) . As for ˜ d k − , (15)-(16) imply ˜ d k − = V ˜ d ,k − + V ˜ d ,k − = P Ni =1 λ i,k − ( − V M C − V M C A e,i )˜ x k − | k − +( − V M + V M C G M ) T P Ni =1 λ i,k − v ik − − V M C P Ni =1 λ i,k − w ik − − V M T P Ni =1 λ i,k v ik . (24) (cid:4) Now, we are ready to prove Theorem 3. First, we define B e,k , Q k − j =0 A e,k − j , C ie,k , Q i − j =0 A e,k − j , t k , Ψ w k − ˜ Lv k (25) for ≤ i ≤ k . Then, from Lemma 3, it follows that k ˜ x k | k k = k B e,k ˜ x | + P ki =1 C ie,k t k − i k≤ k B e,k kk ˜ x | k + k P ki =1 C ie,k t k − i k . (26) Moreover, by similar reasoning, k B e,k k = k Q k − j =0 A e,k − j k ≤ Q k − j =0 k A e,k − j k = Q k − j =0 k ΨΦ ˆ A k − j k = Q k − j =0 k ΨΦ P Ni =1 λ ik − j ( A i − G M C ) k = Q k − j =0 k P Ni =1 λ ik − j ΨΦ( A i − G M C ) k (27) ≤ Q k − j =0 P Ni =1 λ ik − j k ΨΦ( A i − G M C ) k≤ Q k − j =0 θ = θ k , k P ki =1 C ie,k t k − i k ≤ P ki =1 k C ie,k t k − i k≤ P ki =1 k C ie,k kk t k − i k , (28) k C ie,k k = k Q i − j =0 A e,k − j k ≤ Q i − j =0 k A e,k − j k = Q i − j =0 k P Ns =1 λ s,k − j A e,s k ≤ Q i − j =0 θ ≤ θ i − . (29) Furthermore, from the definition of w k and (25) we have w k − i = − Φ( G M P Ns =1 λ s,k − i v s ,k − i − P Ns =1 λ s,k − i w sk − i ) − G M P Ns =1 λ s,k − i v s ,k − i , k t k − i k = k Ψ w k − i − ˜ Lv k − i k = k − ΨΦ G M T P Ns =1 λ s,k − i v sk − i + ΨΦ P Ns =1 λ s,k − i w sk − i − Ψ G M T P Ns =1 λ s,k − i v sk − i − ˜ LT P Ns =1 v sk − i k = k P Ns =1 λ s,k − i (Γ v sk − i + (ΨΦ) w sk − i ) k ≤ η , from which,as well as (26)-(29), we conclude that k ˜ x k | k k ≤ k ˜ x | k θ k + η P ki =1 θ i − = k ˜ x | k θ k + η − θ k − θ , δ xk . (30) As for δ dk − , using Lemma 3 and (24), triangle inequalityand the facts that ≤ λ i,k ≤ , P Ni =1 λ i,k = 1 andsubmultiplicativity of matrix norms, we obtain the result. (cid:4) E. Proof of Theorem 4
Notice that ≤ k A e,i k ≤ θ < for all i ∈ { , , . . . , N } by assumption. So, θ k in (30) vanishes in steady state,which gives us the following steady state estimation radius: lim k →∞ δ xk = lim k →∞ (cid:16) k ˜ x | k θ k + η − θ k − θ (cid:17) = η − θ . Usingthis and starting from the expression for δ dk − in Theorem3, it converges to steady state, as follows: lim k →∞ δ dk − =(lim k →∞ βδ xk − )+ k V M C k η w +( k ( V M C G − V ) M T k + k V M T k ) η v = ηβ − θ + η w k V M C k + η v ( k V M T k + k R k ) ..