Simultaneous Mode, Input and State Set-Valued Observers with Applications to Resilient Estimation Against Sparse Attacks
aa r X i v : . [ ee ss . S Y ] J a n Simultaneous Mode, Input and State Set-Valued Observers withApplications to Resilient Estimation against Sparse Attacks
Mohammad Khajenejad Sze Zheng Yong
Abstract — A simultaneous mode, input and state set-valuedobserver is proposed for hidden mode switched linear systemswith bounded-norm noise and unknown input signals. Theobserver consists of two constituents: (i) a bank of mode-matched observers and (ii) a mode estimator. Each mode-matched observer recursively outputs the mode-matched setsof compatible states and unknown inputs, while the modeestimator eliminates incompatible modes, using a residual-basedcriterion. Then, the estimated sets of states and unknown inputsare the union of the mode-matched estimates over all com-patible modes. Moreover, sufficient conditions to guarantee theelimination of all false modes are provided and the effectivenessof our approach is exhibited using an illustrative example.
I. I
NTRODUCTION
Potential vulnerability of Cyber-Physical Systems (CPS)to adversarial attacks and henceforth their security, areemerging as an important and critical issue. Given thatattackers are often strategic, there are many potential avenuesthrough which they can cause harm, steal information/power,etc. Recent incidents of attacks on CPS, e.g., the Maroochywater system and Ukrainian power grid, [1], [2], highlight aneed for new resilient estimation and control designs.In particular, an adversary’s ability to inject counterfeitdata into sensor and actuator signals (false data injection) orto compromise an unknown subset of vulnerable sensors andactuators (e.g., [3]–[9]) in order to mislead the system opera-tor has been a subject of considerable interest in recent years.This problem can be considered in a more general frameworkof hidden mode switched linear systems with unknown inputsand also has applications in urban transportation systems [6],aircraft tracking and fault detection [10], etc.
Literature review.
The filtering problem of hidden modesystems without unknown inputs have been extensivelystudied (see, e.g., [11], [12] and references therein). Morerecently, an extension to consider unknown inputs has beenproposed in [6] for stochastic systems. However, these meth-ods mainly focus on obtaining point estimates, i.e., the mostlikely or best single estimates, and do not directly apply tobounded-error models, i.e., uncertain dynamic systems withset-valued uncertainties (e.g., bounded-norm noise), wherethe sets of all modes, states and unknown inputs that arecompatible with sensor observations are desired.On the other hand, set-membership or set-valued stateobservers (e.g., [13]–[15]) are capable of estimating the set ofcompatible states and are preferable to stochastic estimationwhen hard accuracy bounds are important, e.g., to guarantee a Mohammad Khajenejad and Sze Zheng Yong are with the Schoolfor Engineering of Matter, Transport and Energy, Arizona State University,Tempe, AZ, USA (e-mail: [email protected], [email protected]). safety. Moreover, a recent extension to also compute the setof unknown input signals in addition to the states has beenintroduced in [16]. However, these approaches do not applyto hidden mode systems that we consider in this paper.In the context of resilient estimation against sparse falsedata injection attacks, numerous approaches were proposed(e.g., [3]–[9]), but they all only obtain point estimates,as opposed to set-valued estimates. Moreover, only sensorattacks have been considered, although actuator attacks arealso a source of concern in CPS security. On the other hand,our prior work in [16], [17] design a fixed-order set-valuedobserver that simultaneously outputs sets of compatible stateand input estimates despite data injection attacks for lineartime-invariant and linear parameter-varying systems, withoutconsidering the hidden modes, i.e., with the assumption thatthe subset of attacked sensors and actuators is known.To consider hidden modes, a common approach is toconstruct residual signals, especially for fault detection [18],where a threshold based on the residual signal is used to dis-tinguish between consistent and inconsistent modes. Usingthis idea, [19] presents a robust control inspired resilient stateestimator for models with bounded-norm noise that consistsof local estimators, residual detectors and a global fusiondetector. However, in their setting, only sensors are attacked,while the existence of the observers are assumed with noobserver design approach nor performance guarantees.
Contributions.
The goal of this paper is to simultaneouslyconsider state and unknown input estimation as well as modedetection for hidden mode switched linear systems withbounded-norm noise and unknown inputs. To address this,we propose a multiple-model approach that leverages theoptimally designed set-valued state and input H ∞ observersin our previous work [16] to obtain a bank of mode-matched set-valued observers in combination with a novelmode observer based on elimination. Our mode eliminationapproach uses the upper bound of the norm of to-be-designedresidual signals to remove inconsistent modes from the bankof observers. In particular, we provide a tractable method tocalculate an upper bound signal for the residual’s norm andprove that the upper bound signal is a convergent sequence.Moreover, we provide sufficient conditions to guarantee thatall false modes will be eventually eliminated. Notation. R n denotes the n -dimensional Euclidean spaceand N nonnegative integers. For a vector v ∈ R n and a matrix M ∈ R p × q , k v k , √ v ⊤ v , k v k ∞ , max ≤ i ≤ n v i and k M k and σ min ( M ) denote their induced -norm and non-trivial leastsingular value, respectively.I. P ROBLEM S TATEMENT
Consider a hidden mode switched linear system withbounded-norm noise and unknown inputs (i.e., a hybridsystem with linear and noisy system dynamics in each mode,and the mode and some inputs are not known/measured): x k +1 = Ax k + Bu qk + G q d qk + w k ,y k = Cx k + Du qk + H q d qk + v k , (1)where x k ∈ R n is the continuous system state and q ∈ Q = { , , . . . , Q } is the hidden discrete state or mode . For each(fixed) mode q , u qk ∈ U qk ⊂ R m is the known input, d qk ∈ R p the unknown but sparse input or attack signal, i.e., everyvector d qk has precisely ρ ∈ N nonzero elements where ρ isa known parameter, y k ∈ R l is the output, whereas w k ∈ R n and v k ∈ R l are process and measurement 2-norm boundeddisturbances with known parameters η w and η v as their 2-norm bounds respectively. The matrices A ∈ R n × n , B ∈ R n × m , G q ∈ R n × p , C ∈ R l × n , D ∈ R l × m and H q ∈ R l × p are known and no prior ‘useful’ knowledge or assumptionof the dynamics of d qk , except sparsity is assumed.More precisely, G q and H q represent the different hy-pothesis for each mode q ∈ Q , about the sparsity pattern ofthe unknown inputs, which in the context of sparse attackscorresponds to which actuators and sensors are attacked ornot attacked. In other words, we assume that G q = G I qG and H q = H I qH for some input matrices G ∈ R n × t a and H ∈ R l × t s , where t a and t s are the number of vulnerable actuatorand sensor signals respectively. Note that ρ qa ≤ t a ≤ m and ρ qs ≤ t s ≤ l , where ρ qa ( ρ qs ) is the number of attacked actuator(sensor) signals and clearly cannot exceed the number ofvulnerable actuator (sensor) signals, which in turn cannotexceed the total number of actuators (sensors). Furthermore,we assume that the total number of unknown inputs/attacksin each mode is known and equals ρ = ρ a + ρ s (sparsityassumption). Moreover, the index matrix I qG ∈ R t a × ρ ( I qH ∈ R t s × ρ ) represents the sub-vector of d k ∈ R ρ that indicatessignal magnitude attacks on the actuators (sensors).Note that the approach in our paper can be easily extendedto handle mode-dependent A , B , C , D , w k , v k , η w and η v but is omitted to simplify the notation. Moreover, throughoutthe paper, we assume, without loss of generality, that foreach possible mode q , the system ( A, G q , C, H q ) is stronglydetectable [16, Definition 1], since this is a necessary andsufficient condition for obtaining meaningful set-valued stateand input estimates when the mode is known.Using the modeling framework above, the simultaneousstate, unknown input and hidden mode estimation problemis threefold and can be stated as follows: Problem 1.
Given a switched linear hidden mode discrete-time bounded-error system with unknown inputs (1) , Design a bank of mode-matched observers that for eachmode optimally finds the set estimates of compatiblestates and unknown inputs in the minimum H ∞ -normsense, i.e., with minimum average power amplification,conditional on the mode being true. Develop a mode observer via elimination and the corre- sponding criterion to eliminate false modes. Find sufficient conditions for eliminating all false modes.
III. P
ROPOSED O BSERVER D ESIGN
In this section, we propose a multiple-model approach forsimultaneous mode, state and unknown input estimation for(1), where the goal of the observer is to find compatible setestimates ˆ D k , ˆ X k and ˆ Q k for unknown inputs, states andmodes at time step k , respectively. A. Overview of Multiple-Model Approach
The multiple-model design approach consists of threecomponents: (i) designing a bank of mode-matched set-valued observers, (ii) designing a mode observer for elimi-nating incompatible modes using residual detectors, and (iii)a global fusion observer that outputs the desired set-valuedmode, input and state estimates.
1) Mode-Matched Set-Valued Observer:
First, we designa bank of mode-matched observers, which consists of Q simultaneous state and input H ∞ set-valued observers basedon the optimal fixed-order observer design in [16], whichwe briefly summarize here. For each mode-matched observercorresponding to mode q , following the approach in [16,Section 3.1], we consider set-valued fixed-order estimatesof the form: ˆ D qk − = { d k − ∈ R p : k d k − − ˆ d qk − k ≤ δ d,qk − } , (2) ˆ X qk = { x k ∈ R n : k x k − ˆ x qk | k k ≤ δ x,qk } , (3)where their centroids are obtained with the following three-step recursive observer that is optimal in H ∞ -norm sense: Unknown Input Estimation : ˆ d q ,k = M q ( z q ,k − C q ˆ x qk | k − D q u qk )ˆ d q ,k − = M q ( z q ,k − C q ˆ x qk | k − − D q u qk )ˆ d qk − = V q ˆ d q ,k − + V q ˆ d q ,k − (4) Time Update : ˆ x qk | k − = A ˆ x qk − | k − + Bu qk − + G q ˆ d q ,k − ˆ x ⋆,qk | k = ˆ x qk | k − + G q ˆ d q ,k − (5) Measurement Update : ˆ x qk | k = ˆ x ⋆,qk | k + ˜ L q ( z q ,k − C q ˆ x ⋆,qk | k − D q u qk ) (6)where ˜ L q ∈ R n × ( l − p Hq ) , M q ∈ R p Hq × p Hq and M q ∈ R ( p − p Hq ) × ( l − p Hq ) are observer gain matrices that are chosenin the following theorem from [16] to minimize the “volume”of the set of compatible states and unknown inputs, quantifiedby the radii δ d,qk − and δ x,qk . Theorem 1. [16, Lemma 2 & Theorem 4] Suppose thesystem ( A, G q , C, H q ) is strongly detectable, M q Σ q = I and M q C q G q = I . Then, for each mode q , there exists a stableand optimal (in H ∞ -norm sense) observer with gain ˜ L q ,where the input and state estimation errors, ˜ d qk − , d qk − − ˆ d qk − and ˜ x qk | k , x k − ˆ x qk | k , are bounded for all k (i.e., theset-valued estimates are bounded with radii δ d,qk − , δ x,qk < ∞ ),and the observer gains and the set estimates are given in [16,Theorem 2 & Algorithm 1]. lgorithm 1 Simultaneous Mode, State and Input Estimation ˆ Q = Q ; for k = 1 to N do for q ∈ ˆ Q k − do ⊲ Mode-Matched State and Input Set-Valued EstimatesCompute T q , M q , M q , ˜ L q , ˆ x ⋆,qk | k , ˆ X qk , ˆ D qk − via Theorem 1; z q ,k = T q y k ; ⊲ Mode Observer via Elimination ˆ Q k = ˆ Q k − ;Compute r qk via Definition 1 and ˆ δ qr,k via Theorem 3; if k r qk k > ˆ δ qr,k then ˆ Q k = ˆ Q k \{ q } ; end if end for ⊲ State and Input Estimates ˆ X k = ∪ q ∈ ˆ Q k ˆ X qk ; ˆ D k = ∪ q ∈ ˆ Q k ˆ D qk ; end for
2) Mode Estimation Observer:
To estimate the set ofcompatible modes, we consider an elimination approach thatcompares residual signals against some thresholds. Specifi-cally, we will eliminate a specific mode q , if k r qk k > ˆ δ qr,k ,where the residual signal r qk is defined as follows and thethresholds ˆ δ qr,k will be derived in Section III-B. Definition 1 (Residuals) . For each mode q at time step k ,the residual signal is defined as: r qk , z q ,k − C q ˆ x ⋆,qk | k − D q u qk .
3) Global Fusion Observer:
Then, combining the outputsof both components above, our proposed global fusion ob-server will provide mode, unknown input and state set-valuedestimates at each time step k as: ˆ Q k = { q ∈ Q k r qk k ≤ ˆ δ qr,k } , ˆ D k − = ∪ q ∈ ˆ Q k D qk − , ˆ X k = ∪ q ∈ ˆ Q k X qk . The multiple-model approach is summarized in Algorithm 1.
B. Mode Elimination Approach
The idea is simple. If the residual signal of a particularmode exceeds its upper bound conditioned on this modebeing true, we can conclusively rule it out as incompatible.To do so, for each mode q , we first compute an upper bound( ˆ δ qr,k ) for the 2-norm of its corresponding residual at time k , conditioned on q being the true mode. Then, comparingthe 2-norm of residual signal in Definition 1 with ˆ δ qr,k , wecan eliminate mode q if the residual’s 2-norm is strictlygreater than the upper bound. This can be formalized usingthe following proposition and theorem. Proposition 1.
Consider mode q at time step k , its residualsignal r qk (as defined in Definition 1) and the unknown truemode q ∗ . Then, r qk = r q |∗ k + ∆ r q | q ∗ k , wherer q |∗ k , z q ∗ ,k − C q ˆ x ⋆,qk | k − D q u qk = T q ∗ y k − C q ˆ x ⋆,qk | k − D q u qk , ∆ r q | q ∗ k , ( T q − T q ∗ ) y k , where r q |∗ k is the true mode’s residual signal (i.e., q = q ∗ ),and ∆ r q | q ∗ k is the residual error .Proof. This follows directly from plugging the above expres-sions into the right hand side term of Definition 1. (cid:4)
Theorem 2.
Consider mode q and its residual signal r qk at time step k . Assume that δ q, ∗ r,k is any signal that satisfies k r q |∗ k k ≤ δ q, ∗ r,k , where r q |∗ k is defined in Proposition 1. Then,mode q is not the true mode, i.e., can be eliminated at time k , if k r qk k > δ q, ∗ r,k . Proof.
To use contradiction, suppose q is the true mode. Byuniqueness of the true mode q = q ∗ , so T q = T q ∗ and byProposition 1, ∆ r q | q ∗ k = 0 and hence k r qk k = k r q |∗ k k ≤ δ q, ∗ r,k , which contradicts with the assumption. (cid:4) C. Tractable Computation of Thresholds
Theorem 2 provides a sufficient condition for mode elim-ination at each time step. To apply this sufficient condition,we need to compute an upper bound for k r q |∗ k k , i.e., our δ q, ∗ r,k signal (cf. Theorem 3) and show that it is bounded inthe following lemmas. Lemma 1.
Consider any mode q with the unknown truemode being q ∗ . Then, at time step k , we have r q |∗ k = C q ˜ x ⋆,qk | k + v q ,k = A qk t k , (7) where t k , (cid:2) ˜ x ⊤ | w ⊤ . . . w ⊤ k − v ⊤ . . . v ⊤ k (cid:3) ⊤ ∈ R ( n + l )( k +1) , A qk , [ C q A q A qek − C q A q A qek − B qe,w C q A q A qek − B qe,w . . .C q A q A qek − − i B qe,w . . . C q A q A qe B qe,w C q B ⋆,qe,w C q A q A qek − B qe,v C q A q A qek − ( B qe,v + A qe B qe,v ) . . .C q A q A qek − − i ( B qe,v + A qe B qe,v ) . . .C q A q A qe ( B qe,v + A qe B qe,v ) C q ( B q,⋆e,v + A q B qe,v ) C q B q,⋆e,v + T q ] ∈ R ( l − p Hq ) × ( n + l )( k +1) , with A q , ( I − G q M q C q )( A − G q M q C q ) , A qe , ( I − ˜ L q C q ) A q , B ⋆,qe,w , ( I − G q M q C q ) , B ⋆,qe,v , − ( I − G q M q C q )( G q M q T q ) , B qe,w , ( I − ˜ L q C q ) B ⋆,qe,w , B qe,v , ( I − ˜ L q C q ) B ⋆,qe,v and B qe,v , ( I − ˜ L q C q ) B ⋆,qe,v − ˜ L q T q , B ⋆,qe,v , − G q M q T q . Proof.
Considering (7), the first equality comes from Defini-tion 1 and z q ,k = C q x k + D q ,k u qk + v q ,k from [16], assumingthat q is the true mode, and the second equality is impliedby the first equality and the fact in [16, Appendix C] that ˜ x ⋆,qk | k = A q A qek − ˜ x | + A q A qek − (cid:2) B qe,w B qe,v (cid:3) ~w + B ⋆,qe,w w k − + ( B ⋆,qe,v + A q B qe,v ) v k − + B ⋆,qe,v v k + P k − i =1 A q A qek − − i (cid:2) B qe,w B qe,v + A qe B qe,v (cid:3) ~w i ,~w k , (cid:2) w ⊤ k v ⊤ k (cid:3) ⊤ . (cid:4) Lemma 2.
For each mode q at time step k , there exists ageneric finite valued upper bound δ qr,k < ∞ for k r q |∗ k k .Proof. Consider the following optimization problem for k r q |∗ k k by leveraging Lemma 1: δ qr,k , max t k k A qk t k k (8) s.t. t k = h ˜ x ⊤ | w ⊤ . . . w ⊤ k − v ⊤ . . . v ⊤ k i ⊤ , k ˜ x | k ≤ δ x , k w i k ≤ η w , k v j k ≤ η v ,i ∈ { , ..., k − } , j ∈ { , ..., k } . he objective 2-norm function is continuous and the con-straint set is an intersection of level sets of lower dimensionalnorm functions, which is closed and bounded, so is compact.Hence, by Weierstrass Theorem [20, Proposition 2.1.1], theobjective function attains its maxima on the constraint setand so a finite-valued upper bound exists. (cid:4) Clearly δ qr,k in Lemma 2 is the tightest possible residualnorm’s upper bound and potentially can eliminate the mostpossible number of modes, so is the best choice if wecan calculate it. But, notice that although it was straightforward to show that a finite-valued δ qr,k exists, but sincethe optimization problem in Lemma 2 is a norm maximiza-tion (not minimization) over the intersection of level setsof lower dimensional norm functions, i.e., a non-concavemaximization over intersection of quadratic constraints, itis an NP-hard problem [21]. To tackle with this complexity,we provide an over-approximation for δ qr,k in the followingTheorem 3, which we call ˆ δ qr,k . Theorem 3.
Consider mode q . At time step k , let ˆ δ qr,k , min { δ q,infr,k , δ q,trir,k } ,δ q,infr,k , k A qk t ⋆k k ,δ q,trir,k , δ x,q k C q A q A qek − k + η w k C q A q A qek − k + P k − i =1 [ η w k C q A q A qei B qe,w k + η v k C q A q A qei ( B qe,v + A qe B qe,v ) k ]+ η v ( k C q A q A qek − B qe,v k + k C q ( B q,⋆e,v + A q B qe,v ) k )+ k C q B q,⋆e,v + T q k ) + η w k C q B ⋆,qe,w k , where t ⋆k is a vertex of the following hypercube: X qk , (cid:8) x ∈ R ( n + l )( k +1) | x ( i ) | ≤ δ x , ≤ i ≤ nη w , n + 1 ≤ i ≤ n ( k + 1) η v , n ( k + 1) + 1 ≤ i ≤ ( n + l )( k + 1) (cid:9) , i.e., t ⋆k ( i ) ∈ {− δ x , δ x } , ≤ i ≤ n, {− η w , η w } , n + 1 ≤ i ≤ n ( k + 1) , {− η v , η v } , n ( k + 1) + 1 ≤ i ≤ ( n + l )( k + 1) . Then, ˆ δ qr,k is an over-approximation for δ qr,k in Lemma 2.Proof. Consider the optimization problem δ q,infr,k , max t k k A qk t k k (9) s.t. t k = h ˜ x ⊤ | w ⊤ . . . w ⊤ k − v ⊤ . . . v ⊤ k i , k ˜ x | k ∞ ≤ δ x , k w i k ∞ ≤ η w , k v j k ∞ ≤ η v , ∀ i ∈ { , ..., k − } , ∀ j ∈ { , ..., k } . Comparing (8) and (9), the two problems have the sameobjective functions, while since k . k ∞ ≤ k . k , the constraintset for (8) is a subset of the one for (9). Hence δ qr,k ≤ δ q,infr,k . Also, it is easy to see that ˆ δ qr,k ≤ δ q,trir,k , usingtriangle and sub-multiplicative inequalities. Moreover, (9) isa maximization of a convex objective function over a convexconstraint (hypercube X qk ). By a famous result [22, Corollary32.2.1], in such a problem, the objective function attains itsmaxima on some of the extreme points of the constraint set, which in this case are the vertices of the hypercube X qk . (cid:4) It can be easily seen as a corollary of Theorem 3 that:
Corollary 1. η tk , k t ⋆k k = q nδ xo + knη w + ( k + 1) lη v . Theorem 3 enables us to obtain an upper bound for k r q |∗ k k , by enumerating the objective function in (9) atvertices of the hypercube X qk and choosing the largest valueas δ q,infr,k . Moreover, we can easily calculate δ q,trir,k ; then, theupper bound is chosen as the minimum of the two as ˆ δ qr,k . Remark 1.
Although simulation results indicate that espe-cially in earlier time steps, δ q,infr,k may have smaller valuesthan δ q,trir,k , but if we only consider δ q,infr,k as the over-approximation and do not use δ q,trir,k , then we will face twodifficulties. First, as time increases, the number of requiredenumerations (i.e., the number of hypercube’s vertices whichis ( n + l )( k +1) ) increases with an exponential rate. Secondand more importantly, as Lemma 3 will indicate later, δ q,infr,k goes to infinity as time increases, so it will be unlikelyto eliminate any mode when the time step is large, i.e.,asymptotically speaking, δ q,infr,k will be useless. In contrast,again by Lemma 3, δ q,trir,k converges to some steady-statevalue, so it can be always used as an over-approximationfor δ qr,k in the mode elimination process. IV. M
ODE D ETECTABILITY
In addition to the nice properties regarding the stability andboundedness of the mode-matched set estimates of state andinput obtained from [16], we now provide some sufficientconditions for the system dynamics, which guarantee thatregardless of the observations, after some large enough timesteps, all the false (i.e., not true) modes can be eliminated,when applying Algorithm 1. To do so, first, we define theconcept of mode detectability as well as some assumptionsfor deriving our sufficient conditions for mode detectability.
Definition 2 (Mode Detectability) . System (1) is called modedetectable if there exists a natural number
K > , such thatfor all time steps k ≥ K , all false modes are eliminated. Assumption 1.
There exist known R y , R x ∈ R such that ∀ k, y k ∈ Y , { y ∈ R l k y k ≤ R y } and x k ∈ X , { x ∈ R n k x k ≤ R x } , i.e., there exist known bounds for the wholeobservation/measurement and state spaces, respectively. Assumption 2.
The unknown input/attack signal has an unlimited energy , i.e., lim k →∞ k d q ∗ k k = ∞ , where d q ∗ k , h d q ∗⊤ k d q ∗⊤ k − . . . d q ∗⊤ i ⊤ . Note that Assumption 2 is not restrictive because other-wise, the unknown input/attack signal must vanish asymp-totically, which means that the true mode (with no unknowninputs) can be inferred asymptotically.In order to derive the desired sufficient conditions formode detectability in Theorem 4, we first present the fol-lowing Lemmas 3–5. For the sake of clarity, the proofs ofthese results are given in the Appendix. emma 3.
For each mode q , lim k →∞ δ q,infr,k = ∞ . (10) lim k →∞ ˆ δ qr,k = lim k →∞ δ q,trir,k ≤ lim k →∞ δ q,trir,k = δ q,trir < ∞ , (11) where δ q,trir,k , δ x,q k C q A q A qek − k + η w k C q A q A qek − k + η w [ k C q A q A qe k k B qe,w k P k − i =0 ( k A qe k i ) + k C q B ⋆,qe,w k ] + η v [ k C q A q A qe k k B qe,v + A qe B qe,v k P k − i =0 k A qe k i ] + η v [ k C q B q,⋆e,v + T q k + k C q ( B q,⋆e,v + A q B qe,v ) k ] + η v k C q A q A qek − B qe,v k , δ q,trir , η w [ k C q B q,⋆e,w k + k C q A q A qe k / (1 − θ q )+ k B qe,w k ]+ η v [ k B qe,v + A qe B qe,v k + k C q B q,⋆e,v + T q k + k C q ( B q,⋆e,v + A q B qe,v ) k ] and θ q , k A qe k , with A q , A qe , B qe,w , B q,⋆e,w , B qe,v , B q,⋆e,v , B qe,v and B q,⋆e,v given in Lemma 1. Lemma 4.
Suppose that Assumption 1 holds. Consider twodifferent modes q = q ′ ∈ Q and their corresponding upperbounds for their residuals’ norms, δ qr,k and δ q ′ r,k , at time step k . At least one of the two modes q = q ′ will be eliminated if k C q ˆ x ⋆,qk | k − C q ′ ˆ x ⋆,q ′ k | k + D q u qk − D q ′ u q ′ k k >δ qr,k + δ q ′ r,k + R q,q ′ z (12) where R q,q ′ z , R y k T q − T q ′ k . Lemma 5.
Consider any mode q with the unknown truemode being q ∗ . Then, at time step k , we have r qk = h T q,q ∗ k B q,q ∗ k D q,q ∗ k i h t ⊤ k u q ∗ ⊤ k d q ∗⊤ k i ⊤ , where u q ∗ k , h u q ∗⊤ k u q ∗⊤ k − . . . u q ∗⊤ i ⊤ , T q,q ∗ k , ( T q ∗ − T q ) (cid:2) CA k CA k − . . . C I (cid:3) + A qk , B q,q ∗ k , ( T q ∗ − T q ) (cid:2) D CB CAB . . . CA k − B (cid:3) , D q,q ∗ k , ( T q ∗ − T q ) (cid:2) H CG CAG . . . CA k − G (cid:3) , with t k given in Lemma 1 and d q ∗ k in Assumption 2. Theorem 4 (Sufficient Conditions for Mode Detectability) . System (1) is mode detectable, i.e., all false modes willbe eliminated after some large enough time step K , usingAlgorithm 1, if the assumptions in Theorem 1 and either ofthe following hold: i) Assumption 1 and ∀ q, q ′ ∈ Q , q = q ′ , σ min ( W q,q ′ ) > δ q,trir + δ q ′ ,trir + R ′ q,q ′ y p R x + η v ; ii) Assumption 2 and T q = T q ′ holds ∀ q, q ′ ∈ Q, q = q ′ ,where W q,q ′ , h ( C q − C q ′ ) ( T q − T q ′ ) − I I D q − D q ′ i . V. S
IMULATION E XAMPLE
We consider a system that has been used as a benchmarkfor many state and input filters/observers (e.g., [6]): A = . . . . . ; G = . . ; H = ; B = 0 × ; C = I ; D = 0 × . The unknown inputs used in this example are as given inFigure 2, while the initial state estimate and noise signalshave bounds δ x = 0 . , η w = 0 . and η v = 10 − . Weassume possible attacks on the actuator and four of fivesensors, i.e., t a = 1 and t s = 4 . Moreover, we assumethat there are ρ = 4 attacks, so we should consider Q = (cid:0) (cid:1) = 5 modes. Table I indicates different modes, their attacklocation(s) and the matrix T q for each mode q , where, ascan be observed, the second set of sufficient conditions inTheorem 4 holds, i.e., T q = T q ′ for all q = q ′ , so we expectthat after some large enough time, all the false modes beeliminated, i.e., at most one (true) mode remains at each timestep, which can be seen in Figure 1, where the number ofeliminated modes at each time step is exhibited. Moreover,for each specific mode q , the signals k r qk k , k r q |∗ k k , δ q,trir,k and δ q,infr,k are depicted in Figure 1. As can be seen, upto some large enough time, at different time intervals fordifferent modes, one of the upper bounds may be tighter thanthe other, or vice-versa, so it is reasonable that we consider aminimum of them as the computed upper bound in our modeelimination algorithm. Furthermore, for all modes, δ q,trir,k iseventually convergent while δ q,infr,k diverges, as we proved inTABLE I: Different modes and their T q . Mode Attack location(s) T q q = 1 Actuator & Sensors 1,2,3 [0.2518 -0.1068 -0.2409 -0.5862 0.7236] ⊤ q = 2 Actuator & Sensors 1,2,4 [0.0080 0.7604 -0.1522 -0.5862 -0.6313] ⊤ q = 3 Actuator & Sensors 1,3,4 [-0.5357 0.7289 0.1984 -0.3774 0.0009] ⊤ q = 4 Actuator & Sensors 2,3,4 [0.7092 -0.5570 -0.1797 -0.3295 0.2143] ⊤ q = 5 Sensors 1,2,3,4 [0.1679 -0.5682 0.5198 -0.4883 0.3747] ⊤ Fig. 1: k r qr,k k , k r q |∗ r,k k and their upper bounds for differentmodes, as well as the number of eliminated modes in timeFig. 2: State and unknown input set-valued estimates.emma 3. So, after some large enough time, δ q,trir,k can beused as our upper-bound, while δ q,infr,k becomes useless. Thecorresponding set-valued estimates are provided in Figure 2.VI. C ONCLUSION
We proposed a residual-based approach for hidden modeswitched linear systems with bounded-norm noise and un-known attack signals. The proposed approach at each timestep, removes the inconsistent modes and their correspondingobservers from a bank of estimators, which includes mode-matched observers. Each mode-matched observer, condi-tioned on its corresponding mode being true, simultaneouslyfinds bounded sets of states and unknown inputs that includethe true state and inputs. Our mode elimination criterionrequired a bounded upper bound for the residual’s norm,for which we proved its existence and computed it by over-approximating the value function of a non-concave NP-hardnorm-maximization problem by expanding its constraint setand converting it into a convex maximization over a convexset with finite number of extreme points. Such a problemcan be solved by enumerating the objective function onthe extreme points of the constraint set and comparing thecorresponding values. Moreover, we proved the convergenceof the upper bound signal and derived sufficient conditionsfor eventually eliminating all false modes using our modeelimination algorithm. Finally, we demonstrated the effec-tiveness of our observer using an illustrative example.R
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ROOFS
Proof of Lemma 3.
To show (10), we first find a lowerbound for δ q,infr,k . Then, we show that the lower bounddiverges and so does δ q,infr,k . Define ˜ t ⋆k , t ⋆k /η tk , where η tk isdefined in Corollary 1. Now consider η tk σ min ( A qk ) = σ min ( η tk A qk ) = min k t k ≤ k η tk A qk t k ≤ k η tk A qk ˜ t ⋆k k = k A qk t ⋆k k = δ q,infr,k , where σ min ( A ) is the least non-trivial singular value ofmatrix A , the first equality holds since σ min ( . ) is a linearoperator, the second equality is a special case of a matrixlower bound [23] when 2-norms are considered, the inequal-ity holds since k ˜ t ⋆k k = 1 by Corollary 1, so ˜ t ⋆k is a feasiblepoint for the minimization in the third statement and thelast equality holds by Theorem 3. So far we have shownthat η tk σ min ( A qk ) is a lower bound for δ q,infr,k . Next, we willprove that η tk σ min ( A qk ) is unbounded. First, it is trivial that η tk is unbounded by its definition in Corollary 1. Second,consider the block matrix A qk in Lemma 1. By the strongdetectability assumption, matrix A qe is stable [16, Theorem3 and Appendix C], so all the block matrices of A qk , exceptthree of them which are constant matrices with respect totime, converge to zero matrices when time goes to infinity.Hence A qk converges to an infinite dimensional sparse matrix,with only three non-zero finite dimensional constant blocksand so the limit matrix has a finite rank and clearly hasa bounded minimum non-trivial singular value. Henceforth, η tk σ min ( A qk ) is unbounded, since the product of the boundedand non-zero σ min ( A qk ) and unbounded η tk is unbounded.As for (11), the first equality holds by definition of ˆ δ qr,k (cf. Theorem 3) and (10), the first inequality holds since δ q,trir,k ≤ δ q,rr,k by triangle and sub-multiplicative inequalitiesand the last equality, i.e., convergence of δ q,trir,k , follows fromstrong detectability assumption which implies the stability of A qe [16, Theorem 3]. (cid:4) Proof of Lemma 4.
Suppose, for contradiction, that none of and q ′ are eliminated. Then k C q ˆ x ⋆,qk | k + D q u qk − C q ′ ˆ x ⋆,q ′ k | k − D q ′ u q ′ k k = k r q ′ k − r qk + z q ,k − z q ′ ,k ) k ≤ k r q ′ k k + k r qk k + k z q ,k − z q ′ ,k k ≤ δ qr,k + δ q ′ r,k + R y k T q − T q ′ k , where the equality holds by Definition 1, the first inequalityholds by triangle inequality and the last inequality holds bythe assumption that none of q and q ′ can be eliminated, aswell as the boundedness assumption for the measurementspace. This last inequality contradicts with the inequality inthe lemma, thus the result holds. (cid:4) Proof of Lemma 5.
The result can be obtained by applyingProposition 1, (7) and the closed-form output signal: y k = ( CA k ) ⊤ ( CA k − ) ⊤ ... C ⊤ I ⊤ H ⊤ ( CG ) ⊤ ( CAG ) ⊤ ... ( CA k − G ) ⊤ ⊤ D ⊤ ( CB ) ⊤ ( CAB ) ⊤ ... ( CA k − B ) ⊤ ⊤ t k d q ∗ k u q ∗ k , which can be derived by using (1) and simple induction. (cid:4) Proof of Theorem 4.
To show that (i) is sufficient for asymp-totic mode detectability, consider Lemma 4 with δ q,trir,k as theupper bound. It suffices to show ∃ K ∈ N , such that (12)holds for k ≥ K, ∀ q = q ′ ∈ Q . Notice that by Definition 1, C q ˆ x ⋆,qk | k = C q x k + T q v k − r q |∗ k . Plugging this into (12), weneed to show ∃ K ∈ N such that: k W q,q ′ s q,q ′ k k > δ q,trir,k + δ q ′ ,trir,k + R q,q ′ z , ∀ k ≥ K, (13) s q,q ′ k , h x ⊤ k v ⊤ k r q |∗⊤ k r q ′ |∗⊤ k u q ⊤ k u q ′ ⊤ k i ⊤ , ∀ q = q ′ ∈ Q . A sufficient condition to satisfy (13) is that ∃ K ∈ N suchthat ∀ k ≥ K , (13) holds for all s q,q ′ k . Equivalently, it suffices min x k ,v k ,r qk ,r q ′ k k W q,q ′ s q,q ′ k k > δ q,trir,k + δ q ′ ,trir,k + R q,q ′ z s.t. k x k k ≤ R x , k v k k ≤ η v , k r q |∗ k k ≤ δ q,trir,k , k r q ′ |∗ k k ≤ δ q ′ ,trir,k , ∀ k ≥ K, ∀ q = q ′ ∈ Q . By expanding the constraint set, it is sufficient to require that ∃ K ∈ N such that: min s q,q ′ k k W q,q ′ s q,q ′ k k > δ q,trir,k + δ q ′ ,trir,k + R q,q ′ z s.t. k s q,q ′ k k ≤ R x + η v +( δ q,trir,k ) +( δ q ′ ,trir,k ) + ( u qk ) + ( u q ′ k ) ∀ k ≥ K, ∀ q = q ′ ∈ Q . Now, by matrix lower bound theorem [23] and similarargument as in the proof of Lemma 3, it is sufficient to besatisfied that ∃ K ∈ N s.t. ∀ k ≥ K, ∀ q = q ′ ∈ Q : σ min ( W q,q ′ ) > ( δ q,trir,k + δ q ′ ,trir,k + R q,q ′ z ) R x + η v +( δ q,trir,k ) +( δ q ′ ,trir,k ) +( u qk ) +( u q ′ k ) . (14) (14) provides us a time-dependent sufficient conditionfor mode detectability. In order to find a time-independent sufficient condition, notice that ( δ q,trir,k + δ q ′ ,trir,k + R q,q ′ z ) R x + η v is anupper bound for the right hand side of (14), since the latter’sdenominator is smaller than the former’s and the numeratorof the latter is an upper bound signal for the former’s by triangle and sub-multiplicative inequalities. So a sufficientcondition for (14) is ∃ K ∈ N s.t. ∀ k ≥ K, ∀ q = q ′ ∈ Q : σ min ( W q,q ′ ) > ( δ q,trir,k + δ q ′ ,trir,k + R q,q ′ z ) R x + η v . (15)Then, for the above to hold, it suffices that σ min ( W q,q ′ ) > lim k →∞ ( δ q,trir,k + δ q ′ ,trir,k + R q,q ′ z ) R x + η v , which is equivalent to (i) by (11). As for the sufficiencyof (ii), notice that by Theorems 2 and 3, Lemma 1 andDefinition 2, for mode detectability, it suffices that for anyspecific mode q , the true mode q ∗ and large enough k , k r qk k = k h T q,q ∗ k B q,q ∗ k D q,q ∗ k i h t ⊤ k u q ∗ ⊤ k d q ∗⊤ k i ⊤ k > δ q,trir,k , with t k given in (9). Since q ∗ is unknown, a sufficientcondition to satisfy the above equality is ∀ q ′ = q ∈ Q : k r qk k = k h T q,q ′ k B q,q ′ k D q,q ′ k i h t ⊤ k u q ′ ⊤ k d q ∗⊤ k i ⊤ k > δ q,trir,k . So it suffices that ∀ q ′ = q ∈ Q, ∃ d ∈ R , such that: min t ′ k k h T q,q ′ k B q,q ′ k D q,q ′ k i t ′ k k > δ q,trir,k s.t. t ′ k = h t ⊤ k u q ′ ⊤ k d q ∗⊤ k i ⊤ , k d q ∗ k k ≥ d,t k = h ˜ x ⊤ | w ⊤ . . . w ⊤ k − v ⊤ . . . v ⊤ k i , k ˜ x | k ∞ ≤ δ x , k w i k ∞ ≤ η w , k v j k ∞ ≤ η v , ∀ i ∈ { , ..., k − } , ∀ j ∈ { , ..., k } . Again by matrix lower bound theorem, a sufficient conditionfor the above inequality to hold is that ∃ d ∈ R , such that: min t k ,d k k t ′ k k > δ q,trir,k σ min h T q,q ′ k B q,q ′ k D q,q ′ k i (16) s.t. t ′ k = h t ⊤ k u q ′ ⊤ k d q ∗⊤ k i ⊤ , k d q ∗ k k ≥ d,t k = h ˜ x ⊤ | w ⊤ . . . w ⊤ k − v ⊤ . . . v ⊤ k i , k ˜ x | k ∞ ≤ δ x , k w i k ∞ ≤ η w , k v j k ∞ ≤ η v , ∀ i ∈ { , ..., k − } , ∀ j ∈ { , ..., k } . Finally, since δ q,trir,k ≤ δ q,trir,k and k t ′ k k = k h t ⊤ k u q ′ ⊤ k d q ∗⊤ k i k ≥ q + 0 + k d q ∗⊤ k k = k d q ∗⊤ k k , then a sufficient condition for (16) is that k d q ∗⊤ k k > δ q,trir,k σ min ( h T q,q ′ k B q,q ′ k D q,q ′ k i ) . (17)Now suppose that T q = T q ′ (otherwise the matrix in thedenominator of (17) is zero and it never holds). Asymptoti-cally speaking, the right hand side of (17) converges to ˜ δ , max { , ( δ q,trir /σ q,q ′ ) } , since δ q,trir,k converges to δ q,trir andthe least singular value in the denominator either diverges orconverges to some steady value σ q,q ′ . So we set d equal toany real number strictly grater than ˜ δ . By unlimited energyassumption for attack signal, after some large enough timestep K , the monotone increasing function k d q ∗ k k , exceeds d and so the system will be mode detectable.and so the system will be mode detectable.