Simultaneous Mode, State and Input Set-Valued Observers for Switched Nonlinear Systems
aa r X i v : . [ ee ss . S Y ] F e b Simultaneous Mode, State and Input Set-ValuedObservers for Switched Nonlinear Systems
Mohammad Khajenejad, Sze Zheng Yong ∗ School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe,AZ, USA
Abstract
In this paper, we study the problem of designing a simultaneous mode, inputand state set-valued observer for a class of hidden mode switched nonlinearsystems with bounded-norm noise and unknown input signals, where the hiddenmode and unknown inputs can represent fault or attack models and exogenousfault/disturbance or adversarial signals, respectively. The proposed multiple-model design has three constituents: (i) a bank of mode-matched set-valuedobservers, (ii) a mode observer and (iii) a global fusion observer. The mode-matched observers recursively find the sets of compatible states and unknowninputs conditioned on the mode being the true mode, while the mode observereliminates incompatible modes by leveraging a residual-based criterion. Then,the global fusion observer outputs the estimated sets of states and unknowninputs by taking the union of the mode-matched set-valued estimates over allcompatible modes. Moreover, sufficient conditions to guarantee the eliminationof all false modes (i.e., mode detectability) are provided and the effectiveness ofour approach is demonstrated and compared with existing approaches using anillustrative example.
Keywords:
Fault detection, Mode estimation, Set-valued observers, Switchedsystems, Nonlinear systems ∗ Corresponding author
Email addresses: [email protected] (Mohammad Khajenejad), [email protected] (SzeZheng Yong)
Preprint submitted to Journal of L A TEX Templates February 23, 2021 . Introduction
Cyber-Physical Systems (CPS), which tightly couple communication andcomputation elements, can enhance the functionality of control systems andimprove their performance. However, these features may also become a sourceof vulnerability to attacks or faults. On the other hand, autonomous systems,e.g., self-driving cars or robots, typically must operate without the direct knowl-edge of the intentions and decisions of other systems/agents. These systems,which can be conveniently considered within the general framework of hiddenmode hybrid/switched systems (HMHS, see, e.g., [1, 2, 3] and references therein)with unknown inputs, are often safety-critical. Thus, the ability to estimate thestates, unknown inputs and modes of such systems is important for monitor-ing these systems as well as for designing feedback controllers with safety andsecurity guarantees.
Literature review.
The problem of designing filters/observers for hiddenmode systems, without considering unknown inputs/faults/data injection at-tacks, has been extensively studied, e.g., in [4, 5] and references therein. Re-cently, the work in [2, 3] proposed an extension to include unknown inputs forstochastic systems, aiming to obtain point estimates, i.e., the most likely or bestsingle estimates. However, probabilistic distributions of uncertainty are oftenunavailable and moreover, it may also be desirable to consider set-valued uncer-tainties, e.g., bounded-norm noise, especially when hard guarantees or boundsare important. In the latter setting, set-membership or set-valued state ob-servers, e.g., [6, 7, 8], have been proposed to estimate the set of compatiblestates, and later, extensions of this framework to include the estimation of un-known inputs have been proposed in [9, 10, 11]. Nonetheless, these approachesare not directly applicable to systems with hidden modes that are considered inthis paper.To consider hidden modes, which can be used to model/represent fault orattack models, a common approach is to construct residual signals (see, e.g.,22, 3, 4, 12, 13, 14]), where a threshold based on the residual signal is usedto distinguish between consistent and inconsistent modes. In the context ofresilient state estimation against sparse data injection attacks, [15] presenteda robust control-inspired approach for linear systems with bounded-norm noisethat consists of local estimators, residual detectors, and a global fusion detector.Similar residual-based techniques have been used for uniformly observable non-linear systems in [16] and some classes of nonlinear systems in [17]. However,these approaches only consider sparse attacks on the sensors, which is a specialcase of a hidden mode system, as was discussed in our previous work for hiddenmode switched linear stochastic systems in [2]. Thus, to our best knowledge,the design of an estimator for hidden mode switched nonlinear systems withunknown inputs and bounded-norm noise remains an open problem. Contributions.
To bridge this gap, this paper considers the problem of simul-taneous mode, state and unknown input estimation for hidden mode switchednonlinear systems with bounded-norm noise, where the hidden mode representsa fault or attack model. To tackle this problem, our preliminary conference pub-lication [18] proposed a multiple-model approach for switched linear systems. Inthis paper, we further extend this approach to hidden mode switched nonlinearsystems with unknown inputs using a similar multiple-model approach, whichconsists of a bank of mode-matched set-valued observers and a novel elimination-based mode observer. The mode-matched set-valued observers are based on theoptimally designed set-valued state and input H ∞ observers in our recent work[11], while the mode observer eliminates inconsistent modes from the bank ofobservers by using the upper bound of the norm of to-be-designed residual sig-nals as a threshold. In particular, we propose a tractable method to calculatean upper bound signal for the residual’s norm by carefully over-approximatingthe value function of a non-concave NP-hard norm-maximization problem witha convex maximization problem over a convex set that has a finite number ofextreme points in a manner that guarantees that no compatible modes are elim-inated. We also prove that the upper bound signal is a convergent sequence.Furthermore, we provide sufficient conditions for mode detectability , i.e., for3uaranteeing that all false modes will be eventually ruled out under some rea-sonable assumptions. Finally, we compare the performance of our proposedapproach with an existing H ∞ observer in the literature. Notation. R n denotes the n -dimensional Euclidean space, and N the set ofnonnegative integers. For a vector v ∈ R n , k v k , √ v ⊤ v and k v k ∞ , max ≤ i ≤ n v i ,and for a matrix M ∈ R p × q , k M k , σ min ( M ) and M ( i : j ) denote the induced2-norm, the smallest non-trivial singular value and the sub-matrix consisting ofthe i -th through j -th columns of M , respectively. Further, 0 n × m denotes an n -by- m zero matrix.
2. Problem Statement
Consider a hidden mode switched nonlinear system with bounded-normnoise and unknown inputs (i.e., a hybrid system with nonlinear and noisysystem dynamics in each mode, where the mode and some inputs are notknown/measured): x k +1 = f q ( x k ) + B q u qk + G q d qk + W q w qk ,y k = C q x k + D q u qk + H q d qk + v qk , (1)where x k ∈ R n is the continuous system state and q ∈ Q = { , , . . . , Q } ⊂ N isthe hidden discrete state or mode . For each q ∈ Q , y k ∈ R l is the measurementoutput signal and w qk ∈ R n and v qk ∈ R l are external process and measurementdisturbances with known ℓ -norm bounds, i.e., k w k k ≤ η w and k v k k ≤ η v ,respectively. Moreover, u qk ∈ U k ⊂ R m is the known input and d qk ∈ R p theunknown input signal (representing, e.g., the input of other agents/robots oradversarially injected data signal). It is worth mentioning that no prior ‘useful’knowledge or assumption of the dynamics of d qk is assumed. For each (fixed)mode q , the mapping f q ( · ) : R n → R n and the matrices B q ∈ R n × m , G q ∈ R n × p , C q ∈ R l × n , D q ∈ R l × m and H q ∈ R l × p are the corresponding mode-dependentknown state vector field and system matrices, respectively.The above modeling framework can capture a very broad range of prob-lems, including intention estimation, fault detection and resilient state estima-4ion against sparse data injection and switching/mode attacks. Specifically, inthe context of intention estimation or fault diagnosis, each mode represents anintent or fault model and the unknown inputs can model the inputs of otheragents/robots or exogenous fault signals. On the other hand, with regard toresilient state estimation, the switching/mode attacks (e.g., attacks on circuitbreakers) can be represented with a set of different f q ( · ), B q , C q and D q , whilethe unknown attack location of sparse data injection attacks can be modeled bya set of different G q and H q that represent the different hypotheses for whichactuators and sensors are attacked or not attacked. Further, the attack signalmagnitudes can be modeled as the unknown inputs in this scenario.In addition, we assume the following: Assumption 1.
There is only one “true” mode, i.e. the true mode q ∗ is con-stant over time. Assumption 2.
For each q ∈ Q , f q ( · ) is twice continuously differentiable andLipschitz continuous on its domain with a known Lipschitz constant L qf > . Using the above modeling framework, the simultaneous state, unknown inputand hidden mode estimation problem based on a multiple-model framework canbe stated as follows:
Problem 1.
Given a hidden mode switched nonlinear discrete-time system withunknown inputs and bounded-norm noise in the form of (1) , (i) Design a bank of mode-matched observers, where each mode-matched ob-server, conditioned on the mode being true, optimally returns the set-valuedestimates of compatible states and unknown inputs in the minimum H ∞ -norm sense, i.e., with minimum average power amplification. (ii) Find a threshold criterion to eliminate false modes and subsequently, de-velop a mode observer via elimination. (iii)
Derive sufficient conditions for the elimination of all false modes. . Proposed Observer Design In this section, we propose a multiple-model approach for simultaneousmode, state and unknown input estimation for the system in (1), with thegoal of recursively finding the sets of states ˆ X k , unknown inputs ˆ D k and modesˆ Q k that are compatible with observed outputs y k . The multiple-model design approach consists of three steps: (i) designinga bank of mode-matched set-valued observers, (ii) developing a mode observerfor eliminating incompatible modes using a residual-based threshold, and (iii)devising a global fusion observer that returns the desired set-valued mode, inputand state estimates.
First, based on the optimal fixed-order observer design in [11], we develop abank of mode-matched observers, which includes Q ∈ N simultaneous state andinput H ∞ set-valued observers, which can be briefly summarized as follows. Foreach mode-matched observer corresponding to mode q , following the approachin [11, Section 4], we consider set-valued fixed-order estimates in the form of ℓ -norm balls: 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 ˆ x qk | k and ˆ d qk − are obtained with the following three-steprecursive observer that is optimal in H ∞ -norm sense (cf. [11, Section 4.2] formore details): 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)6 ime Update : ˆ x qk | k − = f q (ˆ x qk − | k − ) + B q u 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 C q , C q , D q , D q , G q , G q , V q , V q , z q ,k and z q ,k can be computed by apply-ing a similarity transformation described in Appendix A and ˜ 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 thatare chosen via the following Proposition 1. This proposition is a restatement ofthe results in [11] that is tailored to the setting considered in this paper, wherethe main idea is to minimize the “volume” of the set of compatible states andunknown inputs, quantified by the radii δ d,qk − and δ x,qk . Proposition 1. [11, Proposition 5.16, Lemma 5.1 & Theorem 5.13] Considersystem (1) and a bank of Q mode-matched observers in the form of (4) – (6) .Suppose that ∀ q ∈ Q , { , . . . , Q } , rk( C q G q ) = p − p H q and M q , M q arechosen as M q = (Σ q ) − and M q = ( C q G q ) † , where Σ q is obtained by applyingsingular value decomposition on H q (cf. Appendix A for more details). Then,the following statements hold: (a) Given mode q ∈ Q , the following difference equation governs the state esti-mation error dynamics (i.e., the dynamics of ˜ x qk | k , x k − ˆ x qk | k ): ˜ x qk +1 | k +1 = ( I − ˜ L q C q )Φ q (∆ f qk − Ψ q ˜ x qk | k ) + W q ( ˜ L q ) w qk , (7) where ∆ f qk , f q ( x k ) − f q (ˆ x qk ) , Φ q , I − G q M q C q ,w qk , h ( √ ) v q ⊤ k w q ⊤ k ( √ ) v q ⊤ k +1 i ⊤ ,R q , h −√ q G q M q T q − Φ q W q −√ G q M q T q i ,Q q , h ( l − p Hq ) × l ( l − p Hq ) × n −√ T q i , Ψ q , G q M q C q , W q ( ˜ L q ) , ( I − ˜ L q C q ) R q + ˜ L q Q q . Solving the following mixed-integer SDP for each mode q : ( ρ ⋆q ) = min { P ≻ , Γ ≻ , ˜Γ (cid:23) , ˘ Q (cid:23) ,Y, ˘ Z,ρ > , ≤ α ≤ ,ε > ,ε > ,κ> ,κ > ,κ > } ρ s.t. P ˜ Y q ˜ Y q ⊤ ˜ M q (cid:23) , P ˜ Y q ˜ Y q ⊤ ˜ M q (cid:23) , P ˜ Y q ˜ Y q ⊤ ˜ M q (cid:23) , P ˜ Y q ˜ Y q ⊤ ˘ Z (cid:23) , ˜Γ ˘ Z ˘ Z ⊤ Ψ q ⊤ ˘ Q Ψ q (cid:23) , I − Γ 0 00
P Y Y ⊤ I (cid:23) , N q ∗ ∗N q N q ∗N q N q (cid:23) ,κ I (cid:22) P (cid:22) κ I, ∧ (( κ ≥ , κ − κ < ∨ ( κ ≤ , κ > . , we obtain an observer in the form of (4) – (6) with the observer gain ˜ L q =( P q ) − Y q , where ( P q , Y q ) are solutions to the above mixed-integer SDP,that • is quadratically stable, and • guarantees that θ q , k ( I − ˜ L q C q )Φ q k < , (8) and consequently, the upper bound sequences for the radii { δ x,qk , δ d,qk − } ∞ k =1 ,which are computed as: δ x,qk , δ x ( θ q ) k + η q − ( θ q ) k − θ q ,δ dk − , β q δ x,qk − + α q , (9) are convergent to some steady state value δ x,q ∞ , δ d,q ∞ (cf. Appendix Bfor definitions of δ x,q ∞ and δ d,q ∞ , as well as the matrices and parametersin the above SDP and (9) ).3.1.2. Mode Observer To estimate the set of compatible modes, we consider an elimination ap-proach that compares the ℓ -norm of residual signals against some thresholds.8pecifically, we will eliminate a specific mode q , if k r qk k > ˆ δ qr,k , where theresidual signal r qk is defined as follows and the thresholds ˆ δ qr,k will be derived inSection 3.2. Definition 1 (Residuals) . For each mode q at time step k , the residual signalis defined as: r qk , z q ,k − C q ˆ x ⋆,qk | k − D q u qk . Finally, combining the outputs of both components above, our proposedglobal fusion observer 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.
We leverage a relatively simple idea to develop a criterion for elimination offalse modes, as follows. We rule out a particular mode as incompatible, if the ℓ -norm of its corresponding residual signal exceeds its upper bound conditionedon this mode being true. To do so, for each mode q , we first compute an upperbound (ˆ δ qr,k ) for the ℓ -norm of its corresponding residual at time k , conditionedon q being the true mode. Then, comparing the ℓ -norm of residual signal inDefinition 1 with ˆ δ qr,k , mode q can be eliminated if the residual’s ℓ -norm isstrictly greater than the upper bound. The following proposition and theoremformalize this procedure. Proposition 2.
Consider mode q at time step k , its residual signal r qk (asdefined in Definition 1) and the unknown true mode q ∗ . Then, r qk = r q |∗ k + ∆ r q | q ∗ k , 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 Proposition 1; z q ,k = T q y k ; ⊲ Mode Observer via Eliminationˆ Q k = ˆ Q k − ;Compute r qk via Definition 1 and ˆ δ qr,k via Theorem 2; 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 with r 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 expressions into the righthand side term of Definition 1.
Theorem 1.
Consider mode q and its residual signal r qk at time step k . Assumethat δ q, ∗ r,k is any signal that satisfies k r q |∗ k k ≤ δ q, ∗ r,k , where r q |∗ k is defined inProposition 2. Then, mode q is not the true mode, i.e., can be eliminated attime k , if k r qk k > δ q, ∗ r,k . roof. To use contradiction, suppose that k r qk k > δ q, ∗ r,k and let q be the truemode, i.e., q = q ∗ and thus, T q = T q ∗ . By Proposition 2, ∆ 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.By the above theorem, our approach guarantees that the true mode is nevereliminated. However, Theorem 1 only provides a sufficient condition for modeelimination at each time step and the capability of our proposed mode observerto eliminate as many false modes as possible is dependent on the tightness ofthe upper bound, δ q, ∗ r,k . To apply the sufficient condition in Theorem 1, we need a tractable approachto compute the upper bound δ q, ∗ r,k that is finite-valued. This procedure is derivedand described in the following. Lemma 1.
Consider any mode q with the unknown true mode being q ∗ . Then,at time step k , we have r q |∗ k = C q ˜ x ⋆,qk | k + v q ,k = A qk t k , (10) where t k , h ˜ x ⊤ | v q ⊤ . . . v q ⊤ k w q ⊤ . . . w q ⊤ k − ∆ f q ⊤ . . . ∆ f q ⊤ k − i ⊤ ∈ R ( n + l )( k +1)+ nk , A qk , [ A qk J q, k − ( J q, k − + J q, k − ) · · · ( J q, + J q, ) J q, J q, k − . . . J q, F qk − . . . F q ] ,A qk , ( − k (( I − ˜ L q C q )Φ q Ψ q ) k ,J qi , Y q , if i = 0 , − C q Φ q G q M q C q ( I − ˜ L q C q ) i − W q , if ≤ i ≤ k − ,F qi , C q Φ q , if i = 0 , ( − i C q Φ q G q M q C q (( I − ˜ L q C q )Ψ q ) i − ( I − ˜ L q C q )Φ q , if ≤ i ≤ k − , Y q , h −√ C q Φ q G q M q T q C q Φ q W q √ I − C q G q M q ) T q i ,J q, i , J qi (1 : l ) , J q, i , J qi ( l + 1 : 2 l ) , J q, i , J qi (2 l + 1 : 2 l + n ) , i = 1 , . . . , k − . roof. The first equality in (10) comes from Definition 1 and z q ,k = C q x k + D q ,k u qk + v q ,k from (A.3) in Appendix A, assuming that q is the true mode. Toobtain the second equality, note that [11, (A.11)] returns˜ x ⋆,qk | k = Φ q [∆ f qk − − G q M q C q ˜ x qk − | k − ] + ˜ w qk , (11)˜ w qk , − Φ q ( G q M q v q ,k − − W q w qk − ) − G q M q v q ,k . Now, from the first equality and (11), we have r q |∗ k = C q Φ q (∆ f qk − − G q M q C q ˜ x qk − | k − ) + Y q w qk − . (12)On the other hand, by iteratively applying (7), we obtain:˜ x qk | k = i − X i =1 [(( I − ˜ L q C q )Ψ q ) i − ( I − ˜ L q C q )Φ q ∆ f qk − i + ( I − ˜ L q C q ) i − W q w qk − i +1 ]+ ( − k (( I − ˜ L q C q )Φ q Ψ q ) k ˜ x q | . (13)Combining (12) and (13) yields r q |∗ k = A qk ˜ x q | + k − X i =0 F qi ∆ f qk − − i + J qi w qk − i , which is equivalent to the second equality in (10). Lemma 2.
For each mode q at time step k , there exists a finite-valued upperbound δ qr,k < ∞ for k r q |∗ k k .Proof. Consider the following optimization problem for k r q |∗ k k by leveragingLemma 1: δ qr,k , max t k k A qk t k k (14) s.t. t k = h ˜ x ⊤ | v q ⊤ . . . v q ⊤ k w q ⊤ . . . w q ⊤ k − ∆ f q ⊤ . . . ∆ f q ⊤ k − i ⊤ , k ˜ x | k ≤ δ x , k v qi k ≤ η qv , k w qj k ≤ η qw , k ∆ f qj k ≤ L qf δ x,qj ≤ L qf δ x,q ,i ∈ { , ..., k } , j ∈ { , ..., k − } . The objective ℓ -norm function is continuous and the constraint set is an inter-section of level sets of lower dimensional norm functions, which is closed and12ounded, so is compact. Hence, by the Weierstrass Theorem [19, Proposition2.1.1], the objective function attains its maxima on the constraint set and so afinite-valued upper bound exists.Clearly, δ qr,k in Lemma 2, if computable, is the tightest possible upper boundfor the norm of the residual signal and using this as the threshold can eliminatethe most possible number of false modes. However, note that although the ex-istence proof of a finite-valued δ qr,k is straightforward, the optimization problemin Lemma 2 is NP-hard [20], since it is a norm maximization (not minimization)over the intersection of level sets of lower dimensional norm functions, i.e., itis a non-concave maximization over intersection of quadratic constraints. Totackle this complexity, through the following Theorem 2, we propose a tractableover-approximation/upper bound for δ qr,k , which we call ˆ δ qr,k and is used insteadas the elimination threshold. Theorem 2.
Consider mode q . At time step k , let ˆ δ qr,k , min { δ q,trir,k , δ q,infr,k } ,δ q,trir,k , k − X i =0 L qf k F qi k δ x,qk − − i + 1 √ η qv ( k J q, i k + k J q, i k ) + η qw k J q, i k (15)+ ( k A qk k + L qf k F qk − k ) δ x + 1 √ η qv ( k J q, k − k + k J q, k − k ) + η qw k J q, k − k ,δ q,infr,k , k A qk t ⋆k k , where t ⋆k , arg max t k ∈T k k A qk t k k and T k is the set of all vertices of the followinghypercube: qk , (cid:8) x ∈ R ( n + l )( k +1)+ nk | x ( i ) | ≤ δ x , ≤ i ≤ n,η qv , n + 1 ≤ i ≤ n + l ( k + 1) ,η qw , n + l ( k + 1) + 1 ≤ i ≤ ( n + l )( k + 1) ,L qf δ x , ( n + l )( k + 1) + 1 ≤ i ≤ ( n + l )( k + 1) + n, ... L qf δ x,qj , ( n + l )( k + 1) + jn + 1 ≤ i ≤ ( n + l )( k + 1) + n ( j + 1) , ... L qf δ x,qk − , ( n + l )( k + 1) + ( k − n + 1 ≤ i ≤ ( n + l )( k + 1) + nk. (cid:9) . Then, ˆ δ qr,k is an over-approximation for δ qr,k in Lemma 2, i.e., ˆ δ qr,k ≥ δ qr,k .Proof. Consider the following optimization problem: δ q,infr,k , max t k k A qk t k k (16) s.t. t k = t k = h ˜ x ⊤ | v q ⊤ . . . v q ⊤ k w q ⊤ . . . w q ⊤ k − ∆ f q ⊤ . . . ∆ f q ⊤ k − i ⊤ , k ˜ x | k ∞ ≤ δ x , k v qi k ∞ ≤ η qv , k w qj k ∞ ≤ η qw , k ∆ f qj k ∞ ≤ L qf δ x,qj ∀ i ∈ { , ..., k } , ∀ j ∈ { , ..., k − } . Comparing (14) with (16), the two problems have the same objective functions.Then, since k . k ∞ ≤ k . k , the constraint set for (14) is a subset of the one for(16). Hence δ qr,k ≤ δ q,infr,k . Also, it is easy to see that δ qr,k ≤ δ q,trir,k , which is ob-tained using triangle inequality and the sub-multiplicative property of norms.Moreover, (16) is a maximization of a convex objective function over a con-vex constraint (hypercube X qk ). By a famous result [21, Corollary 32.2.1], insuch a problem, the objective function attains its maxima on some of the ex-treme points of the constraint set, which in this case are the vertices T k of thehypercube X qk .Theorem 2 enables us to obtain an upper bound for k r q |∗ k k , by enumeratingthe objective function in (16) for all vertices of the hypercube X qk and choosing14he largest value as 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.
The reason for not only using δ q,infr,k is two-fold. First, as timeincreases, the number of required enumerations for δ q,infr,k (i.e., the cardinality of T k ) can be shown to be |T k | = 2 ( n + l )( k +1)+ kn , which increases at an exponentialrate. Second and more importantly, as will be shown later in Lemma 3, δ q,infr,k goes to infinity as time increases, which renders it ineffective in the limit. Onthe other hand, Lemma 3 will show that δ q,trir,k converges to some steady-statevalue, so it can always be used as an over-approximation for δ qr,k in the modeelimination process. Nonetheless, we chose to use the minimum of the twobounds, since our simulation results in Section 5 show that δ q,infr,k is generallysmaller than δ q,trir,k in the initial time steps. Further, the following result that we will make use of later can be easilyobtained as a corollary of Theorem 2.
Corollary 1. t ⋆k (defined in Theorem 2) has the following norm: η tk , k t ⋆k k = vuut n ((1 + L qf ) δ x + kη qw + L qf k − X j =1 δ x,qj ) + l ( k + 1) η qv .
4. Mode Detectability
In addition to the nice properties regarding the quadratic stability andboundedness of the mode-matched set-valued estimates of the state and un-known input obtained from [11], we are interested in guaranteeing the effec-tiveness of our mode elimination algorithm. Thus, in the following, we searchfor some sufficient conditions based on the properties/structures of the systemdynamics and/or unknown input signals for guaranteeing that the applicationof Algorithm 1 can eliminate all false (i.e., not true) modes after some largeenough number of time steps.To achieve this, we first define the concept of mode detectability .15 efinition 2 (Mode Detectability) . System (1) is called mode detectable ifthere exists a natural number
K > , such that for all time steps k ≥ K , allfalse modes are eliminated. Moreover, we consider two different sets of assumptions that we will use forderiving our sufficient conditions for mode detectability.
Assumption 3.
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 knownbounds for the whole observation/measurement and state spaces, respectively. Assumption 4.
The state space X is bounded and the unknown input signal has 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 the unlimited energy condition in Assumption 4 is not restrictiveif f ( · ), B , C and D are mode-independent, since otherwise, the unknown inputsignal must vanish asymptotically, which means that we effectively have a non-switched system in the limit and the mode estimation would be trivial.Next, in order to derive the desired sufficient conditions for mode-detectabilityin Theorem 3, we first present the following Lemmas 3–5. Lemma 3.
For each mode q , lim k →∞ δ q,infr,k = ∞ . (17)lim k →∞ ˆ δ qr,k = lim k →∞ δ q,trir,k < ∞ , (18) Proof.
To show (17), we first find a lower bound for δ q,infr,k . Then, we prove thatthe lower bound diverges and so does δ q,infr,k . Define ˜ t ⋆k , t ⋆k η tk , where η tk is definedin 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 smallest non-trivial singular value of matrix A . The firstequality holds since σ min ( . ) is a linear operator and the second equality is aspecial case of the matrix lower bound [22] when ℓ -norms are considered. The16nequality holds since k ˜ t ⋆k k = 1 by Corollary 1, so ˜ t ⋆k is a feasible point forthe minimization problem (i.e., min k t k ≤ k η tk A qk t k ) and the last equality holdsby Theorem 2. So far we have shown that η tk σ min ( A qk ) is a lower bound for δ q,infr,k . Next, we will prove that η tk σ min ( A qk ) is unbounded. First, it is trivialto observe that η tk grows unbounded by its definition in Corollary 1. Second, σ min ( A qk ) ≤ σ min ( A qk +1 ), since the latter is an augmentation of the former withadditional columns. Hence, η tk σ min ( A qk ) grows unbounded, since the productof the unbounded and positive σ min ( A qk ) and the unbounded and positive η tk isunbounded.To prove (18), we show that { δ q,trir,k } ∞ k =1 is a convergent sequence. Then, thisfact, as well as (17) and the fact that ˆ δ qr,k , min { δ q,trir,k , δ q,infr,k } by Theorem 2,imply (18). To show the convergence of { δ q,trir,k } ∞ k =1 , starting from (15), we firstshow that ∀ q ∈ Q , S q ,k , P k − i =0 L qf k F qi k δ x,qk − − i + √ η qv ( k J q, i k + k J q, i k ) + η qw k J q, i k on the right hand side of (15) converges to some steady state value.Note that k F qi k ≤ R q θ qi by the sub-multiplicative property of norms, where R q , L qf k C q Φ q G q M q C q k k Ψ q k k Φ q k and θ q is given in (8). Combining this and (9) implies that k − X i =0 L qf k F qi k δ x,qk − − i ≤ R q ( δ x − η q − θ q )( k − θ q ) k − + η q − θ q − ( θ q ) k − − θ q ! , and the upper bound tends to R q η q (1 − θ q ) as k tends to ∞ , since 0 < θ q < k →∞ k ( θ q ) k = 0 when 0 < θ q <
1. Moreover, it follows fromthe definitions of J qi and θ q (cf. Proposition 1 and Lemma 1), as well as thesub-multiplicative property of norms that:1 √ η qv ( k J q, i k + k J q, i k ) + η qw k J q, i k ≤ O q , i = 0 ,S q θ qi , i ≥ , where O q , η qw ( k C q Φ q G q M q T q k + k ( I − C q G q M q ) T q k ) + η qv k C q Φ q W q k and S q , ( η qw k C q Φ q G q M q C q k ( k Φ q G q M q T q k + k G q M q T q k )+ η qv k Φ q W q k ). Com-17ining this and (8) results in k − X i =0 √ η qv ( k J q, i k + k J q, i k ) + η qw k J q, i k ≤ O q + S q θ q − θ qk − − θ q , where the upper bound tends to S q θ q − θ q as k tends to ∞ . Next, it is straightforwardto observe that all constitutent terms in S q ,k , ( k A qk k + L qf k F qk − k ) δ x + √ η qv ( k J q, k − k + k J q, k − k ) + η qw k J q, k − k (on the right hand side of (15)) are alldecreasing to zero as k increases, since they are all upper bounded by some termsinvolving ( θ q ) k by their definitions (cf. Lemma 1) and the sub-multiplicativeproperty. Hence, lim k →∞ δ q,trir,k = lim k →∞ ( S q ,k + S q ,k ) = lim k →∞ S q ,k < ∞ . Lemma 4.
Suppose that Assumption 3 holds. Consider two different modes q = q ′ ∈ Q and their corresponding upper bounds 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 eliminatedif 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 , (19) where R q,q ′ z , R y k T q − T q ′ k .Proof. Suppose, for contradiction, that none of q 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 inequality holds by triangleinequality and the last inequality holds by the assumption that none of q and q ′ can be eliminated, as well as the boundedness assumption for the measurementspace. This last inequality contradicts with the inequality in the lemma, thusthe result holds. Lemma 5.
Consider any mode q with the unknown true mode being q ∗ . Supposewithout loss of generality that f q (0) = 0 . Then, at time step k , we have r qk = A qk t qk + α q ∗ k + ǫ q ∗ k , (20)18 ith ε q ∗ k being an error term that satisfies ∃ ξ , . . . , ξ k ∈ X, s.t. k ε q ∗ k k ≤ k X i =1 k J q ∗ f, k k − i k x i − k k H q ∗ f ( ξ i ) k , (21) where α q ∗ k , ( T q − T q ∗ )( C q ∗ f,k x + C q ∗ d,k d q ∗ k + C q ∗ u,k u q ∗ k + C q ∗ ˜ w,k ˜ w q ∗ k ) C q ∗ d,k , h H q ∗ C q ∗ G q ∗ C q ∗ J q ∗ f, G q ∗ . . . C q ∗ ( J q ∗ f, ) k − G q ∗ i ,C q ∗ u,k , h D q ∗ C q ∗ B q ∗ C q ∗ J q ∗ f, B q ∗ . . . C q ∗ ( J q ∗ f, ) k − B q ∗ i ,C q ∗ ˜ w,k , h I C q ∗ W q ∗ C q ∗ J q ∗ f, W q ∗ . . . C q ∗ ( J q ∗ f, ) k − W q ∗ i ,d q ∗ k , h d q ∗ ⊤ k . . . d q ∗ ⊤ i ⊤ , u q ∗ k , h u q ∗⊤ k . . . u q ∗⊤ i ⊤ , C q ∗ f,k , C q ∗ ( J q ∗ f, ) k , ˜ w q ∗ k , h v q ∗ ⊤ k w q ∗ ⊤ k − . . . w q ∗ ⊤ i ⊤ , ǫ q ∗ k , ( T q − T q ∗ ) ε q ∗ k , and J q ∗ f, and H q ∗ f ( ξ ) are the Jacobian and Hessian matrices of the vector field f q ∗ ( · ) at and ξ , respectively.Proof. Recall from Proposition 2, Lemma 1 and (1) that: r qk = A qk t qk + ( T q − T q ∗ )( C q ∗ x k + H q ∗ d q ∗ k + D q ∗ u q ∗ k + v q ∗ k ) . (22)On the other hand, by applying Taylor series expansion to (1) we obtain: x k = J q ∗ f, x k − + B q ∗ u q ∗ k − + G q ∗ d q ∗ k − + W q ∗ w q ∗ k − + ( H.O.T ) q ∗ k , (23)where ( H.O.T ) q ∗ k is an error term that satisfies k ( H.O.T ) q ∗ k k ≤ H q ∗ f ( ξ k ) forsome ξ k ∈ X . Then, by applying (23) at time steps k, k − , . . . ,
1, pluggingthem into (22) and augmentating the results, we obtain (20).
Theorem 3 (Sufficient Conditions for Mode Detectability) . System (1) is modedetectable, i.e., by applying Algorithm 1, all false modes will be eliminated atsome large enough time step K , if the assumptions in Proposition 1 and eitherof the following hold: Assumption 3 holds 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 , where W q,q ′ , h ( C q − C q ′ ) ( T q − T q ′ ) − I I D q − D q ′ i . ii. Assumption 4 holds and T q = T q ′ holds ∀ q, q ′ ∈ Q, q = q ′ . Moreover, H q ∗ f ( · ) is bounded on X and k J q ∗ f, k < .Proof. To show that (i) is sufficient for asymptotic mode detectability, considerLemma 4 with δ q,trir,k as the upper bound. It suffices to show that ∃ K ∈ N ,such that (19) 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 . Hence, by plugging this into (19), we need toshow that ∃ 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, ∀ q = q ′ ∈ Q , (24)where s q,q ′ k , h x ⊤ k v ⊤ k r q |∗⊤ k r q ′ |∗⊤ k u q ⊤ k u q ′ ⊤ k i ⊤ . A sufficient conditionto satisfy (24) is that ∃ K ∈ N such that ∀ k ≥ K , (24) holds for all s q,q ′ k .Equivalently, it suffices that: W q,q ′ k > δ q,trir,k + δ q ′ ,trir,k + R q,q ′ z , ∀ k ≥ K, ∀ q = q ′ ∈ Q , where W q,q ′ k , min x k ,v k ,r qk ,r q ′ k k W q,q ′ s q,q ′ k k 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 . Finally, by expanding the constraint set, it suffices to require that ∃ K ∈ N suchthat: W q,q ′ k > δ q,trir,k + δ q ′ ,trir,k + R q,q ′ z , ∀ k ≥ K, ∀ q = q ′ ∈ Q , where W q,q ′ k , min s q,q ′ k k W q,q ′ s q,q ′ k k 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 ) . matrix lower bound theorem [22] and a similar argument to theproof of Lemma 3, it is sufficient to require that ∃ K ∈ N such that ∀ 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 ) . (25)The result in (25) provides us a time-dependent sufficient condition for modedetectability. In order to find a time-independent sufficient condition, noticethat ( δ q,trir,k + δ q ′ ,trir,k + R q,q ′ z ) R x + η v is an upper bound for the right hand side of (25), sincethe latter’s denominator is smaller than the former’s and the numerator of thelatter is an upper bound signal for the former’s by triangle inequality and thesub-multiplicative property of norms. So, a sufficient condition for (25) is that ∃ K ∈ N such that ∀ 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 . (26)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 (18).As for the sufficiency of (ii), we show that the sufficient conditions in (ii)imply that if q = q ∗ , then the residual signal r qk grows unbounded. Then, sincewe showed in Lemma 3 that the computed upper bound signal ˆ δ qr,k is bounded,so there must exist a time step K such that r qk > ˆ δ qr,k for k ≥ K , and hence,mode q will be eliminated after time step K and therefore, mode detectabilityholds. To do so, we show that if (ii) holds, then the right hand side of (20)grows unbounded, and so does r qk . First, note that by Lemma 3, the first termin the right hand side of (20), i.e., A qk t qk , is bounded. Moreover, (21) and thefacts that the state space is bounded and k J q ∗ f, k < ǫ q ∗ k , i.e., thethird term in the right hand side of (20), is bounded.Next, we show that the second term in the right hand side of (20), i.e. α q ∗ k ,grows unbounded. Consequently, the summation of the two bounded terms21 qk t qk and ǫ q ∗ k as well as the unbounded term α q ∗ k will be unbounded. To showthat α q ∗ k grows unbounded, it suffices to show that for any c >
0, any specificmode q with the true mode being q ∗ , there exists a large enough K such that: k α q ∗ K k = (cid:13)(cid:13)(cid:13)(cid:13)h T q,q ∗ K C q,q ∗ u,K C q,q ∗ d,K i h ζ ⊤ K u q ∗ ⊤ k d q ∗⊤ K i ⊤ (cid:13)(cid:13)(cid:13)(cid:13) > c, with T q,q ∗ K , ( T q − T q ∗ ) h C q ∗ x,K C q ∗ ˜ w,K i , C q,q ∗ u,K , ( T q − T q ∗ ) C q ∗ u,K , C q,q ∗ d,K , ( T q − T q ∗ ) C q ∗ d,K and ζ K , h x ⊤ ˜ w q ∗ ⊤ K i ⊤ . Since q ∗ is unknown, a sufficientcondition to satisfy the above equality is that ∀ c > , ∀ q ′ = q ∈ Q, ∃ K ∈ N suchthat: (cid:13)(cid:13)(cid:13)(cid:13)h T q,q ′ K C q,q ′ u,K C q,q ′ d,K i h ζ ⊤ K u q ′ ⊤ K d q ∗⊤ k i ⊤ (cid:13)(cid:13)(cid:13)(cid:13) > c. So it suffices that ∀ c > , ∀ q ′ = q ∈ Q, ∃ d ∈ R , ∃ K ∈ N , such that: T q,q ′ k > c, where T q,q ′ k , min ζ ′ k (cid:13)(cid:13)(cid:13)h T q,q ′ K C q,q ′ u,K C q,q ′ d,K i ζ ′ K (cid:13)(cid:13)(cid:13) s.t. ζ ′ K = h x ⊤ ˜ w q ∗ ⊤ K u q ′ ⊤ K d q ∗⊤ K i ⊤ , k d q ∗ K k ≥ d, k w i k ∞ ≤ η w , k v j k ∞ ≤ η v , i ∈ { , ..., K − } , j ∈ { , ..., K } . Once again, by the matrix lower bound theorem, a sufficient condition for theabove inequality to hold is that ∃ d ∈ R , ∃ K ∈ N , such that: T q,q ′ k > cσ min ( h T q,q ′ K C q,q ′ u,K C q,q ′ d,K i ) , where T q,q ′ k , min ˜ w q ∗ K ,d q ∗ K k ζ ′ K k (27) s.t. ζ ′ K = h x ⊤ ˜ w q ∗ ⊤ K u q ′ ⊤ k d q ∗⊤ K i ⊤ , k d q ∗ K k ≥ d, k w i k ∞ ≤ η w , k v j k ∞ ≤ η v , i ∈ { , ..., K − } , j ∈ { , ..., K } . k ζ ′ K k = (cid:13)(cid:13)(cid:13)h x ⊤ ˜ w q ∗ ⊤ k u q ′ ⊤ K d q ∗⊤ K i(cid:13)(cid:13)(cid:13) ≥ q + 0 + 0 + k d q ∗⊤ K k = k d q ∗⊤ K k , then a sufficient condition for (27) to hold is that k d q ∗⊤ K k > cσ min ( h T q,q ′ K C q,q ′ u,K C q,q ′ d,K i ) . (28)Now, suppose that T q = T q ′ (otherwise the matrix in the denominator of (28) iszero and it never holds). Then, the right hand side of (28) converges asymptot-ically to ˜ δ , max { , cσ q,q ′ } , since the smallest singular value in the denominatoreither diverges, or converges to some steady value σ q,q ′ . So we set d to be equalto any real number that is strictly greater than ˜ δ . By the unlimited energyassumption for the unknown input signal, at some large enough time step K ,the monotonely increasing function k d q ∗ k k will exceed d and so, the system willbe mode detectable.
5. Simulation Results
In this section, we evaluate the effectiveness of our Simultaneous Mode,Input, and State Set-Valued Observer (SMIS), by comparing its performancewith the LMI-based H ∞ -observer in [23] that obtains point state estimates. Forcomparison, we apply the two observers on a modified version of the discrete-time nonlinear switched system in [23], where we increase the number of modes(subsystems) to five, i.e., Q = 5. The considered system is in the form of (1),with the following parameters: n = l = 2 , m = p = 1 and ∀ q = 1 , . . . , B q = D q = 0 × , f q ( x ) = ˜ A q γ ( x ) + ˆ A q x, γ ( x ) , h sin( x ) sin( x ) i ⊤ . Moreover,ˆ A = . . − . , ˜ A = . − . . − . , C = . . . . , H = . . , G = . − . , ˆ A = − . − . , ˜ A = . − . . − . , C = . − . . − . , H = . − . , G = − . . , ˆ A = . − . − . . , ˜ A = . − . − . − . , C = . . − . . , H = − . . , G = . . , ˆ A = − . − . . . , ˜ A = − . . . − . , C = . − . . − . , H = − . . , G = . . , ˆ A = − . . − . . , ˜ A = − . . . − . , C = − . − . − . , H = − . . , G = . . . The initial state estimate and noise signals satisfy k x k ≤ δ x = 0 . k w k k ≤ η w = 0 .
02 and k v k k ≤ η w = 0 .
02. Furthermore, we assume that ˆ x | = h . . i ⊤ .We consider two scenarios for the unknown input. In the first (Scenario I),the unknown input is a random signal with bounded norm, i.e., k d k k ≤ . d k in the second scenario (Scenario II) is a time-varying signal that be-comes unbounded as time increases. As is demonstrated in Figure 1, in the firstscenario, i.e., with bounded unknown inputs, the set estimates of our approach(i.e., SMIS estimates) converge to steady-state values and the point estimatesof the approach in [23] are within the predicted upper bounds and exhibit con-vergent behavior. More interestingly, considering the second scenario, i.e., withunbounded unknown inputs, Figure 2 shows that our set-valued estimates stillconverge, i.e., our observer remains stable, while the estimates of the approachin [23] exceed the boundaries of the compatible sets of states and inputs of ourapproach after some time steps and display a divergent behavior (cf. Figure 2).Further, Tables 1 and 2 show the matrix T q for each mode q for Scenarios Iand II, respectively. It can be verified that the second set of sufficient conditionsin Theorem 3 holds, i.e., T q = T q ′ for all q = q ′ , for both scenarios. Hence, we24 igure 1: Actual states x , x , and their estimates, as well as the unknown input d and its estimates, and the number of non-eliminated modes at each time step inthe bounded unknown input scenario (Scenario I), when applying the observer in [23](Zhen-Xu-Zhang Estimate) and our proposed observer (SMIS Estimate). Figure 2:
Actual states x , x , and their estimates, as well as the unknown input d and its estimates, and the number of non-eliminated modes at each time step in theunbounded unknown input scenario (Scenario II), when applying the observer in [23](Zhen-Xu-Zhang Estimate) and our proposed observer (SMIS Estimate). able 1: Different modes and their T q in Scenario I (i.e., with bounded d k ). Mode T q q = 1 [0.3629 -0.2179 ] ⊤ q = 2 [0.1191 0.8715 ] ⊤ q = 3 [-0.6468 0.8390 ] ⊤ q = 4 [0.8103 -0.6681 ] ⊤ q = 5 [0.2780 -0.6793 ] ⊤ Table 2:
Different modes and their T q in Scenario II (i.e., with unbounded d k ). Mode T q q = 1 [0.4730 -0.3280 ] ⊤ q = 2 [0.2202 0.9826 ] ⊤ q = 3 [-0.7579 0.9401 ] ⊤ q = 4 [0.9214 -0.7792 ] ⊤ q = 5 [0.3891 -0.7804 ] ⊤ expect that all false modes are eliminated, i.e., exactly one (true) mode remains,after some large enough time in both scenarios, which is indeed what we observein Figures 1 and 2, where the number of non-eliminated modes at each time stepis shown.Moreover, for each mode q , the signals k r qk k , k r q |∗ k k , δ q,trir,k and δ q,infr,k aredepicted in Figures 3 and 4 for Scenarios I and II, respectively. In both scenarios,we observe that δ q,infr,k is smaller than δ q,trir,k up until approximately 40 timesteps, after which δ q,trir,k is smaller/tighter. This is one of the main reasonswe considered the minimum of both as the threshold in our mode eliminationalgorithm (also see Remark 1). Furthermore, for all modes, δ q,trir,k is eventuallyconvergent while δ q,infr,k diverges, as proven in Lemma 3. So, after some largeenough time, δ q,trir,k can be used as our upper bound threshold, while δ q,infr,k becomes ineffective. 26 igure 3: k r qr,k k , k r q |∗ r,k k and their upper bounds for different modes in the boundedunknown input scenario (Scenario I). Figure 4: k r qr,k k , k r q |∗ r,k k and their upper bounds for different modes in the un-bounded unknown input scenario (Scenario II). . Conclusion This paper introduced a novel multiple-model approach for simultaneousmode, unknown input and state estimation for hidden mode switched nonlinearsystems with bounded-norm noise and unknown inputs. The proposed approachconsists of a bank of mode-matched state and unknown input observer that isoptimal in the H ∞ sense and a mode observer, which eliminates inconsistentmodes and their corresponding observers at each time step. The proposed modeelimination criterion is based on the use of a provably finite-valued upper boundfor the norm of a residual signal as the threshold. Moreover, we provided atractable approach to compute the threshold signal and proved the convergenceof the upper bound/threshold signal as well as derived sufficient conditionsfor eventually eliminating all false modes when using our mode eliminationalgorithm. Finally, we demonstrated the effectiveness of our observer using anillustrative example, where we compared our approach with an existing H ∞ observer in the literature under two different scenarios. Acknowledgments
This work is partially supported by the National Science Foundation, USAgrants CNS-1943545 and CNS-1932066.
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For q ∈ Q , let p H q , rk( H q ). Using singular value decomposition, werewrite the direct feedthrough matrix H q as H q = h U q U q i Σ q
00 0 V q ⊤ V q ⊤ ,where Σ q ∈ R p Hq × p Hq is a diagonal matrix of full rank, U q ∈ R l × p Hq , U q ∈ R l × ( l − p Hq ) , V q ∈ R p × p Hq and V q ∈ R p × ( p − p Hq ) , while U q , h U q U q i and V q , h V q V q i are unitary matrices. When there is no direct feedthrough, Σ q , U q and V q are empty matrices , and U q and V q are arbitrary unitary matrices,while when p H q = p = l , U q and V q are empty matrices, and U q and Σ q areidentity matrices. Then, we decouple the unknown input into two orthogonalcomponents and since V q is unitary, we obtain: d q ,k = V q ⊤ d qk , d q ,k = V q ⊤ d qk , d qk = V q d q ,k + V q d q ,k . (A.1)So, we can represent system (1) as: x qk +1 = f q ( x k ) + B q u qk + G q d q ,k + G q d q ,k + W q w qk ,y k = C q x k + D qk u qk + H q d q ,k + v qk , (A.2)where G q , G q V q , G q , G q V q and H q , H q V q = U q Σ q . Next, the output y k is decoupled using a nonsingular transformation T q = h T q ⊤ T q ⊤ i ⊤ , U q ⊤ = h U q U q i ⊤ to obtain z q ,k ∈ R p Hq and z q ,k ∈ R l − p Hq given by z q ,k , T q y k = U q ⊤ y k = C q x k + Σ q d q ,k + D qk, u qk + v q ,k ,z q ,k , T q y k = U q ⊤ y k = C q x k + D qk, u qk + v q ,k , (A.3) Based on the convention that the inverse of an empty matrix is an empty matrix and theassumption that operations with empty matrices are possible. C q , U q ⊤ C q , C q , U q ⊤ C q , D qk, , U q ⊤ D qk , D qk, , U q ⊤ D qk , v q ,k , U q ⊤ v qk and v q ,k , U q ⊤ v qk . This transformation is also chosen such that (cid:13)(cid:13)(cid:13)(cid:13)h v q ,k ⊤ v q ,k ⊤ i ⊤ (cid:13)(cid:13)(cid:13)(cid:13) = k U q ⊤ v qk k = k v qk k . Appendix B. Matrices and Parameters in Proposition 1 ˜ Y q , ( P − Y C q )Φ q , ˜ Y q , − ( P − Y C q )Φ q Ψ q , ˜ M , − κI − ˘ Q, ˜ M , − κ ( L qf ) I + (1 − α ) P − ˜Γ , ˜ M , κI, N q , Ψ q ⊤ Φ q ⊤ ( P R q − Y Ω q − C q ⊤ Y ⊤ R q ) , N q , ρ I + 2 R q ⊤ Y Ω q − R q ⊤ P R q − Ω q ⊤ (Γ + ( ε − + ε − ) I )Ω q , N q , Φ q ⊤ ( Y Ω q + C q ⊤ Y ⊤ R q − P R q ) , N q , − ε Φ q ⊤ C q ⊤ C q Φ q + I, N q , − I + αP − ε Ψ q ⊤ Φ q ⊤ C q ⊤ C q Φ q Ψ q − L qf I,δ x ∞ , δ x,q ∞ , , if θ q < , θ q ≥ ,δ x,q ∞ , , if θ q ≥ , θ q < , min( δ x,q ∞ , , δ q,x ∞ , ) , if θ q < , θ q < , ,δ x,q ∞ , , ρ ⋆q s η qw + η qv λ min ( P q )(1 − θ q ) ,δ x,q ∞ , , η q − θ q ,θ q , | λ max ( P q ) − | λ min ( P q ) ,θ q , ( L qf + k Ψ q k ) k ( I − ˜ L q C q )Φ q k ,η q , kℜ q k η qv + k Ψ q Φ q W q k η qw , Ω q , C q R q − Q q ,δ d,q ∞ , β q δ x,q ∞ + α q , ℜ q , − (Ψ q Φ q G q M q T q + Ψ q G q M q T q + ˜ L q T q ) ,β q , k V q M q C q − V q M q C q Ψ q k + L qf k V q M q C q k ,α q , k V q M q C q k η qw + (cid:2) k ( V q M q C q G q − V q ) M q T q k + k V q M q T q k (cid:3) η qv ..