Observer Design for Linear Aperiodic Sampled-Data Systems: A Hybrid Systems Approach
aa r X i v : . [ ee ss . S Y ] F e b Observer Design for Linear Aperiodic Sampled-Data Systems: AHybrid Systems Approach
Francesco Ferrante and Alexandre Seuret
Abstract — Observer design for linear systems with aperiodicsampled-data measurements is addressed. To solve this problem,a novel hybrid observer is designed. The main peculiarity ofthe proposed observer consists of the use two output injectionterms, one acting at the sampling instants and one providing anintersample injection. The error dynamics are augmented witha timer variable triggering the arrival of a new measurementand analyzed via hybrid system tools. Using Lyapunov theory,sufficient conditions for the convergence of the observer areprovided. Relying on those conditions, an optimal LMI-baseddesign is proposed for the observer gains. The effectiveness ofthe approach is illustrated in an example.
I. I
NTRODUCTION
A. Motivation
State estimation is a fundamental problem in systems andcontrol theory. Indeed, since state variables can be difficult orimpossible to measure, having access to reliable estimates ofthe plant state is paramount for fault detection, monitoring,and control. The pervasive use of data networks in moderncontrol systems has led to several major difficulties in thedesign of reliable observers for networked systems. Indeed,when the plant output is accessed through a data network, thetypical assumption of continuously or periodically measuringis unrealistic; see, e.g., [10], [12], [21] and [11] for a recentsurvey on aperiodic sampled-data systems. In this paper, weare interested in the design of state observers in the pres-ence of sporadically available measurements. The fact thatmeasurements are available only at some aperiodic isolatedtimes requires the use of observer schemes that are able tohandle this intermittent stream of information to generatesuitable innovation terms. This naturally leads to the use ofhybrid observers, whose dynamics evolve continuously whenno measurements are available and experience instantaneouschanges when a new measurement gets available.
B. Problem Formulation
In this paper, we consider the problem of estimating thestate of a continuous-time linear time-invariant plant in thepresence of intermittent measurements. In particular, weconsider a plant of the form: (cid:26) ˙ z = Azy = Cz (1) Francesco Ferrante is with Univ. Grenoble Alpes, CNRS, GIPSA-lab,F-38000 Grenoble, France. Email: [email protected] Seuret is with LAAS-CNRS, Universit´e de Toulouse, CNRS,Toulouse, France. Email: [email protected] by Francesco Ferrante is partially funded by ANR via projectHANDY, number ANR-18-CE40-0010. where z ∈ R n z is the lant state and y ∈ R n y is theplant output. Matrices A and C are known, constant and ofappropriate dimensions. The plant output y is assumed to beavailable only at some time instants t k , k ∈ N , not known apriori . We assume that the sequence { t k } k ∈ N is unbounded,in addition we suppose that there exist two positive realscalars T ≤ T such that ≤ t ≤ T , T ≤ t k +1 − t k ≤ T , ∀ k ∈ N . (2)The lower bound in condition (2) prevents the existence ofaccumulation points in the sequence { t k } k ∈ N , and, hence,avoids the existence of Zeno behaviors, which are typicallyundesired in practice. In fact, T defines a strictly positiveminimum time in between consecutive measurements. Fur-thermore, T defines the Maximum Allowable Transfer Time(MATI) [17].
C. Related work
The design of the observers in the presence of sporadicmeasurements has been largely studied by researchers overthe last two decades and several observer design strategieshave been proposed in the literature. Such strategies essen-tially belong to two main families. The first one pertainsto observers whose state is entirely reset whenever a newmeasurement is available and that run in open-loop inbetween such events [1], [3], [4], [16], [19], i.e., continuous-discrete observers: (cid:26) ˙ˆ z ( t ) = A ˆ z ( t ) if t = t k , k ∈ N , ˆ z ( t + ) = ˆ z ( t ) + F ( y ( t ) − C ˆ z ( t )) if t = t k , k ∈ N , (3)where F is a gain to be designed, which can be potentiallyselected to be dependent on the time elapsed in betweenmeasurements; see [1], [3], [19]. The working principle ofthe above observer is as follows, when no plant measurementis available, the observer behaves as a copy of the plant.When a new measurement gets available, the observer stateis instantaneously reset. The main advantage of this class ofobservers is that it allows to achieve fast convergence rate.On the other hand, fast convergence rate typically comesat the price of enforcing large changes of the observerstate at the sampling times. This can be unsuitable whenemploying an observer-based control. Indeed, large jumps inthe estimate may lead to overly large discontinuities in thecontrol input, which can jeopardize the safety of the actuator.In addition, the fact that the observer runs in “open-loop”may lead to poor intersample behaviors. completely different paradigm has been proposed byKarafyllis and Kravaris in [13]. In [13], the proposed archi-tecture is composed by a so-called output predictor whosestate is reset to the value of the plant output at the samplingtimes and used as an intersample injection to feed a Luen-berger like observer. This idea has been later generalized in[5]. The main advantage of this class of observers is that itavoids the occurrence of jumps in the estimate. Moreover,the above mentioned intersample injection can be tunedto conveniently shape the transient response. However, thisclass of observers typically exhibit less aggressive transientperformance when compared to the scheme in (3). D. Outline of the Proposed Solution
With the objective of achieving a tradeoff between con-vergence speed and transient performance, while avoidingoverly large jumps in the plant estimate, in this paper weblend the architecture (3) with that in [5] and propose a newclass of hybrid observers for aperiodic sampled-data systems.In particular, we consider the following hybrid observer: (cid:26) ˙ˆ z ( t ) = A ˆ z ( t ) + Lθ ( t )˙ θ ( t ) = Hθ ( t ) if t = t k , k ∈ N , (cid:26) ˆ z ( t + ) = ˆ z ( t ) + F ( y ( t ) − C ˆ z ( t )) θ ( t + ) = ( I − CF )( y ( t ) − C ˆ z ( t )) if t = t k , k ∈ N , (4)where the observer gains L , F and H are real matrices ofappropriate dimensions to be designed. Variable ˆ z representsthe estimate of z provided by the observer. The observer in(4) generalizes several existing architectures for state estima-tion in the presence of sampled-data aperiodic measurements.In particulalr, selecting H = 0 and F = 0 leads to classicalsampled-data observers with zero-order hold output injection[15], [18]. If only F is set to zero, the resulting observerresumes to the observer presented in [5]. If H and L areboth set equal to zero, one recovers (3). E. Contribution and organization
The main contribution of this paper consists of sufficientconditions for the design of the observer (4) to ensure globalexponential stability of the estimation error with tunabletransient performance. Compared to the previous schemesin this area, the observer contains three correction terms tobe designed. More precisely, a blend of injections duringflows and jumps. The paper is organized as follows. Sec-tion II presents a hybrid model of the error dynamics and asufficient conditions to exponential stability of the estimationerror. The main contributions of the paper are presented inSection III, where computationally affordable conditions forthe design of the observer gains are provided. These resultsare illustrated through an example in Section IV.
F. Notation
The symbol N stands for the set of positive integersincluding zero, R ≥ ( R > ) denotes the set of nonnegative(positive) reals, R n is the Euclidean space of dimension n , R n × m is the set of n × m real matrices, and S n + is the setof n × n symmetric positive definite matrices. The identity matrix is denoted by I . For a matrix M ∈ R n × m , M T denotes the transpose of M . When M is a square matrix, He( M ) refers to M + M T . For a symmetric matrix M , M ≻ , M (cid:23) , M ≺ , and M (cid:22) denote positivedefiniteness, positive semidefiniteness, negative definiteness,and negative semidefiniteness of M , respectively. Given,symmetric matrices M and M of the same size, we use thenotation M (cid:22) M to denote M − M (cid:22) . In partitionedsymmetric matrices, the symbol • stands for symmetricblocks. For a vector x ∈ R n , | x | denotes its Euclideannorm. Given two vectors x, y , we use the equivalent notation ( x, y ) = [ x T y T ] T . Given a vector x ∈ R n and a nonemptyset A ⊂ R n , the distance of x to A is defined as | x | A =inf y ∈A | x − y | . For any function z : R → R n , we denote z ( t + ) := lim s → t + z ( s ) , when it exists. G. Preliminaries on Hybrid Dynamical Systems
In this paper we consider hybrid dynamical systems in theframework [9] represented as: H (cid:26) ˙ x = f ( x ) , x ∈ C ,x + ∈ G ( x ) , x ∈ D . (5)where x ∈ R n is the state vector, f : R n → R n denote the flow map and G : R n ⇒ R n the (set valued) jump map ,while the sets C ⊂ R n and D ⊂ R n refer to the flow andthe jump sets . A set E ⊂ R ≥ × N is a hybrid time domain if it is the union of a finite or infinite sequence of intervals [ t j , t j +1 ] × { j } , with the last interval (if existent) of theform [ t j , T ) with T finite or T = ∞ . A function φ definedover a hybrid time domain is a hybrid arc if t φ ( t, j ) is locally absolutely continuous for each j . Given a hybridarc φ , dom t φ := { t ∈ R ≥ : ∃ j ∈ N s.t. ( t, j ) ∈ dom φ } and dom j φ := { j ∈ N : ∃ t ∈ R ≥ s.t. ( t, j ) ∈ dom φ } .Given a hybrid arc φ , s ∈ dom t φ , and i ∈ dom j φ , j ( s ) = min { j ∈ N : ( s, j ) ∈ dom φ } and t ( i ) = min { t ∈ R ≥ : ( t, i ) ∈ dom φ } . A solution φ to H is maximal if itsdomain cannot be extended and it is complete if its domainis unbounded. Given a set M , we denote by S H ( M ) the setof all maximal solutions φ to H with φ (0 , ∈ M . If noset M is mentioned, S H is the set of all maximal solutionsto H . The following notion of global exponential stability inconsidered in the paper. Definition 1: (Global exponential stability [20]) Let
A ⊂ R n be closed. The set A is said to be globally exponentiallystable (GES) for hybrid system H if there exist strictlypositive real numbers λ, k such that every maximal solution φ to H is complete and it satisfies for all ( t, j ) ∈ dom φ | φ ( t, j ) | A ≤ ke − λ ( t + j ) | φ (0 , | A . (6) ⋄ We invite the reader to check [9] for more details on theconsidered framework for hybrid systems.II. H
YBRID M ODELING AND S TABILITY A NALYSIS
A. Hybrid Modeling
Let us first introduce the following change of variables ε := z − ˆ z, ˜ θ := C ( z − ˆ z ) − θ, hich defines, respectively, the estimation error and thedifference between the output estimation error and θ . Inparticular, the dynamics of those estimation errors read: ( " ˙ ε ( t )˙˜ θ ( t ) = F (cid:20) ε ( t )˜ θ ( t ) (cid:21) if t = t k , k ∈ N (cid:26) (cid:20) ε ( t + )˜ θ ( t + ) (cid:21) = G (cid:20) ε ( t )˜ θ ( t ) (cid:21) if t = t k , k ∈ N (7)where F := (cid:20) A − LC LCA − CLC − HC CL + H (cid:21) , G := (cid:20) I − F C
00 0 (cid:21) . (8)The fact that the observer experiences jumps, when a newmeasurement is available and evolves according to a differen-tial equation in between updates, suggests that the updatingprocess of the error dynamics can be described via a hybridsystem. Hence, we represent the whole system composed bythe plant (1), the observer (4), and the logic triggering jumpsas a hybrid system. The proposed hybrid systems approachalso models the hidden time-driven mechanism triggeringthe jumps of the observer. To this end, and as in [4], [14],[5], we augment the state of the system with an auxiliarytimer variable τ that keeps track of the duration of flows andtriggers a jump whenever a certain condition is verified. Thisadditional state allows to describe the time-driven triggeringmechanism as a state-driven triggering mechanism, therebyleading to a model that can be efficiently represented byrelying on the framework for hybrid systems in [9]. Moreprecisely, we make τ decrease as ordinary time t increasesand, whenever τ = 0 , reset it to any point in [ T , T ] , soas to enforce (2). After each jump, we allow the systemto flow again. The whole system composed by the states ε and ˜ θ , and the timer variable τ can be represented by thefollowing hybrid system, which we denote by H e , with state x = ( ε, ˜ θ, τ ) ∈ R n x where n x := n z + n y + 1 : H e (cid:26) ˙ x = f ( x ) , x ∈ C ,x + ∈ G ( x ) , x ∈ D , (9a)where f ( x ) := F (cid:20) ε ˜ θ (cid:21) − , ∀ x ∈ C , (9b) G ( x ) := G (cid:20) ε ˜ θ (cid:21) [ T , T ] , ∀ x ∈ D , (9c)and the flow set C and the jump set D are defined as follows C := R n z + n y × [0 , T ] , D := R n z + n y × { } . (9d)The set-valued jump map allows to capture all possiblesampling events occurring within T or T units of time fromeach other. Specifically, the hybrid model in (9) is able tocharacterize not only the behavior of the analyzed system fora given sequence { t k } ∞ k =1 , but for any sequence satisfying(2). Concerning existence of solutions to system (9), byrelying on the concept of solution proposed in [9, Definition 2.6], it is straightforward to check that any maximal solutionto (9) is complete. Thus, completeness of the maximalsolutions to (9) is guaranteed for any choice of the gains L, H , and F . In addition, we can characterize the domain ofthese solutions. In particular, from the definition of the sets C and D , it follows that for any maximal solution φ to H e , dom φ = [ j ∈ N ([ t j , t j +1 ]) × { j } , with t = 0 , ≤ t ≤ T ,and t j +1 − t j ∈ [ T , T ] , for all j ∈ N .To solve the considered state estimation problem, ourapproach is to design gains L, F , and H in (9) such thatthe set wherein the estimation error is zero is globallyexponentially stable for (9). To this end, we consider thefollowing closed set A = { } × { } × [0 , T ] , (10)and provide sufficient conditions to ensure globally exponen-tially stability of A for system H e . B. Sufficient conditions for exponential stability
In this section, sufficient conditions for observer designare provided. To this end, let us consider the followingassumption, which is somehow driven by [8, Example 27]and whose role will be clarified later via Theorem 1.
Assumption 1:
There exist two continuously differentiablefunctions V : R n z +1 → R , V : R n y +1 → R , positive realnumbers α , α , ω , ω , χ c , and ̟ d ∈ [0 , such that(A1) α | ε | ≤ V ( ε, τ ) ≤ α | ε | ∀ x ∈ C ;(A2) ω | ˜ θ | ≤ V (˜ θ, τ ) ≤ ω | ˜ θ | ∀ x ∈ C ;(A3) for each ε ∈ R n z , ν ∈ [ T , T ] V (( I − F C ) ε, ν ) ≤ (1 − ̟ d ) V ( ε, , (11)(A4) for each x ∈ C , the function x V ( x ) := V ( ε, τ ) + V (˜ θ, τ ) is such that h∇ V ( x ) , f ( x ) i ≤ − χ c V ( x ) . (12) △ The following result provides sufficient conditions forglobal exponential stability of the set A defined in (10). Theorem 1:
Let Assumption 1 hold. Then, the set A in (10) is globally exponentially stable for H e . Proof:
Using items (A2) and (A3) in Assumption 1,one has that for all x ∈ D , g ∈ G ( x ) V ( g ) ≤ e χ d V ( x ) , (13)where χ d := ln(1 − ̟ d ) . Let φ be any maximal solutionto H e . Then, by integrating ( t, j ) ( V ◦ φ )( t, j ) andusing item (A4) in Assumption 1 and (13), one has, for all ( t, j ) ∈ dom φ , V ( φ ( t, j )) ≤ e − χ c t + χ d j ) V ( φ (0 , , whichby using items (A1) and (A2) in Assumption 1 yields: | φ ( t, j ) | A ≤ e − ( χ c t + χ d j ) ρ ρ | φ (0 , | A , ∀ ( t, j ) ∈ dom φ, (14)here ρ := min { α , ω } and ρ := max { α , ω } . Toconclude, using [6, Lemma 1], it follows that there existsome solution independent positive real numbers ̺ and λ such that for all ( t, j ) ∈ dom φ , − χ c t ≤ ̺ − λ ( t + j ) .Hence, by using the bound in (14), one gets yields, for all ( t, j ) ∈ dom φ , | φ ( t, j ) | A ≤ e − λ ( t + j ) e ̺ ρ ρ | φ (0 , | A . Thisconcludes the proof. C. Quadratic conditions
A possible construction for the functions V and V inTheorem 1 is illustrated in the result given next. Theorem 2:
Let
L, H , and F be given. Assume that thereexist P ∈ S n z + , P ∈ S n y + , δ > , and η > such that thefollowing conditions hold M ( µ i ) ≺ ∀ i ∈ { , } , (15) (cid:20) − P P − C T F T P • − e δT P (cid:21) (cid:22) , (16)where, for all µ ∈ R , M ( µ ) is defined in (19) (at the top ofthe next page) and µ := η and µ := (1 + η ) e δT − . Then,functions ( ε, τ ) V ( ε, τ ) = e − δτ ε T P ε, ( ε, ˜ θ ) V (˜ θ, τ ) = (1 + η − e − δτ )˜ θ T P ˜ θ, (17)satisfy Assumption 1 and the set A in (10) is globallyexponentially stable for H e . Proof:
As a first step, notice that V and V satisfyitems (A1) and (A2) in Assumption 1 with: α = λ max ( P ) , α = e − δT λ min ( P ) , ω = (1 − e − δT + η ) λ max ( P ) and ω = ηλ min ( P ) . Straightforward calculations show that forall x ∈ Ch∇ V ( x ) , f ( x ) i = e − δτ (cid:20) ε ˜ θ (cid:21) T M ( µ ( τ )) (cid:20) ε ˜ θ (cid:21) , (18)where, for all τ ∈ [0 , T ] , µ ( τ ) := (1 + η ) e δτ − and M ( · ) isdefined in (19) (at the top of the next page). Since M ( µ ( τ )) is affine with respect to µ ( τ ) , it is also convex with respectto it. In addition, notice that range µ = [ η, (1 + η ) e δT −
1] = : [ µ , µ ] . Therefore, the following equivalence holds: M ( µ ( τ )) ≺ , ∀ τ ∈ [ T , T ] ⇔ M ( µ ) ≺ , µ ∈ { µ , µ } . Hence, it follows that (15) implies item (A4) in Assump-tion 1. To conclude the proof, it remains to show thatinequality (16) implies the satisfaction of item (A3) inAssumption 1. To this end, notice that for all ε ∈ R n z , ν ∈ [ T , T ] , V (( I − F C ) ε, ν ) − V ( ε,
0) = ε T (cid:0) e − δν ( I − F C ) T P ( I − F C ) − P (cid:1) ε ≤ ε T Q ε where Q := e − δT ( I − F C ) T P ( I − F C ) − P . Hence, if Q (cid:22) , it follows that item (A3) in Assumption 1 holdswith any ̟ d ∈ h , | λ max ( Q ) | α i ∩ [0 , . At this stage noticethat by simple congruence transformations and by Schurcomplement, (16) is equivalent to Q (cid:22) . Hence, (16) impliesthat item (A3) in Assumption 1 holds. The proof is concludedby application of Theorem 1. It is straightforward to check that α − | λ max ( Q ) | ∈ [0 , . III. O
BSERVER D ESIGN
A. Guaranteed Cost Observer Design
The objective of this section is to transform the stabilitycondition of Theorem 2 into constructive ones. This meansthat the observer gains appears now as additional decisionvariables. In this situation, the conditions are no longer LMI.However, the use of simple manipulations inspired from [5]allows to alleviate this drawback. In addition another aspectof this section is to illustrate how the proposed architecturelends itself to a guaranteed cost design, this is not the casefor (3).Let φ be any solution to H e , consider the following costfunctional [7]: J ( φ ) := Z dom t φ q c ( φ ( s, j ( s ))) ds + dom j φ X j =1 q d ( φ ( t ( j ) , j − , where for all x = ( ε, ˜ θ, τ ) ∈ C , q c ( x ) := ε T Q F ε and q d ( x ) := ε T Q J ε , with Q F , Q J ∈ S n z + . In particular, for any ξ ∈ C , we consider the following cost associated to H e : J ⋆ ( ξ ) = sup φ ∈S H e ( ξ ) J ( φ ) . The following result is established.
Theorem 3:
Suppose that there exist P ∈ S n z + , P ∈ S n y + , Y ∈ R n z × n y , X ∈ R n y × n y , and Z ∈ R n z × n y , δ > , and η > such that the following conditions hold: R i ≺ , ∀ i ∈ { , } , (20) (cid:20) − P + Q J P − C T Z T • − e δT P (cid:21) (cid:22) , (21)where R i is defined in (22) (at the top of the next page) with µ := η , µ := (1 + η ) e δT − , ˜ µ := 1 , and ˜ µ := e δT .Let L = P − Y, H = P − X − CP − Y, F = P − Z, (23)Then, the following items hold: ( i ) the set A defined in (10) is globally exponentially stablefor hybrid system H e ; ( ii ) For any initial condition ξ = ( ξ ε , ξ ˜ θ , ξ τ ) ∈ C , inequality J ⋆ ( ξ ) ≤ e − δξ τ ξ T ε P ξ ε + (1 + η − e − δξ τ ) ξ T ˜ θ P ξ ˜ θ holds. Proof:
Thanks to the definition of the observer gainsin (23), we have P L = Y, P ( H + CL ) = X and P F = Z . Therefore, due to Q F and Q J being positivedefinite, a few calculations allow to show that (20) and (21)imply, respectively, (15) and (16). Hence, item ( i ) followsdirectly from Theorem 2. To conclude, let V be defined asin Assumption 1 with V and V as in (17). By followinganalogous steps as in the proof of Theorem 2, it can be easilyshown that the satisfaction of (20) implies for all x ∈ C , h∇ V ( x ) , f ( x ) i + ε T Q f ε ≤ . Similarly, the satisfaction of(21) can be easily shown to imply for all x ∈ D , g ∈ G ( x ) , V ( g ) − V ( x ) + ε T Q J ε ≤ . Thus, since from item ( i ) maximal solutions to H e converge to the set A in (10), ( µ ) := (cid:20) He( P ( A − LC ))+ δP P L + µ ( CA − CLC − HC ) T P • µ He( P ( CL + H )) − δP (cid:21) . (19) R i := (cid:20) He( P A − Y C )+ δP + ˜ µ i Q F Y + µ i ( P CA − XC ) T • µ i He( X ) − δP (cid:21) . (22) V is positive definite with respect to A and continuouslydifferentiable on R n x , direct application [7, Corollary 1]yields ( ii ) . Hence, the result is established. B. Optimal Design and Numerical Issues
As mentioned in the introduction, one of the main ob-jective of the proposed observer consists of reducing thevariation of the plant state estimate across jumps. To achievethis goal, it appears relevant to consider additional constraintsthroughout the design of the observer gains, and more inparticular on the gain F . The result stated next provides apossible approach towards this goal. Proposition 1:
Consider matrices P ∈ S n z + , P ∈ S n y + , Y ∈ R n z × n y , X ∈ R n y × n y , and Z ∈ R n z × n y , and positivereal numbers γ and γ , δ , and η , such that (20) and (21)hold and (cid:20) P Y • γ I n y (cid:21) ≻ , (cid:20) P Z • γ I n y (cid:21) ≻ . (24)Then, under the selection of the observer gains given in(23) the following items hold: ( i ) the set A defined in (10) is globally exponentially stablefor hybrid system H e ; ( ii ) For any initial condition ξ = ( ξ ε , , ξ τ ) ∈ C , inequality J ⋆ ( ξ ) ≤ ξ T ε P ξ ε holds. ( iii ) The norm of the observer gains F and L is constrained. Proof:
Items ( i ) and ( ii ) follow directly from item ( ii ) of Theorem 3 whenever ξ ˜ θ = 0 . To show item ( iii ) , let L and F be selected as in (23). Then, direct application of the Schurcomplement lemma reveals that the two matrix inequalitiesin (24) are equivalent to L ⊤ P L (cid:22) γ I n y , F ⊤ P F (cid:22) γ I n y .Proposition 1 can be embedded into the following optimiza-tion problem to perform an optimal design of the observer: minimize P ,P ,X,Y,Z,γ trace( P )+ α γ + α γ subject to (20) , (21) , (24) (25)In particular, minimizing trace( P ) + α γ + α γ allowsto simultaneously bound the observer gains F and L and,in the light of item ( ii ) in Proposition 1, to minimize thecost J ⋆ ( ξ ) (with ξ ˜ θ = 0 ) uniformly with respect to ξ . Theparameters α and α are introduced to enable a tradeoffbetween the constraints on the observers gains F and L .Those parameters need to be tuned a priori. The impact ofthis tuning is discussed in Section IV. Cases
L F H trace( P ) (I): (25) α =0 α =0 . (cid:2) − (cid:3) (cid:2) . − . (cid:3) − . (II): (24) α =100 α =0 . . (cid:2) . − . (cid:3) (cid:2) . − . (cid:3) − .
93 354 . (III): (24) α =100 α =1 . (cid:2) . − . (cid:3) (cid:2) . − . (cid:3) − .
47 357 . (IV): Hybrid obs. [4] L = 0 , H = 0 (cid:2) (cid:3) (cid:2) . − . (cid:3) ∅ ∗ (V): Hybrid obs. [4] L = 0 , H = 0 (cid:2) (cid:3) (cid:2) . − . (cid:3) ∅ ∗ TABLE I: Different selections of the observer gains.IV. N
UMERICAL EXAMPLE
The objective of this section is to showcase the effective-ness of the proposed hybrid observer. Consider the followingdata for (1): A = (cid:20) . − .
011 0 (cid:21) , C T = (cid:20) . − (cid:21) , T = 0 . , T = 1 . . Table I gathers the gains obtained by solving optimizationproblem (1) with δ = 0 . and η = 10 − and for severalvalues of α and α . Noticing that the case α = α = 0 refers to the situation in which no constraints on the observergains are imposed, which leads to overly large gains.Moreover, to show the benefit of the proposed observer inensuring convergence speed while limiting the variation ofthe estimate across jumps, we compare it with the observer(3). Specifically, we consider a “large” (4th row in Table I)and a “small” gain (5th row in Table I), both gains aredesigned via the conditions in [4]. Indeed, for (3), theamplitude of the variations of the estimate can be limitedby minimizing the norm of the gain F . Notice that sinceobserver (3) runs in open-loop in between measurements,for such a scheme it is not possible to perform a guaranteedcost design as done for observer proposed in this paper viaTheorem 3. This explains why the last column of Table Icontains ∅ . To show the effectiveness of the proposed design,in Fig. 1 we compare the evolution of the estimation error ε with the observer gains presented in Table I from the initialcondition z (0 ,
0) = [ ] , ˆ z (0 ,
0) = [ ] , θ (0 ,
0) = Cz (0 ,
0) = 5 and τ (0 ,
0) = 0 . In these simulations, the value of τ at jumpsis selected as τ ( t, j + 1) = T − T sin( t ) + T + T .Fig. 1 clearly shows that for the observer (3) limitingthe norm of the gain F induces poor convergence time.On the other hand, the proposed observer enables to limit PSfrag replacements t [sec] ε ε PSfrag replacements t [sec] ε ε Fig. 1: Evolution of the estimation error ( ε and ε ) (pro-jected onto ordinary time) for the observers provided inTable I. The green, black, blue, purple and red lines referto cases (I) to (V), respectively, with the same order.the variation of the estimation error across jumps (this ismostly visible in the evolution of ε ) while maintaining afast convergence rate. It is worth notice that the additionalconstraints on the observer gains introduced in Theorem 1do not impair the performance of the observer, which ispractically preserved; see green (barely visible), blue andblack lines in Fig. 1. V. C ONCLUSION
In this paper, a novel observer design for linear systemssubject to aperiodic sampled-data measurements has beenpresented. The estimation error dynamics are modeled as ahybrid dynamical systems. By employing a Lyapunov ap-proach, sufficient conditions for global exponential stabilityof a closed set wherein the estimation error is zero areobtained. Guaranteed cost optimal design of the observergains is presented as the solution of an LMI optimizationproblem. The potential of this new hybrid observer is illus-trated through an academic example.This paper can be seen as a first step towards the derivationof more general observers for systems subject to aperiodicsampled-data measurements. One of the main features ofthe proposed architecture consists of combining two typesof injections. The use of this additional degree of freedomprovides more flexibility in the design of the observer andmay potentially lead to better tradeoff between robustnessto measurement noise and convergence speed. In addition,with the objective of limiting the variation of the stateacross jumps, we envision to explore the use of explicithard bounds on the injection term of the observer. Another interesting direction pertains to the use of less conservativeclock-dependent Lyapunov functions for the analysis of theestimation error dyanamics. In this setting, an adaptationof the results in [2] to the observer design problem seemspromising. R
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