OBSERVER DESIGN FOR DISCRETE-TIME LINEAR SWITCHING SYSTEMS 1

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1 OBSERVER DESIGN FOR DISCRETE-TIME LINEAR SWITCHING SYSTEMS 1 E. De Santi, M.D. Di Benedetto Department of Electrical Engineering and Computer Science, Univerity of L Aquila. (deanti,dibenede@ing.univaq.it Abtract: Obervability wa previouly introduced and characterized for the cla of linear witching ytem. In the dicrete-time cae, detection require a finite, non-zero, number of time intant, while in continuou-time detection can be achieved in an arbitrarily mall amount of time. To addre thi iue, obervability wa introduced, which i baed on the recontruction of the hybrid tate at leat in ome - but not necearily all - time interval. In thi paper, we build upon thee reult to give a definition of a hybrid oberver for dicrete-time witching ytem and to propoe an oberver deign technique. Copyright c 2007 IFAC Keyword: linear witching ytem, obervability, hybrid tate recontruction, hybrid oberver 1. INTRODUCTION Different notion of obervability of hybrid ytem have been given in the literature, depending on the cla of ytem under conideration and on the knowledge that i aumed at the output (ee e.g. (Vidal et al., 2003, (De Santi et al., 2003, (Babaali and Egertedt, 2004, (De Santi et al., 2006, (Bemporad et al., The deign of hybrid oberver wa invetigated in (Balluchi et al., 2002, where a methodology for dynamical oberver of hybrid ytem with linear continuou-time dynamic wa propoed. In thi approach, the complete tate (dicrete location and continuou tate i recontructed from the knowledge of the input and output of a hybrid plant. The ynthei of hybrid oberver 1 Thi work wa partially upported by the HY- CON Network of Excellence, contract number FP6-IST , European Commiion under STREP project n. TREN/07/FP6AE/S / IFLY and by Minitero dell Itruzione, dell Univerita e della Ricerca under Project MACSI and SCEF (PRIN05 wa addreed in a et of cae characterized by the amount of information available on the current location ranging from the cae of complete knowledge (i.e. the hybrid plant produce a dicrete output it current location, treated for intance in (Aleandri and Coletta, 2001, to the cae of abence of dicrete output information (i.e. the hybrid plant produce no dicrete output, conidered e.g. in (Bemporad et al., 2000; Ferrari- Trecate et al., A definition of a hybrid oberver for the cla of autonomou piecewie affine ytem wa given in (Collin and van Schuppen, In (De Santi et al., 2007 the definition and reult of (De Santi et al., 2003 were adapted to dicrete-time linear witching ytem and condition were given to tet obervability. In the dicrete-time cae, detection require a finite, nonzero, number of time intant, while in continuoutime detection can be achieved in an arbitrarily mall amount of time. To addre thi iue, obervability wa introduced, which i baed

2 on the recontruction of the hybrid tate at leat in ome - but not necearily all - time interval. In thi paper, we build upon the reult of (De Santi et al., 2007 to give a definition of a hybrid oberver for dicrete-time witching ytem and to propoe an oberver deign technique. The paper i organized a follow. In Section 2, we introduce dicrete-time linear witching ytem and the notion of obervability. In Section 3, an oberver deign technique i preented. Finally, Section 4 offer ome concluding remark. Notation. We denote by I the et of integer and by R the et of real. For any a, b I for which a b we et [a, b] = {z I : a z b}. The ymbol Im (M denote the range pace of ome matrix M. For a function f : I R n, the ymbol f [a,b] with a b denote the vector f(a f(a 1.. Rn(b a1 f(b 2. PRELIMINARY DEFINITIONS AND RESULTS In thi ection, we recall ome definition and reult for the cla of dicrete time linear witching ytem (De Santi et al., The input of a linear witching ytem are a dicrete and unknown diturbance σ and a continuou control input u. The hybrid tate ξ i compoed of two component: the dicrete tate i belonging to a finite et Q and the continuou tate x belonging to the linear pace R n i, whoe dimenion n i depend on i. The hybrid output ha a dicrete and a continuou component a well, the former aociated to the dicrete tate and the latter aociated to the continuou tate. The evolution of the dicrete tate i governed by a Finite State Machine; a tranition e = (i, σ, h may occur at time t from the dicrete tate i to the dicrete tate h, if the dicrete diturbance σ occur at time t. The evolution of the continuou tate i decribed by a et of dicrete time linear dynamical ytem, controlled by the continuou input u, and whoe matrice depend on the current dicrete tate i. Whenever a tranition e occur, the continuou tate x i intantly reet to a new value R(ex, where R(e i a matrix depending on the tranition e. For implicity, in thi paper we aume that no dicrete ignal i available at the output of the witching ytem, and that a tranition i defined for each pair i, h. More formally, Definition 1. A dicrete time linear witching ytem S i a tuple (Ξ, Θ,R l,s, E, R, (1 where: Ξ = i i Q {i} Rn i i the hybrid tate pace, where: Q = {1, 2,..., N} i the dicrete tate pace; R n i i the continuou tate pace aociated with the dicrete tate i Q; Θ = Σ R m i the hybrid input pace, where: Σ = {σ j, j J} i the dicrete diturbance pace, J = {1, 2,..., N 1 }; R m i the continuou control input pace; R l i the continuou output pace; S i a map aociating to each dicrete tate i Q the linear dynamical control ytem: { x(t 1 = Ai x(t B S(i : i u(t, y(t = C i x(t, where x(t R ni i the continuou tate, u(t R m i the continuou control input, y(t R l i the continuou output, A i R n i n i, B i R ni m and C i R l n i ; E Q Σ Q i a collection of tranition; R i the reet function aociating to every e = (i, σ, h E the reet matrix R(e R n h n i. We now define the emantic of linear witching ytem. We aume that the dicrete diturbance i not available for meaurement. Following (Lygero et al., 1999, a hybrid time bai τ i an infinite or finite equence of et I j = {t I : t j t t j }, with t j > t j and t j = t j1; let be card(τ = L1. If L <, then t L can be finite or infinite. A hybrid time bai τ i aid to be finite, if L < and t L < and infinite, otherwie. Given a hybrid time bai τ, time intant t j are called witching time. Denote by T the et of all hybrid time bae. The witching ytem temporal evolution i defined a follow. Definition 2. (Linear witching ytem execution An execution χ of a linear witching ytem S i a collection: (ξ 0, τ, σ, u, ξ, η, with hybrid initial tate ξ 0 Ξ, hybrid time bai τ T, dicrete diturbance σ : I Σ, control input u : I R m, hybrid tate evolution ξ : I I Ξ and output evolution η : I I R l. The hybrid tate evolution ξ i defined a follow: ξ (t 0, 0 = ξ 0, ξ (t, j = (q (j, x(t, j, t I j, j = 0,..., L, ξ (t j1, j 1 = (q (j 1, R(e j x(t j, j, j = 0,..., L 1, where q : I Q and for any j = 0, 1,..., L, e j = (q (j, σ (j, q (j 1 E and x (t, j i

3 the (unique olution at time t I j of the dynamical ytem S (q (j, with initial time t j, initial condition x (t j, j and control law u. The output evolution of S i pecified by the function η : I R l, which for any j = 0, 1,..., L i defined a: η (t = C i x (t j, j, t [t j, t j 1], (2 where C i i the output matrix aociated with the current dicrete tate q(j = i. In linear ytem theory, obervability deal with the recontruction of the tate, on the bai of the knowledge of the continuou input and of the continuou output that i acceible from the environment. In the following, we generalize thoe notion to the cla of linear witching ytem. According to the definition of the function η, the tranition from one dicrete tate to another may not be viible from the output. The definition of obervability we propoe i baed on the exitence of an input output experiment uch that the hybrid tate i recontructed, at leat in ome time interval of the time bai. Definition 3. Given a nonnegative integer, a linear witching ytem S i obervable if there exit a control input û : I R m and a function ξ : R l( 1 R m 2 Ξ uch that ξ = {ξ (t, j} t [ t j, t j 1], for any execution χ with control input û, for any j {0, 1,..., L} uch that t j t j 1. Moreover, S i aid to be obervable if there exit a nonnegative integer uch that S i obervable. The available information at ome t [ t j, t j 1], given by the input û [t,t 1] and the obervation η [t,t], could be compatible with more than one current hybrid tate. The above definition require that the current hybrid tate i unambiguouly determined at each time intant in the interval [ tj, t j 1]. Since in general a ytem cannot be obervable with = 0, becaue in that cae only the value η(t would be available at the output, we aume 1 without lo of generality. By pecializing Definition 3 to the recontruction of the dicrete component of the hybrid tate only, the following definition i obtained. Definition 4. Given an integer 1, a linear witching ytem S i location obervable for an input function û : I R m, if there exit a function q : R l( 1 R m 2 Q uch that q = {q (j} t [ t j, t j 1], for any execution χ with control input û, for any j {0, 1,..., L} uch that t j t j 1. The ytem S i called location obervable if there exit an input function û for which it i location obervable. The ytem S i called location obervable if there exit for which it i location obervable. For a given, U denote the et of input function for which S i location obervable. A well known aumption on the behavior of witching ytem i the exitence of a dwell time (More, Formally, a non negative δ I i aid to be a dwell time for a witching ytem S if any execution χ = (ξ 0, τ, σ, u, ξ, η generated by S i characterized by the following property: t j t j δ, for any [t j, t j ] τ. If a obervable witching ytem S ha a dwell time δ 1, then each execution i uch that t j t j 1, for any j {0, 1,..., L}. Hence, in that cae, the condition of Definition 4 hold for any execution with control input û U, for any j {0, 1,..., L}. We can tate the following: Theorem 1. (De Santi et al., 2007Given a linear witching ytem S i S i location obervable if and only if (i, h J J, k I, 0 k < n i n h : C i A k i B i C h A k h B h. ii S i obervable if and only if it i location obervable and (A i, C i i obervable i J. iii if S i location obervable, then S i obervable for any max i J n i. If a ytem i obervable, an intereting quetion i how to contruct a function q that recontruct the current dicrete tate at leat in the time interval [ t j, t j 1]. To do thi, we need to ue the available information on the input and obervable output to determine all the dicrete tate that are compatible with it. However, over ome time interval of length, the dynamic of a node i may be inditinguihable from the dynamic reulting from a witching that occurred in that interval between node i and another node h. Then, ince the witching time t j are not known a priori, it would not be poible to undertand whether the dicrete tate i i or h. Thi motivate the following definition: Definition 5. Given a linear witching ytem S, a function ξ : R l( 1 R m 2 Ξ i called

4 hybrid tate oberver of S, for an input function û, if for any execution χ with control input û, for any j {0, 1,..., L}, uch that t j t j 1, the following three condition hold: i ξ = {ξ (t, j}, t [ tj, t j 1] ; ii there exit a time t j [t j, t j ] at which the occurrence of a witching i detected; iii if ξ = {(q, x}, for ome t [ t j, t j 1], then ξ (t, j = (q, x. A function q : R l( 1 R m 2 Q i called dicrete tate oberver of S, for an input function û, if for any execution χ with control input û, for any j {0, 1,..., L}, uch that t j t j 1, the following three condition hold: i q = {q (j}, t [ tj, t j 1] ; ii there exit a time t j [t j, t j ] at which the occurrence of a witching i detected; iii if q = {i}, for ome t [ t j, t j 1], then q (j = i. Roughly peaking, a witching may not be intantaneouly identified. However, it ha to be detected at ome time in the interval [t j, t j ]. Hence, the dicrete oberver may return an erroneou etimation of the current tate if the current data are compatible with the dicrete tate before the witching. After detection of a witching, if the function q return a ingleton, then thi ingleton ha to be equal to the current dicrete tate. For the exitence of a hybrid tate oberver, the ytem S ha to be obervable. Converely, obervability doe not imply the exitence of a hybrid tate oberver. A an example, conider a ytem S with one dicrete mode q, with E = {e = (q, σ, q}, and obervable dynamic ytem S(q decribed by x(t 1 = Ax(t y(t = x(t Sytem S i obviouly obervable, for any 0. However, condition ii may not hold for any function ξ atifying the obervability condition: in fact, even if R(e i not the identity, it may happen that x(t j, j = R(ex(t j, j = x(t j1, j 1 and the available information doe not allow the recontruction of the witching that occurred at time t j. A another example, conider a obervable witching S with no inloop tranition i.e. E = {e = (q i, σ, q h E : q i = q h } =. Then, for a ufficiently large, condition i and ii (rep. i and ii are atified while condition iii (rep. iii may not hold if t j [t j, t j 1]. 3. OBSERVER DESIGN A done in (Balluchi et al., 2002, we decompoe the deign of a hybrid oberver of S into two tep: we firt deign the dicrete tate oberver (i.e. the function q introduced in Definition 5 and then the hybrid tate oberver. In the next ubection, we will decribe each component. 3.1 The dicrete tate oberver In what follow, we alway aume that S i location obervable and that the input function i in U. We alo aume that there are no in loop tranition. Then, for any input function in U, if a witching occurred at ome time t from a known tate q i to an unknown tate q h, at ome time t [t, t ] the obervation η [t,t] are guaranteed to indicate that a witching occurred and that the witching time wa in the interval [t, t]. location obervability implie - location obervability, for all. An etimation of a lower bound for wa given in (De Santi et al., Thi etimation doe not play any particular role here, it i therefore omitted. For a given poitive integer d, define the matrice F (d,i = diag {C i } M (d,i R (d1l n i and G (d,i = diag {C i } N (d,i R dl dni, where I M (d,i A i =. N (d,i = A d i B i A i B i B i B i A d 2 i B i... A i B i B i A d 1 i and the et Ft d { i Q : η [t d,t] G (d,i u [t d,t 1] Im (F (d,i} (3 which i the et of all dicrete tate compatible with the obervation η [t d,t] and the input u [t d,t 1], under the hypothei that no witching occurred in the time interval [t d, t] from tate i to another dicrete tate. If thi hypothei doe not hold, then the et Ft d can be empty. Set F 0 t = {i Q : η(t Im (C i }

5 Define the function φ : I 2 Q and f : I {0, 1} a follow: t = 0, f (0 = 1 while t <, φ (t = Ft 0, t = t 1, t = 1 Define while F t t and F t t φ (t = Ft t f (t = 0 t = t 1 t = min {t t, } endwhile f (t = 1 t = t t = t endwhile. F t 1 t 1 q = φ(t (4 The next reult tate that q atifie the condition of Definition 5, and the time intant t at which f (t = 1 are the time at which the witching are detected. Propoition 1. Let the ytem S be location obervable and aume there i a dwell time δ 1, = 2. The function q defined in 4 i a dicrete tate oberver of S, for any u U. Proof. Since S i location obervable, at each t [ t, t j 1] the obervation η [t,t] and the input u [t,t 1] are compatible only with the dicrete tate q(j. In [ t, t j 1] no witching occurred, and ince by definition φ(t i the et of all dicrete tate compatible with the obervation η [t,t] and the input u [t,t 1], φ (t = {q(j}, t [ t, t j 1]. Hence φ(t = {q(j}, t [ t j 2, t j 1]. We now prove that given t, the updated value t in the definition of φ i the time intant at which from the output it i poible to deduce [ that a witching ] occurred in the time interval t t, t 1, t = min {t t, }. Suppoe that t I j i the time intant at which it i poible to deduce from the output that a witching occurred in the interval [t t, t ]. Then, in the interval [t, t ] no witching can occur, and at any t [t, t ], F t t F t 1 t 1 and F t t. If φ return a ingleton at ome time in the interval [t, t ], thi ingleton i the current dicrete tate. Moreover, at any t [ t, t j 1] the function φ return the correct value for the tate, and it return the ame value for t t j, until uch value i no more compatible with the available information, in which cae Ft t = or Ft t F t 1 t 1. At thi point t i equal to t, and we are ure [ that a witching ] occurred in the time interval t t, t. Since t i initially et at 0 and we can uppoe that 0 i a witching time, then by induction the reult follow, i.e. t i the time intant at which it i poible to deduce from the output [ that a ] witching occurred in the interval t t, t 1 and q atifie the condition of Definition 5. The etimation of for which it i poible to define a dicrete oberver can be refined, with repect to the etimation we have conidered here, i.e. 2. Moreover, a different and more ophiticated deign for the dicrete oberver i poible, for example to increae the peed of convergence to the actual dicrete tate. The quetion i then to find the bet compromie between the performance and the implicity of the oberver. 3.2 The hybrid tate oberver Let q be a dicrete tate oberver for S, with a given, and aume a dwell time δ 1, = max {, max i Q n i }. The dicrete tate oberver at time t give the value of φ (t and f(t, a decribed in the previou ubection. We aume that (A i, C i i obervable, i Q. The deign of the hybrid tate oberver leverage the information returned by the dicrete tate oberver, a decribed below. Conider the function ξ O : I 2 Ξ, defined a: ξ O (t = {i} X i (t where, if f(t = 1, then i φ(t X i (t = {x R n i : C i x = η (t} and if f(t = 0 then X i (t i the et of continuou tate compatible with the dynamic ytem S i, the input u [t,t 1] and the obervation η [t,t], = min {t t, max i Q n i }, where t < t i the lat time intant d at which f(d i equal to 1. Formally, X i (t = M (,i X0(t i N (,i u [t,t 1] } X0(t i = {x : F (,i x = η [t,t] G (,i u [t,t 1] Given φ (t and f(t, et ξ = ξ O (t (5 We now how that ξ, defined in 4, atifie the requirement of Definition 5, for a uitable value of. Propoition 2. Let the ytem S be location obervable and aume there i a dwell time δ 1, = 2 max i Q n i. The function ξ, defined in 5 i a dicrete tate oberver of S, for any u U.

6 Proof. Conider any execution with input function u U, and a time interval [ t j, t j]. By aumption, t j t j, for all j. Let t be a time in [ ] tj, t j uch that f(t = 1. Since S i location obervable, then t exit, i unique, and belong to the interval [t j, t j ]. At time t j 2, the current dicrete tate ha been recontructed, at time t max i Q n i, which by aumption belong to [ t, t j 1], the value for the current continuou tate ha been recontructed, and, if the function ξ O give a ingleton at ome time in [ t, t j 1], uch a ingleton coincide with the current hybrid tate, ince the dynamic are obervable, and no witching occurred in the interval [ t, t j 1]. The hybrid oberver we propoed above imply compute at each tep all the hybrid tate compatible with the information obtained from the dicrete oberver and with the obervation window. When a witching i detected, i.e. when f(t = 1, the ize of the obervation window i reet to zero, to be ure that the meaurement that are input to the oberver are produced by the ame dynamical ytem, o that they are not ubject to uncertaintie due to a non exact recontruction of the witching time. The propoed technique can be improved. For example, we can compute at time t an etimation {q i } X i (t of the et of tate com- i φ(t patible with the pat obervation, rather than auming that at t the et of poible tate i bounded by {q i } X i (t, X i (t = i φ(t {x R n i : C i x = η (t }. Thi refinement implie a better etimation of the tranient continuou tate evolution and poibly a fater convergence of the hybrid oberver. 4. CONCLUSIONS In thi paper, we propoed a definition of a hybrid oberver for dicrete-time witching ytem and we preented an algorithm for oberver deign. Future work include the development of cutomized technique according to problem-pecific requirement of peed of convergence, preciion, or implicity. Poible extenion can be conidered by removing the aumption of obervability on the dynamical ytem, and addreing detectability of the witching ytem. REFERENCES Aleandri, A. and P. Coletta (2001. Switching oberver for continuou-time and dicretetime linear ytem. Proceeding of the 2001 American Control Conference, Arlington, VA, USA 3, Babaali, M. and M. Egertedt (2004. Obervability of witched linear ytem. Hybrid Sytem: Computation and Control 2004, Lecture Note in Computer Science, R. Alur and G.J. Pappa, Ed. pp Balluchi, A., L. Benvenuti, M.D. Di Benedetto and A.L. Sangiovanni-Vincentelli (2002. Deign of oberver for hybrid ytem. Hybrid Sytem: Computation and Control 2002, Lecture Note in Computer Science, C.J. Tomlin and M.R. Greenreet, Ed. 2289, Bemporad, A., G. Ferrari-Trecate and M. Morari (2000. Obervability and controllability of piecewie affine and hybrid ytem. IEEE Tranaction on Automatic Control 45(10, Collin, P. and J.H. van Schuppen (2004. Obervability of piecewie affine hybrid ytem. Hybrid Sytem: Computation and Control 2004, Lecture Note in Computer Science, R. Alur and G.J. Pappa, Ed. 2993, De Santi, E., M.D. Di Benedetto and G. Pola (2003. On obervability and detectability of continuou-time linear witching ytem. 42nd IEEE Conference on Deciion and Control (CDC 03, Maui, Hawaii, USA pp De Santi, E., M.D. Di Benedetto and G. Pola (2007. Obervability of dicrete time linear witching ytem. Proceeding IFAC Workhop on Dependable Control of Dicrete Sytem, June 13-15, ENS Cachan, France pp De Santi, E., M.D. Di Benedetto, S. Di Gennaro, A. D Innocenzo and G. Pola (2006. Critical obervability of a cla of hybrid ytem and application to air traffic management. Lecture Note on Control and Information Science, H.A.P. Blom and J. Lygero Ed. 337, Ferrari-Trecate, G., D. Mignone and M. Morari (2000. Moving horizon etimation for hybrid ytem. Proceeding of the American Control Conference, Chicago, IL, USA, 3, Lygero, J., C. Tomlin and S. Satry (1999. Controller for reachability pecication for hybrid ytem. Automatica, Special Iue on Hybrid Sytem. More, A.S. (1996. Stability uperviory control of familie of linear et point controller part 1: exact matching. IEEE Tran. on Automatic Control 41(10, Vidal, R., A. Chiuo, S. Soatto and S. Satry (2003. Obervability of linear hybrid ytem. Hybrid Sytem: Computation and Control, Lecture Note in Computer Science, A. Pnueli and O. Maler, Ed. 2623,