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Set-valued solutions to impulsive differential inclusions

Set-valued solutions to impulsive differential inclusions Mathematical and Computer Modelling of Dynamical Systems Vol. 11, No. 2, June 2005, 149 – 158 Set-valued solutions to impulsive differential inclusions TATIANA F. FILIPPOVA Institute of Mathematics and Mechanics, Russian Academy of Sciences, 16 S. Kovalevskaya St., GSP-384, Ekaterinburg 620219, Russia. This paper deals with the state estimation problem for impulsive control systems described by differential inclusions with measures. The problem is studied under uncertainty conditions with set-membership description of uncertain variables which are taken to be unknown but bounded with given bounds. Such problems arise from mathematical models of dynamical and physical systems for which we have an incomplete description of their generalized coordinates (e.g. the model may contain unpredictable errors without their statistical description). In this setting instead of an isolated trajectory of the dynamical control system we have a tube of such trajectories and the phase state vector should be replaced by the set of its possible values. The techniques of constructing the trajectory tubes and their cross-sections that may be considered as set-valued state estimates to differential inclusions with impulses are studied. Keywords: Uncertainty, control, differential inclusions, impulsive system. 1. Introduction In this paper the impulsive control problem for a dynamical systems under uncertainty conditions is studied. In many applications related to control problems the evolution of the dynamic control system depends not only on the current system state but also on uncertain disturbances or errors in modelling. There are many publications devoted to different treatments of uncertain dynamical systems, e.g. [1 – 5]. The model of uncertainty considered here is deterministic, with a set-membership description of uncertain items which are taken to be unknown but bounded with given bounds. We consider a dynamic control system described by a differential equation with measure [1,7 – 13] dxðtÞ¼ fðt; xðtÞ; uðtÞÞdt þ Bðt; xðtÞÞdvðtÞ; ð1Þ x 2 R ; t  t  T; with unknown but bounded initial condition *Email: ftf@imm.uran.ru Mathematical and Computer Modelling of Dynamical Systems ISSN 1387-3954 print/ISSN 1744-5051 online ª 2005 Taylor & Francis Group Ltd http://www.tandf.co.uk/journals DOI: 10.1080/13873950500068542 150 T. F. Filippova 0 0 0 xðt  0Þ¼ x ; x 2 X : ð2Þ Here u(t) is the usual (measurable) control with constraint uðtÞ2 U; U R ; and v(t) is an impulsive control function which is continuous from the right, with Var vðtÞ m: t2½t ;T It is well-known that this control system can be modelled by a differential inclusion dx 2 Fðt; xÞdt þ Bðt; xÞdvðtÞ; x 2 R ; ð3Þ with unknown but bounded initial condition 0 0 0 xðt  0Þ¼ x ; x 2 X ; ð4Þ and with certain control variables represented by vector measures dv(t) (generalized or impulsive controls). In such problems the trajectories x(t) are discontinuous and belong to a space of functions with bounded variation. Among many problems related to the treatment of dynamical systems of this type let us mention the results devoted to the precise definition of a solution to (3) [13] and publications on optimal control problems [10,12 – 14]. In the estimation problems the so-called measurement equation is also considered yðtÞ¼ gðt; x; xðtÞÞ ð5Þ with x(t) being the unknown but bounded ‘noise’ or disturbance. The latter equation may be expressed as the state (‘viability’ [2]) constraint: 0 2 Gðt; xÞ; ð6Þ where G is a given set-valued map. One of the principal points of interest in the theory of control under uncertainty conditions is to study the set of all solutions x(t) to (3) – (6). The ‘guaranteed’ estimation problem consists in describing the set Xjtj = [{xjtj} that is actually the reachable set (the information domain) of the system at instant t. The set X(t) may be treated as the unimprovable set-valued estimate of the unknown state x(t) of the system (3 – 6). The mathematical background for investigations of set-valued estimates X(t) of the states of ordinary differential inclusions (without impulsive components) may be found in [16,17]. In this paper we discuss the set-membership approach to the description of the information states for a nonlinear system with impulsive disturbances. 2. The estimation problem In this section we apply the set-membership (bounding) approach to the estimation of unknown states for a system of type (1), (2) but in an autonomous case and without the restriction (6). Consider a dynamic control system Set-valued solutions 151 dxðtÞ¼ fðxðtÞ; uðtÞÞdt þ BðxðtÞ; uðtÞÞdvðtÞ; ð7Þ x 2 R ; t  t  T; with initial condition 0 0 0 xðt  0Þ¼ x ; x 2 X : ð8Þ Following the idea of [1] the information sets are treated here as level sets of the generalized solutions V(t,x) to the HJB (Hamilton – Jacobi – Bellman) equation, where V(t,x) is the value function of type Vðt; xÞ¼ infffðt ; x½t Þjx½  ¼ xð ; t ; x Þ; 0 0 0 x½ ð9Þ x½  is a solution to (7) s.t. x½t¼ xg; 2 0 0 with f being a given function (e.g. f(t ,x)= d (x,X )with X defined in (8) where d(x,M) is the distance function from x to M R ). We assume that the Lipschitz condition jjfðx ; uÞ fðx ; uÞjj þ jjBðx ; uÞ Bðx ; uÞjj 1 2 1 2 4Ljjx  x jj; 8u 2 U; 1 2 is true and jjfðx; uÞjj þ jjBðx; uÞjj  Kð1þjjxjjÞ with some constants L, K4 0. Assume also that the sets fðx; UÞ¼ [ffðx; uÞju 2 Ug; Bðx; UÞ¼[fBðx; uÞlju 2 U;jjljj  1g are convex. Let us introduce a control system of type x_ðtÞ¼ fðxðtÞ; uðtÞÞ þ BðxðtÞ; uðtÞÞwðtÞ; ð10Þ v_ ¼jjwðtÞjj; uðtÞ2 U; jjwðtÞjjdt  m; ð11Þ with state variables x,v and control functions u(t), w(t). Definition [10]. A function x( ) with bounded variation and continuous from the right is called a generalized trajectory to (7)–(8) if there exist a function v ( )also continuous from the right, with bounded variation, and a sequence of controls (u ( ), w ( )) for the n n system (10) – (11) such that the sequence of respective solutions (x (t), v (t)) of (10) – n 1n (11) tends to {x(t),v (t)] at every point t of continuity of {x( ), v ( )}. 1 1 152 T. F. Filippova The set of all such pairs {x( ), v ( )} is a weak *-closure of the set of classical solutions to (10) – (11). n 1 For all s 2 [0,T + m], y2 R , z,Z2 R let us introduce the value function 2 0 2 2 Vðs; y; z; ZÞ¼ minfd ðyð0Þ; X Þþ z ð0ÞþþZ ð0Þg; ð12Þ where the minimum is taken over all solutions fyð Þ; zð Þ; Zð Þg to the auxiliary control system ([10]): y_ðsÞ¼ aðsÞfðyðsÞ; nðsÞÞþð1  aðsÞÞBðyðsÞ; nðsÞÞeðsÞ; z_ðsÞ¼ ð1  aðsÞÞjjeðsÞjj; : ð13Þ ZðsÞ¼ aðsÞ; with terminal conditions yðsÞ¼ y; zðsÞ¼ z; ZðsÞ¼ Z; and with ordinary (measurable) control functions a, n, e such that a 2½0; 1; n 2 U;jjejj  1: The proof of the next theorem follows from the results of [13]. Theorem 1. The cross-section X½T of the trajectory tube Xð Þ to the system (7) – (8) is a subset of the following set X½T p L ðVÞ; ð14Þ y E 0Em ~ ~ L ðVÞ¼ffy; z; ZgjVðT þ E; y; z; ZÞ 0gð15Þ where p M denotes the projection p M ¼fj yj9z; Z s: t: fy; z; Zg2 Mg: Remark 1. It should be mentioned here that the value function V in the optimization problem (12) can be found through the techniques of viscosity ([18,19]) or minimax ([20]) solutions of the corresponding HJB equation ~ ~ ~ @V @V @V þ max a fðy; nÞþð1  aÞ Bðy; nÞe @t @y @y ð16Þ ~ ~ @V @V þð1  aÞ jjejj þ a a 2½0; 1; n 2 U;jjejj  1 ¼ 0; @z @Z with boundary condition Set-valued solutions 153 2 0 2 2 Vðt ; y; z; ZÞ¼ d ðy; X Þþ z þ Z : ð17Þ Theorem 1 gives us the possibility of producing other upper estimates for the information sets X(t) through the comparison principle that allows us to connect the given approach to the techniques of ellipsoidal or box-valued calculus developed for systems with linear structure ([5,21]). Consider the variational inequality @o ~ @o ~ @o ~ þ max a fðy; nÞþð1  aÞ Bðy; nÞe @t @y @y ð18Þ @o ~ @o ~ þð1  aÞ e þ a a 2½0; 1; n 2 U;jjejj  1  0; @z @Z with boundary condition 2 2 2 0 o ~ðt ; y; z; ZÞ d ðy; X Þþ jjzjj þjjZjj : ð19Þ Theorem 2. If there exists a continuously differentiable function o ~ðt; e; z; ZÞ such that the inequalities (18) – (19) are satisfied then the inequality Vðt; y; z; ZÞ o ~ðt; y; z; ZÞð20Þ is valid. The proof of this theorem is based on the verification function techniques applied to the HJB equation (16) [5,18,19]. Theorem 2 produces many estimates for the value function V if a series of different functions o is taken. The following result is a direct consequence of Theorems 1 and 2. Theorem 3. The cross-section X[T] of the trajectory tube X( ) to the system (1) – (2) is a subset of the projection of the level set taken for the value function o ~ : X½T p Lðo ~Þ¼ p [ffy; z; Zgj o ~ðT; y; z; ZÞ 0g y y ¼[fy j9z; Z s:t: o ~ðT; y; z; ZÞ 0g; for any function o that satisfies (18) – (19). 3. The viability and the estimation problems under state constraints In this section we consider the control system of type (3) – (4) dx 2 Fðt; xÞdt þ BðtÞduðtÞ; ð21Þ xðt Þ2 X ; t  t  t 0 0 0 1 with state constraints 0 2 Gðt; xðtÞÞ; t  t  t ; ð22Þ 0 1 154 T. F. Filippova where x 2 R , B(t) is a continuous matrix function, u(t) is a control function with bounded variation, F, and G(t,x) are continuous multivalued maps n n F; G : ½t ; t  R ! convR 0 1 that satisfy the Lipschitz condition with constant L4 0, namely hðFðt; xÞ; Fðt; yÞÞ  L k x  y k;8x; y 2 R ; hðGðt; xÞ; Gðt; yÞÞ  L k x  y k;8x; y 2 R ; and also the condition (for all t,x and some constant c4 0) Fðt; xÞ cð1þk x kÞS; Gðt; xÞ cð1þk x kÞS; where S={x 2 R jjxjj 4 1}. Definition 2. A function x[t]= x(t,t ,x ) will be called a solution to (21) if 0 0 t t Z Z x½t¼ x þ cðtÞdt þ BðtÞduðtÞ; t  t  t ; ð23Þ 0 0 1 t t 0 0 where a function cð Þ 2 L ½t ; t  is a selector of F 0 1 cðtÞ2 Fðt; x½tÞ a:e: The last integral in (23) is taken as the Riemann – Stieltjes integral. Following the scheme of the proof of the well-known Caratheodory theorem we can prove the existence of the solution x[t] = x(t,t ,x ) for all x 2 X 2 compR . 0 0 0 0 n m m Let P be a convex closed cone in R with a vertex at 0 2 R . Denote m m U ¼fu 2 BV ½t ; t  : Var ½u½t ; t  m; 0 1 0 1 p p ð24Þ uðtÞ2 P; t  t  t g; m > 0: 0 1 Assume that there exists at least one solution x ½t¼ x ðt; t ; x Þ ( together with a starting point x ½t ¼ x 2 X and u2 U) that satisfies the condition (22). 0 0 Let X( ,t ,X ) be the set of all solutions to the inclusion (21) that emerge from X (the 0 0 0 ‘solution bundle’) with some u2U. Let X½t ¼ Xðt ; t ; X Þ 1 1 0 0 be its cross-section at instant t . It is not difficult to observe that X[t ] is actually the 1 1 attainability domain (or the ‘reachable set’) at instant t for the differential inclusion (21) with state constraint (22) constructed over all admissible x and u( ). Denote the restriction F (t,x) of the map F(t,x) to the map G by Fðt; xÞ; 0 2 Gðt; xÞ F ðt; xÞ¼ ;; 0 62 Gðt; xÞ: Lemma 1 [17]. A function x(t) defined on the interval [t ,t ] with x 2 X is a solution to 0 1 0 0 (21), (22), (24) if and only if there exists u2 U such that x(t) is a solution to Set-valued solutions 155 dxðtÞ2 F ðt; xÞdt þ BðtÞduðtÞ: We represent F as the intersection of some set-valued functions based on the following auxiliary assertion. Lemma 2 [22]. Suppose A is a bounded set, B a convex closed set, both in R . Then A; 0 2 B fA þ LBjL 2< g¼ ;; 0 62 B: From the lemmas we obtain the following theorem. Theorem 4. A function x( ) defined on an interval [t ,t ]with x(t )2 X is a solution to 0 1 0 0 (21), (22), (24) iff the inclusion dxðtÞ2 ðFðt; xÞdt þ Lðt; xÞGðt; xÞdt þ BðtÞduðtÞÞ Lðt; xÞ2< ; Lð ; Þ is continuous is true for some u2 U and all t2 [t ,t ]. 0 1 We introduce a set of differential inclusions that depend on the matrix function L(t,x). These are given by dz 2ðFðt; zÞþ Lðt; zÞGðt; zÞÞdt þ BðtÞduðtÞ; ð25Þ zðt  0Þ2 X ; t  t  t : 0 0 0 1 Denote by z[ ]= z( ,t ,z ,L) the solution to (25) defined on the interval [t ,t ] with 0 0 0 1 z[t ]= z 2 X and with a function u 2 U. Also denote 0 0 0 Zð ; t ; X ; LÞ¼ fZð ; t ; z ; LÞj z 2 X g 0 0 0 0 0 0 where Z( ,t ,z ,L) is the bundle of all the trajectories z[ ] issued at time t from point z 0 0 0 0 with all admissible u(t) and defined on [t ,t ]. The cross-sections of the set Z( ,t ,X ,L) 0 1 0 0 at time t are then denoted as Z(t ,t ,X ,L). 1 1 0 0 Following the schemes of the proofs of related results in [22,23,17] devoted to the uncertain problems for differential systems with usual control functions we obtain the following characterization of the trajectory tubes. Theorem 5. The following equality is true Xð ; t ; X Þ¼ fZð ; t ; X ; LÞj 0 0 0 0 ð26Þ Lðt; xÞ2 < ; Lð ; Þ is continuousg: Corollary. The following inclusion is true X½t ¼ Xðt ; t ; X Þ fZðt ; t ; X ; LÞj 1 1 0 0 1 0 0 ð27Þ L 2< ; Lð ; Þ is continuousg: Remark 2. For a linear differential impulsive system the relation (27) is actually an equality (the proof of this fact may be seen by using a similar scheme as in [17]). 156 T. F. Filippova 4. Impulsive systems with ellipsoidal constraints Let us consider the linear control system dx ¼ AðtÞxdt þ BðtÞdu; ð28Þ xð0Þ2 X ; 0  t  T with impulsive control u( ) restricted by a set U that will be defined later; X is convex and compact in R . Let m 0 E ¼fl 2 R jl Ml  1g be an ellipsoid in R where M is a symmetric positive definite matrix. Denote m 0 E ¼fyð Þ 2 C j y ðtÞMyðtÞ 1 8t 2½0; Tg ¼fyð Þ 2 C j yðtÞ2 E 8t 2½0; Tg and let us take U = E* where E* is the conjugate ellipsoid to E. We assume in this section that the admissible controls u satisfy the restriction uð Þ 2 U: ð29Þ In particular it follows from (29) that the jumps Du(t )= u(t )– u(t ) of the i i+1 i admissible controls have to belong to an ellipsoid m 0 1 E ¼fl 2 R j l M l  1g: The following theorem concerns the structure of the cross-section of the trajectory tube and generalizes the results of [1]. Theorem 6. The reachable set X(t,t ,X ) is convex and compact for all t2 [0,T]. Every 0 0 state vector x 2 X(t,t ,X ) may be generated by a solution x( ) (i.e. x(t)= x) to (28) with 0 0 the piecewise constant control u( ) whose (n + 1) jumps belong to the set E . 5. Properties of set-valued states Based on the techniques of approximation of the discontinuous generalized trajectory tubes to (1) – (2) by the solutions of the usual differential systems without measure terms [7,16] it is possible to study the dependence of generalized trajectory tubes and their cross-sections (reachable sets) on parameters that define the restrictions on uncertain values (initial data, a variation of impulses, constraints on measurable controls). Let us mention here that by using this method for the problem without state constraints studied in Section 1 we can prove the parameter continuity of the solution tubes under not very restrictive assumptions on the problem data. But it is not difficult to observe that if the state constraints are assumed to be involved in the problem then the reachable sets for impulsive differential systems may become semicontinuous with respect to Hausdorff metrics. Set-valued solutions 157 6. Conclusions The set-valued estimates for the tubes of solutions of a differential inclusion with impulsive components are given. The techniques of constructing the trajectory tubes and their cross-sections that may be considered as set-valued current ‘state vectors’ (the information states) to impulsive differential inclusions under uncertainty are studied. We discuss in this paper the set-membership approach to the description of s for a nonlinear differential system with impulsive disturbances or controls. The schemes developed here may be connected to the techniques of set-valued estimating by ellipsoids or polytopes for linear control systems and to the techniques of level sets for the generalized (viscosity) solutions of the Hamilton – Jacobi – Bellman equation. Acknowledgements The research was supported by the Russian Foundation for Basic Research (RFBR) under project No. 03-01-00528, by the grant ‘Russian Scientific Schools’, No. 1889.2003.1, and by the Russian Academy of Sciences under project ‘The Program for Basic Researches, No. 19’. References [1] Baras, J.S. and Kurzhanski, A.B., 1995, In: A.J. Krener and D.Q. Mayne (Eds) Proc. IFAC NOLCOS Conference, Tahoe, CA (New York: Plenum Press). [2] Chernousko, F.L., 1994, State Estimation for Dynamic Systems (Boca Raton: CRC Press). [3] Krasovskii, N.N. and Subbotin, A.I., 1974, Positional Differential Games (Moscow: Nauka). [4] Kurzhanski, A.B., 1977, Control and Observation under Conditions of Uncertainty. (Moscow: Nauka). [5] Kurzhanski, A.B. and Valyi, I., 1997, Ellipsoidal Calculus for Estimation and Control (Boston: Birkhauser). [6] Anan’ina, T.F., 1975, Different Uravneniya, 11, 595 – 603. [7] Filippova, T.F., 2001, In: A.L. Fradkov and A.B. Kurzhanski (Eds) Proc. IFAC Symposium NOLCOS’2001, St. Petersburg, Russia (Oxford: Elsevier). [8] Gusev, M.I., 1975, In: Differential Games and Control Problems: Proceedings of the Institute of Mathematics and Mechanics of the Academy of Sciences of the USSR, Sverdlovsk, 15, 64 – 112. [9] Kurzhanski, A.B., 1975, In: Differential Games and Control Problems: Proceedings of the Institute of Mathematics and Mechanics of the Academy of Sciences of the USSR, Sverdlovsk, 15, 131 – 166. [10] Miller, B.M., 1989, Automatica i Telemekhanika, 6, 22 – 34. [11] Miller, B.M. and Rubinovich, E.Ya., 2002, Impulsive Control in Continuous and Discrete – Continuous Systems (Foundations of Hybrid Systems Theory) (New York: Kluwer Academic/Plenum). [12] Vinter, R.B. and Pereira, F.M.F.L., 1988, SIAM Journal of Control and Optimization, 26, 155 – 167. [13] Zavalischin, S.T. and Sesekin, A.N., 1991, Impulsive Processes. Models and Applications (Moscow: Nauka). [14] Silva, G.N. and Vinter, R.B., 1996, Journal of Mathematical Analysis and Applications, 202, 727 – 746. [15] Aubin, J.-P. and Frankowska, H., 1990, Set-Valued Analysis (Boston: Birkhauser). [16] Filippova, T.F., 2001, In: J.L. Martin de Carvalho, F.A.C.C. Fortes and M.D.R. de Pinho (Eds), Proc. European Control Conference ECC’2001, Porto, Portugal (Porto: FEUP). [17] Kurzhanski, A.B. and Filippova, T.F., 1993, In: A.B. Kurzhanski (Ed.) Advances in Nonlinear Dynamics and Control: a Report from Russia. Progress in Systems and Control Theory, 17, pp. 122 – 188 (Boston: Birkhauser). [18] Fleming, W.H. and Soner, H.M., 1993, Controlled Markov Processes and Viscosity Solutions (New York: Springer-Verlag). [19] Lions, P.-L. and Souganidis, P.E., 1995, SIAM Journal of Control and Optimization, 23, 566 – 583. [20] Subbotin, A.I., 1995, Generalized Solutions of First-Order PDE’s: The Dynamic Optimization Perspective. Ser. Systems and Control, (Boston: Birkhauser). [21] Kostousova, E.K. and Kurzhanski, A.B., 1996, In: CESA’96 IMACS Multiconference: Computational Engineering in Systems Applications, Lille, France, July 9 – 12. In: P. Borne, G. Dauphin-Tanguy, G. Sueur and E.L. Khattabi (Eds), Proc. Symposium on Modelling Analysis and Simulation, 2 (1997), 849 – 854 (Lille: IMACS/IEEE). 158 T. F. Filippova [22] Filippova, T.F., 2001, Nonlinear Analysis, 47, 5909 – 5920. [23] Filippova, T.F., Kurzhanski, A.B., Sugimoto, K. and Valyi, I., 1995, In: M. Milanese, J. Norton, H. Piet- Lahanier and E. Walter (Eds) Bounding Approaches to System Identification (New York: Plenum Press). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical and Computer Modelling of Dynamical Systems Taylor & Francis

Set-valued solutions to impulsive differential inclusions

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Mathematical and Computer Modelling of Dynamical Systems Vol. 11, No. 2, June 2005, 149 – 158 Set-valued solutions to impulsive differential inclusions TATIANA F. FILIPPOVA Institute of Mathematics and Mechanics, Russian Academy of Sciences, 16 S. Kovalevskaya St., GSP-384, Ekaterinburg 620219, Russia. This paper deals with the state estimation problem for impulsive control systems described by differential inclusions with measures. The problem is studied under uncertainty conditions with set-membership description of uncertain variables which are taken to be unknown but bounded with given bounds. Such problems arise from mathematical models of dynamical and physical systems for which we have an incomplete description of their generalized coordinates (e.g. the model may contain unpredictable errors without their statistical description). In this setting instead of an isolated trajectory of the dynamical control system we have a tube of such trajectories and the phase state vector should be replaced by the set of its possible values. The techniques of constructing the trajectory tubes and their cross-sections that may be considered as set-valued state estimates to differential inclusions with impulses are studied. Keywords: Uncertainty, control, differential inclusions, impulsive system. 1. Introduction In this paper the impulsive control problem for a dynamical systems under uncertainty conditions is studied. In many applications related to control problems the evolution of the dynamic control system depends not only on the current system state but also on uncertain disturbances or errors in modelling. There are many publications devoted to different treatments of uncertain dynamical systems, e.g. [1 – 5]. The model of uncertainty considered here is deterministic, with a set-membership description of uncertain items which are taken to be unknown but bounded with given bounds. We consider a dynamic control system described by a differential equation with measure [1,7 – 13] dxðtÞ¼ fðt; xðtÞ; uðtÞÞdt þ Bðt; xðtÞÞdvðtÞ; ð1Þ x 2 R ; t  t  T; with unknown but bounded initial condition *Email: ftf@imm.uran.ru Mathematical and Computer Modelling of Dynamical Systems ISSN 1387-3954 print/ISSN 1744-5051 online ª 2005 Taylor & Francis Group Ltd http://www.tandf.co.uk/journals DOI: 10.1080/13873950500068542 150 T. F. Filippova 0 0 0 xðt  0Þ¼ x ; x 2 X : ð2Þ Here u(t) is the usual (measurable) control with constraint uðtÞ2 U; U R ; and v(t) is an impulsive control function which is continuous from the right, with Var vðtÞ m: t2½t ;T It is well-known that this control system can be modelled by a differential inclusion dx 2 Fðt; xÞdt þ Bðt; xÞdvðtÞ; x 2 R ; ð3Þ with unknown but bounded initial condition 0 0 0 xðt  0Þ¼ x ; x 2 X ; ð4Þ and with certain control variables represented by vector measures dv(t) (generalized or impulsive controls). In such problems the trajectories x(t) are discontinuous and belong to a space of functions with bounded variation. Among many problems related to the treatment of dynamical systems of this type let us mention the results devoted to the precise definition of a solution to (3) [13] and publications on optimal control problems [10,12 – 14]. In the estimation problems the so-called measurement equation is also considered yðtÞ¼ gðt; x; xðtÞÞ ð5Þ with x(t) being the unknown but bounded ‘noise’ or disturbance. The latter equation may be expressed as the state (‘viability’ [2]) constraint: 0 2 Gðt; xÞ; ð6Þ where G is a given set-valued map. One of the principal points of interest in the theory of control under uncertainty conditions is to study the set of all solutions x(t) to (3) – (6). The ‘guaranteed’ estimation problem consists in describing the set Xjtj = [{xjtj} that is actually the reachable set (the information domain) of the system at instant t. The set X(t) may be treated as the unimprovable set-valued estimate of the unknown state x(t) of the system (3 – 6). The mathematical background for investigations of set-valued estimates X(t) of the states of ordinary differential inclusions (without impulsive components) may be found in [16,17]. In this paper we discuss the set-membership approach to the description of the information states for a nonlinear system with impulsive disturbances. 2. The estimation problem In this section we apply the set-membership (bounding) approach to the estimation of unknown states for a system of type (1), (2) but in an autonomous case and without the restriction (6). Consider a dynamic control system Set-valued solutions 151 dxðtÞ¼ fðxðtÞ; uðtÞÞdt þ BðxðtÞ; uðtÞÞdvðtÞ; ð7Þ x 2 R ; t  t  T; with initial condition 0 0 0 xðt  0Þ¼ x ; x 2 X : ð8Þ Following the idea of [1] the information sets are treated here as level sets of the generalized solutions V(t,x) to the HJB (Hamilton – Jacobi – Bellman) equation, where V(t,x) is the value function of type Vðt; xÞ¼ infffðt ; x½t Þjx½  ¼ xð ; t ; x Þ; 0 0 0 x½ ð9Þ x½  is a solution to (7) s.t. x½t¼ xg; 2 0 0 with f being a given function (e.g. f(t ,x)= d (x,X )with X defined in (8) where d(x,M) is the distance function from x to M R ). We assume that the Lipschitz condition jjfðx ; uÞ fðx ; uÞjj þ jjBðx ; uÞ Bðx ; uÞjj 1 2 1 2 4Ljjx  x jj; 8u 2 U; 1 2 is true and jjfðx; uÞjj þ jjBðx; uÞjj  Kð1þjjxjjÞ with some constants L, K4 0. Assume also that the sets fðx; UÞ¼ [ffðx; uÞju 2 Ug; Bðx; UÞ¼[fBðx; uÞlju 2 U;jjljj  1g are convex. Let us introduce a control system of type x_ðtÞ¼ fðxðtÞ; uðtÞÞ þ BðxðtÞ; uðtÞÞwðtÞ; ð10Þ v_ ¼jjwðtÞjj; uðtÞ2 U; jjwðtÞjjdt  m; ð11Þ with state variables x,v and control functions u(t), w(t). Definition [10]. A function x( ) with bounded variation and continuous from the right is called a generalized trajectory to (7)–(8) if there exist a function v ( )also continuous from the right, with bounded variation, and a sequence of controls (u ( ), w ( )) for the n n system (10) – (11) such that the sequence of respective solutions (x (t), v (t)) of (10) – n 1n (11) tends to {x(t),v (t)] at every point t of continuity of {x( ), v ( )}. 1 1 152 T. F. Filippova The set of all such pairs {x( ), v ( )} is a weak *-closure of the set of classical solutions to (10) – (11). n 1 For all s 2 [0,T + m], y2 R , z,Z2 R let us introduce the value function 2 0 2 2 Vðs; y; z; ZÞ¼ minfd ðyð0Þ; X Þþ z ð0ÞþþZ ð0Þg; ð12Þ where the minimum is taken over all solutions fyð Þ; zð Þ; Zð Þg to the auxiliary control system ([10]): y_ðsÞ¼ aðsÞfðyðsÞ; nðsÞÞþð1  aðsÞÞBðyðsÞ; nðsÞÞeðsÞ; z_ðsÞ¼ ð1  aðsÞÞjjeðsÞjj; : ð13Þ ZðsÞ¼ aðsÞ; with terminal conditions yðsÞ¼ y; zðsÞ¼ z; ZðsÞ¼ Z; and with ordinary (measurable) control functions a, n, e such that a 2½0; 1; n 2 U;jjejj  1: The proof of the next theorem follows from the results of [13]. Theorem 1. The cross-section X½T of the trajectory tube Xð Þ to the system (7) – (8) is a subset of the following set X½T p L ðVÞ; ð14Þ y E 0Em ~ ~ L ðVÞ¼ffy; z; ZgjVðT þ E; y; z; ZÞ 0gð15Þ where p M denotes the projection p M ¼fj yj9z; Z s: t: fy; z; Zg2 Mg: Remark 1. It should be mentioned here that the value function V in the optimization problem (12) can be found through the techniques of viscosity ([18,19]) or minimax ([20]) solutions of the corresponding HJB equation ~ ~ ~ @V @V @V þ max a fðy; nÞþð1  aÞ Bðy; nÞe @t @y @y ð16Þ ~ ~ @V @V þð1  aÞ jjejj þ a a 2½0; 1; n 2 U;jjejj  1 ¼ 0; @z @Z with boundary condition Set-valued solutions 153 2 0 2 2 Vðt ; y; z; ZÞ¼ d ðy; X Þþ z þ Z : ð17Þ Theorem 1 gives us the possibility of producing other upper estimates for the information sets X(t) through the comparison principle that allows us to connect the given approach to the techniques of ellipsoidal or box-valued calculus developed for systems with linear structure ([5,21]). Consider the variational inequality @o ~ @o ~ @o ~ þ max a fðy; nÞþð1  aÞ Bðy; nÞe @t @y @y ð18Þ @o ~ @o ~ þð1  aÞ e þ a a 2½0; 1; n 2 U;jjejj  1  0; @z @Z with boundary condition 2 2 2 0 o ~ðt ; y; z; ZÞ d ðy; X Þþ jjzjj þjjZjj : ð19Þ Theorem 2. If there exists a continuously differentiable function o ~ðt; e; z; ZÞ such that the inequalities (18) – (19) are satisfied then the inequality Vðt; y; z; ZÞ o ~ðt; y; z; ZÞð20Þ is valid. The proof of this theorem is based on the verification function techniques applied to the HJB equation (16) [5,18,19]. Theorem 2 produces many estimates for the value function V if a series of different functions o is taken. The following result is a direct consequence of Theorems 1 and 2. Theorem 3. The cross-section X[T] of the trajectory tube X( ) to the system (1) – (2) is a subset of the projection of the level set taken for the value function o ~ : X½T p Lðo ~Þ¼ p [ffy; z; Zgj o ~ðT; y; z; ZÞ 0g y y ¼[fy j9z; Z s:t: o ~ðT; y; z; ZÞ 0g; for any function o that satisfies (18) – (19). 3. The viability and the estimation problems under state constraints In this section we consider the control system of type (3) – (4) dx 2 Fðt; xÞdt þ BðtÞduðtÞ; ð21Þ xðt Þ2 X ; t  t  t 0 0 0 1 with state constraints 0 2 Gðt; xðtÞÞ; t  t  t ; ð22Þ 0 1 154 T. F. Filippova where x 2 R , B(t) is a continuous matrix function, u(t) is a control function with bounded variation, F, and G(t,x) are continuous multivalued maps n n F; G : ½t ; t  R ! convR 0 1 that satisfy the Lipschitz condition with constant L4 0, namely hðFðt; xÞ; Fðt; yÞÞ  L k x  y k;8x; y 2 R ; hðGðt; xÞ; Gðt; yÞÞ  L k x  y k;8x; y 2 R ; and also the condition (for all t,x and some constant c4 0) Fðt; xÞ cð1þk x kÞS; Gðt; xÞ cð1þk x kÞS; where S={x 2 R jjxjj 4 1}. Definition 2. A function x[t]= x(t,t ,x ) will be called a solution to (21) if 0 0 t t Z Z x½t¼ x þ cðtÞdt þ BðtÞduðtÞ; t  t  t ; ð23Þ 0 0 1 t t 0 0 where a function cð Þ 2 L ½t ; t  is a selector of F 0 1 cðtÞ2 Fðt; x½tÞ a:e: The last integral in (23) is taken as the Riemann – Stieltjes integral. Following the scheme of the proof of the well-known Caratheodory theorem we can prove the existence of the solution x[t] = x(t,t ,x ) for all x 2 X 2 compR . 0 0 0 0 n m m Let P be a convex closed cone in R with a vertex at 0 2 R . Denote m m U ¼fu 2 BV ½t ; t  : Var ½u½t ; t  m; 0 1 0 1 p p ð24Þ uðtÞ2 P; t  t  t g; m > 0: 0 1 Assume that there exists at least one solution x ½t¼ x ðt; t ; x Þ ( together with a starting point x ½t ¼ x 2 X and u2 U) that satisfies the condition (22). 0 0 Let X( ,t ,X ) be the set of all solutions to the inclusion (21) that emerge from X (the 0 0 0 ‘solution bundle’) with some u2U. Let X½t ¼ Xðt ; t ; X Þ 1 1 0 0 be its cross-section at instant t . It is not difficult to observe that X[t ] is actually the 1 1 attainability domain (or the ‘reachable set’) at instant t for the differential inclusion (21) with state constraint (22) constructed over all admissible x and u( ). Denote the restriction F (t,x) of the map F(t,x) to the map G by Fðt; xÞ; 0 2 Gðt; xÞ F ðt; xÞ¼ ;; 0 62 Gðt; xÞ: Lemma 1 [17]. A function x(t) defined on the interval [t ,t ] with x 2 X is a solution to 0 1 0 0 (21), (22), (24) if and only if there exists u2 U such that x(t) is a solution to Set-valued solutions 155 dxðtÞ2 F ðt; xÞdt þ BðtÞduðtÞ: We represent F as the intersection of some set-valued functions based on the following auxiliary assertion. Lemma 2 [22]. Suppose A is a bounded set, B a convex closed set, both in R . Then A; 0 2 B fA þ LBjL 2< g¼ ;; 0 62 B: From the lemmas we obtain the following theorem. Theorem 4. A function x( ) defined on an interval [t ,t ]with x(t )2 X is a solution to 0 1 0 0 (21), (22), (24) iff the inclusion dxðtÞ2 ðFðt; xÞdt þ Lðt; xÞGðt; xÞdt þ BðtÞduðtÞÞ Lðt; xÞ2< ; Lð ; Þ is continuous is true for some u2 U and all t2 [t ,t ]. 0 1 We introduce a set of differential inclusions that depend on the matrix function L(t,x). These are given by dz 2ðFðt; zÞþ Lðt; zÞGðt; zÞÞdt þ BðtÞduðtÞ; ð25Þ zðt  0Þ2 X ; t  t  t : 0 0 0 1 Denote by z[ ]= z( ,t ,z ,L) the solution to (25) defined on the interval [t ,t ] with 0 0 0 1 z[t ]= z 2 X and with a function u 2 U. Also denote 0 0 0 Zð ; t ; X ; LÞ¼ fZð ; t ; z ; LÞj z 2 X g 0 0 0 0 0 0 where Z( ,t ,z ,L) is the bundle of all the trajectories z[ ] issued at time t from point z 0 0 0 0 with all admissible u(t) and defined on [t ,t ]. The cross-sections of the set Z( ,t ,X ,L) 0 1 0 0 at time t are then denoted as Z(t ,t ,X ,L). 1 1 0 0 Following the schemes of the proofs of related results in [22,23,17] devoted to the uncertain problems for differential systems with usual control functions we obtain the following characterization of the trajectory tubes. Theorem 5. The following equality is true Xð ; t ; X Þ¼ fZð ; t ; X ; LÞj 0 0 0 0 ð26Þ Lðt; xÞ2 < ; Lð ; Þ is continuousg: Corollary. The following inclusion is true X½t ¼ Xðt ; t ; X Þ fZðt ; t ; X ; LÞj 1 1 0 0 1 0 0 ð27Þ L 2< ; Lð ; Þ is continuousg: Remark 2. For a linear differential impulsive system the relation (27) is actually an equality (the proof of this fact may be seen by using a similar scheme as in [17]). 156 T. F. Filippova 4. Impulsive systems with ellipsoidal constraints Let us consider the linear control system dx ¼ AðtÞxdt þ BðtÞdu; ð28Þ xð0Þ2 X ; 0  t  T with impulsive control u( ) restricted by a set U that will be defined later; X is convex and compact in R . Let m 0 E ¼fl 2 R jl Ml  1g be an ellipsoid in R where M is a symmetric positive definite matrix. Denote m 0 E ¼fyð Þ 2 C j y ðtÞMyðtÞ 1 8t 2½0; Tg ¼fyð Þ 2 C j yðtÞ2 E 8t 2½0; Tg and let us take U = E* where E* is the conjugate ellipsoid to E. We assume in this section that the admissible controls u satisfy the restriction uð Þ 2 U: ð29Þ In particular it follows from (29) that the jumps Du(t )= u(t )– u(t ) of the i i+1 i admissible controls have to belong to an ellipsoid m 0 1 E ¼fl 2 R j l M l  1g: The following theorem concerns the structure of the cross-section of the trajectory tube and generalizes the results of [1]. Theorem 6. The reachable set X(t,t ,X ) is convex and compact for all t2 [0,T]. Every 0 0 state vector x 2 X(t,t ,X ) may be generated by a solution x( ) (i.e. x(t)= x) to (28) with 0 0 the piecewise constant control u( ) whose (n + 1) jumps belong to the set E . 5. Properties of set-valued states Based on the techniques of approximation of the discontinuous generalized trajectory tubes to (1) – (2) by the solutions of the usual differential systems without measure terms [7,16] it is possible to study the dependence of generalized trajectory tubes and their cross-sections (reachable sets) on parameters that define the restrictions on uncertain values (initial data, a variation of impulses, constraints on measurable controls). Let us mention here that by using this method for the problem without state constraints studied in Section 1 we can prove the parameter continuity of the solution tubes under not very restrictive assumptions on the problem data. But it is not difficult to observe that if the state constraints are assumed to be involved in the problem then the reachable sets for impulsive differential systems may become semicontinuous with respect to Hausdorff metrics. Set-valued solutions 157 6. Conclusions The set-valued estimates for the tubes of solutions of a differential inclusion with impulsive components are given. The techniques of constructing the trajectory tubes and their cross-sections that may be considered as set-valued current ‘state vectors’ (the information states) to impulsive differential inclusions under uncertainty are studied. We discuss in this paper the set-membership approach to the description of s for a nonlinear differential system with impulsive disturbances or controls. The schemes developed here may be connected to the techniques of set-valued estimating by ellipsoids or polytopes for linear control systems and to the techniques of level sets for the generalized (viscosity) solutions of the Hamilton – Jacobi – Bellman equation. Acknowledgements The research was supported by the Russian Foundation for Basic Research (RFBR) under project No. 03-01-00528, by the grant ‘Russian Scientific Schools’, No. 1889.2003.1, and by the Russian Academy of Sciences under project ‘The Program for Basic Researches, No. 19’. References [1] Baras, J.S. and Kurzhanski, A.B., 1995, In: A.J. Krener and D.Q. Mayne (Eds) Proc. IFAC NOLCOS Conference, Tahoe, CA (New York: Plenum Press). [2] Chernousko, F.L., 1994, State Estimation for Dynamic Systems (Boca Raton: CRC Press). [3] Krasovskii, N.N. and Subbotin, A.I., 1974, Positional Differential Games (Moscow: Nauka). [4] Kurzhanski, A.B., 1977, Control and Observation under Conditions of Uncertainty. (Moscow: Nauka). [5] Kurzhanski, A.B. and Valyi, I., 1997, Ellipsoidal Calculus for Estimation and Control (Boston: Birkhauser). [6] Anan’ina, T.F., 1975, Different Uravneniya, 11, 595 – 603. [7] Filippova, T.F., 2001, In: A.L. Fradkov and A.B. Kurzhanski (Eds) Proc. IFAC Symposium NOLCOS’2001, St. Petersburg, Russia (Oxford: Elsevier). [8] Gusev, M.I., 1975, In: Differential Games and Control Problems: Proceedings of the Institute of Mathematics and Mechanics of the Academy of Sciences of the USSR, Sverdlovsk, 15, 64 – 112. [9] Kurzhanski, A.B., 1975, In: Differential Games and Control Problems: Proceedings of the Institute of Mathematics and Mechanics of the Academy of Sciences of the USSR, Sverdlovsk, 15, 131 – 166. [10] Miller, B.M., 1989, Automatica i Telemekhanika, 6, 22 – 34. [11] Miller, B.M. and Rubinovich, E.Ya., 2002, Impulsive Control in Continuous and Discrete – Continuous Systems (Foundations of Hybrid Systems Theory) (New York: Kluwer Academic/Plenum). [12] Vinter, R.B. and Pereira, F.M.F.L., 1988, SIAM Journal of Control and Optimization, 26, 155 – 167. [13] Zavalischin, S.T. and Sesekin, A.N., 1991, Impulsive Processes. Models and Applications (Moscow: Nauka). [14] Silva, G.N. and Vinter, R.B., 1996, Journal of Mathematical Analysis and Applications, 202, 727 – 746. [15] Aubin, J.-P. and Frankowska, H., 1990, Set-Valued Analysis (Boston: Birkhauser). [16] Filippova, T.F., 2001, In: J.L. Martin de Carvalho, F.A.C.C. Fortes and M.D.R. de Pinho (Eds), Proc. European Control Conference ECC’2001, Porto, Portugal (Porto: FEUP). [17] Kurzhanski, A.B. and Filippova, T.F., 1993, In: A.B. Kurzhanski (Ed.) Advances in Nonlinear Dynamics and Control: a Report from Russia. Progress in Systems and Control Theory, 17, pp. 122 – 188 (Boston: Birkhauser). [18] Fleming, W.H. and Soner, H.M., 1993, Controlled Markov Processes and Viscosity Solutions (New York: Springer-Verlag). [19] Lions, P.-L. and Souganidis, P.E., 1995, SIAM Journal of Control and Optimization, 23, 566 – 583. [20] Subbotin, A.I., 1995, Generalized Solutions of First-Order PDE’s: The Dynamic Optimization Perspective. Ser. Systems and Control, (Boston: Birkhauser). [21] Kostousova, E.K. and Kurzhanski, A.B., 1996, In: CESA’96 IMACS Multiconference: Computational Engineering in Systems Applications, Lille, France, July 9 – 12. In: P. Borne, G. Dauphin-Tanguy, G. Sueur and E.L. Khattabi (Eds), Proc. Symposium on Modelling Analysis and Simulation, 2 (1997), 849 – 854 (Lille: IMACS/IEEE). 158 T. F. Filippova [22] Filippova, T.F., 2001, Nonlinear Analysis, 47, 5909 – 5920. [23] Filippova, T.F., Kurzhanski, A.B., Sugimoto, K. and Valyi, I., 1995, In: M. Milanese, J. Norton, H. Piet- Lahanier and E. Walter (Eds) Bounding Approaches to System Identification (New York: Plenum Press).

Journal

Mathematical and Computer Modelling of Dynamical Systems – Taylor & Francis

Published: Jun 1, 2005

Keywords: Uncertainty; control; differential inclusions; impulsive system

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