Dedicate to memory of Uwe Helmke and Rudolf E.Kalman. Damir Z.Arov. South-Ukrainian National Pedagogical University, Odessa

Transcription

1 Dedicate to memory of Uwe Helmke and Rudolf E.Kalman Damir Z.Arov South-Ukrainian National Pedagogical University, Odessa Based on joint work with Olof Staffans Sde Boker

2 A linear time-invariant passive s/s (state/signal) system Σ = V, X, W has a Hilbert (state) space X, a Krein (signal) space W and a maximal nonnegative (generating) subspace V of the Krein (node) space K = X X W with indefinite inner product on K which respective quadratic form is defined by formula z x ω, z x ω =-2Re(z, x) X + [ω, ω] W (1) and for which V is the graph of an (closed) linear operator G: X W X. For a such system the subspace X o = {xεx: z x ω εv} (2) 2 is dense in X

3 The subspace V of K is called generating, since the set of classical trajectories (x(t),ω(t)) on the intervals I of system,i.e. that are solutions xεc 1 (I, X) and ωεc(i,w) of the equation x t = G( x t ω t ),or, equivalently, ( x t, x t, ω t ) τ εv,tεi (3) The set of generalized trajectories on I of system that are limits of classical trajectories in the spaces C(I,X)xL 2,loc I, W is also defined by V. A passive s/s system has enough rich sets of classical and generalized trajectories for any interval I. In particular,such a system is uniquely solvable, i.e. if (z o, x o, ω o ) τ εv, then there exists at least one classical future (on the interval I=R + = [0, )) trajectory of the system with x 0, x 0, ω(0)) = (z o, x o, ω o ) and any generalized future trajectory of the system is uniquely defined by its initial state x 0 = x o and signal component ω. The set M Σ + of signal components of future generalized trajectories of system with zero initial state and ωεl 2 R +, W is called future behavior of system. 3

4 It is a maximal nonnegative right shift- invariant subspace in the Krein space K 2 (R +, W),that as a topological vector space coincides with L 2 (R +, W), but has indefinite inner product, defined by inner product in W. Any maximal nonnegative right shift- invariant subspace of K 2 (R +, W) is called the future passive behavior on the Krein (signal) space W. It is denoted by M + W Analogicaly are defined past and two -side passive behaviors M W and M(W),resp. Thus,the future behavior of any passive s/s system is a future passive behavior. 4

5 Moreover, any future passive behavior on a Krein space W may be realized as future behavior by each of following three typs passive systems : (a) simple conservative, (b) controllable energy preserving and (c) observable co-energy preserving, that are defined by future behavior up to unitary similarity. These three tips classes of passive systems will be defined below. Analogicaly, by consideration closure subspace in K 2 (R, W) of signal components of past (on I = R = (, 0]) generalized trajectories of system with support on finite intervals [a,0] is defined the past behavior M Σ of system,that is shift-invariant maximal nonnegative subspace if the Krein space K 2 (R, W). Analogicaly, the two-side behavior M Σ of system is defined, that is a maximal nonnegative shift invariant subspace of the Krein space K 2 R, W. By one of these three behaviors of system two others ore uniquely defined by certain reception. 5

6 . The closure in X of the state components x(t) of classical future trajectories of system with zero initial state is called reachable subspace and it is denoted by R Σ c ; system Σ is called controllable, if R Σ c = X A future generalized trajectory (x(t),0) of system is called unobservable and the set of all states of all unobservable future trajectories of system is a closed subspace of X that is called unobservable subspace of system. It is denoted by U Σ and its orthogonal complement is called observable subspace of system and it is denoted by R Σ o and s/s is called observable, if R Σ o = X. 6

7 A passive s/s system is called (a) conservative, if V=V [ ], (b) energy preserving,if V V [ ] and (c) co-energy preserving,, if V [ ] V.A conservative system is called simple if R Σ o R Σ c = X. The passivity condition of a passive s/ s system means that all classical trajectories of this system satisfy the power inequality d dt x t X 2 [ω t, ω(t)] W 7

8 and that same property has adjoint system Σ =(V,X,-W) with generating subspace V = I x I X 0 V [ ], 0 0 J (W, W) where J (W, W) - identity operator from W onto its anti-space W. A s/s system is conservative if it and its adjoin satisfies the power equality condition instead inequality. For any s/s system R c Σ = U Σ,R Σ o =U Σ 8

9 Remark1. According to above definitions (x t, ω (t)) is a classical or generalized future trajectory of Σ for a conservative s/s system Σ if and only if (x t, ω ( t)) is a past classical or generalized trajectory of Σ. Thus, U Σ for such a system is the (closed) subspace of such states x o,that there exist generalized past trajectories (x(t),0) of Σ with x(0)=x o. 9

10 . A passive s/s system Σ = V, X, W is called (orthogonal) dilation of a passive s/s system Σ 1 = (V 1, X 1, W) if X 1 is a closed subspace of X and for any future generalized trajectory (x(t),ω(t)) of Σ with initial state from X 1 the pair (P X1 x t, ω(t)) is a future generalized trajectory of Σ 1 and any future generalized trajectory of Σ 1 may be obtained on such way. In this case also Σ 1 is called (orthogonal) compression of Σ from X onto X 1. If above property holds for all future generalized trajectories (not only with initial state from X 1 ) then Σ 1 is called (orthogonal) projection of Σ onto X 1. A compression Σ 1 of Σ is called restriction of Σ onto X 1 of any future trajectory of Σ with initial state from X 1 is a future trajectory of Σ 1 and converse. If Σ 1 is a compression of Σ then R c Σ1 = P X1 R c Σ and U Σ1 = X 1 U Σ. 10

11 A conservative dilation of a passive s/s system Σ 1 is called minimal if it has not a proper conservative compression, that is also the conservative dilation of Σ 1. A such dilation is defined by original passive s/s system up to unitary similarity, restriction of which onto state space of original system is identity. A passive s/s system is called with minimal losses if its minimal conservative dilation is a simple conservative system. Why it calls with minimal losses will be explained below: this is connected with losses of scattering channels,that are looking for simple conservative system as losses internal channels and for a passive s/s system, that is obtained as a compression of a simple conservative s/s system, as losses external scattering channels. 11

12 Theorem1. Let Σ =(V,X,W) be a conservative s/s system. Let D + = U Σ,D = U Σ.Then for any x ± o εd ± there exist unique unobservable generalized future trajectories of Σ and Σ with initial states x 0 = x + o and x 0 = x o, respectively,their state components x(t) and x t are uniquely defined by their values at any fixed time t and by formulas V + t x + o = x(t), V t x o = x (t) the C 0 - semigroup of isometries V + (t) and V (t) are defined on D + and D,respectively. Moreover, following conditions are equivalent: a Σ is simple, (b ) t>0 V + t D + = {0},(c) t>0 V t D = {0} 12

13 Definition. If Σ is a simple conservative s/s system with D + {0} ( D {0} ) then the pair (D +, V + t ) (resp. (D, V (t)) is called losses internal outgoing ( resp.,incoming) scattering channel for Σ Remark2 In view of remark1, D is the subspace of all states x o,for which exist (unique) past generalized trajectories (x(t),0) of considered conservative system Σ with x(0)=x o and for this trajectory x(t)=v ( t)x 0, tεr. Moreover, since considered here operators are isometries, x o = V t x(t),tεr.this explain why (D, V (t)) is called as incoming internel scattering channel of considering conservative s/s system. Theorem2 Any passive s/s system has a minimal conservative dilation,that is defined by original system up to unitary similarity with the restriction on the state space of original passive system equal identity. Remark3. Since a passive s/s system and its compression have same future (past, two-side ) behavior, a passive s/s system and its conservative dilation have same future (past, two-side) behavior. 13

14 But a minimal conservative dilation of considered passive s/s system not necessary is simple and it is simple if and only if original passive s/s system is with minimal losses. Theorem3 Let Σ 1 = V 1, X 1, W be a passive s/s system, that is a compression of a conservative s/s system Σ = (V, X, W) with internal channels (D ±, V ± (t))then there exist orthogonal decompositions X=Z c X 1 Z such, that Z is a closed unobservable subspace for Σ i.e Z D + and V + t Z Z,tεR + L=X 1 Z is a strongly invariant subspace for Σ, i.e. if x o εl and (x(t),ω(t)) is a future generalized trajectory of Σ with x(0)=x o,then x(t) εl for every tεr +. Moreover,along such decompositions there exist unique that have maximal and minimal subspaces Z along all others Z=Z max and Z=Z min ( for all other Z: Z min Z Z max 14

15 Furthermore,if a conservative s/s system Σ has an orthogonal decomposition of state space X with considered above properties (a) and (b), then there exists unique passive s/s system that is the compression of Σ onto X 1 Remark3. Since R Σ c is a minimal closed strongly invariant subspace for Σ,R Σ c L and hence Z c = L (R Σ c ) = U Σ = D In view of the property (a),v + t inhere a continuous semihgroup of izometrical operatoers V + o t = V + (t)i Z.The pair (Z,V + o (t)) we interpretate as external losses outgoing scattering channels for Σ 1. Moreover, if Σ is a (minima;)conservative dilation of a passive s/s system Σ 1,in this and only in this case Σ is a (minimal)conservative dilation of (Σ 1 ) and Σ is simple if and only if Σ is simple. By consideration adjoint systems we will come to the notion of losses external incoming channels for compression Σ 1 15

16 Theorem4. Let M + (W) be a future passive behavior on a Krein (signal) space W. 1.There exists a simple conservative s/s system Σ = (V, X, W) with M + Σ = M + (W) and it is defined up to unitary similarity by its future behavior. 2.There exists a controllable energy preserving (passive) s/s system Σ cep = (V cep, X cep, W) with same future behavior and it is defined up to unitary similarity. Moreover, it may be obtained as restriction of Σ onto R Σ c,i.e. with X cep = R Σ c.thus,if Σ is controllable and only in this case Σ cep = Σ.In other case Σ has non trivial losses incoming internal channels (D, V (t)) with D = U Σ,that is external losses channel for Σ cep. 3.There exists an observable co-energy preserving (passive) s/s system Σ ocep = (V ocep, X ocep W) with same future behavior and it is defined up to unitary similarity. Moreover,it may be obtained as the projection of Σ onto R Σ o and hence it coincide with Σ if last is observable. In other case Σ has a losses outgoing internal channels (D +, V + (t)), that are losses external channels for Σ ocep. 16

17 A passive s/s system Σ o = (V o, X o, W) is called optimal,if for any other passive s/s system Σ 1 = (V 1, X 1, W) with same future behavior for any future generalized trajectories (x o (t),ω(t)) and (x 1 t, ω(t)) of these systems (with same signal component ω(t) and with initial zero states x o t Xo x 1 t X1 for every tεr + A passive observable s/s system Σ o = V o, X o, W is called *-optimal if for any other passive observable s/s system with same future behavior for any their future generalized trajectories with same signal components and zero initial state the states components satisfy the condition x o (t) X o x 1 (t) X1 for every t R _ 17

18 Theorem5. In the setting of theorem4,there exists a minimal (i.e. controllable and observable optimal passive s/s system Σ mo =(V mo, X mo, W) and a minimal *-optimal passive s/s system Σ m o = (V m o, X m o, W) with given passive behavior and they are defined by unitary similarity by this behavior. Such system may be obtained as the compressions of simple conservative s/s system Σ with considered future behavior onto subspaces X mo = X o = P RΣ or Σ c and X m o = X. =P RΣ c R Σ o,respectively. o First system is the projection onto X o = R Σcep of controllable energy preserving s/s system Σ cep and the second is the restriction onto X. = c R Σo ep of observable *-energy preserving s/s system Σ o ep that were obtained in the Theorem4. 18

19 All above considered notions and results on passive s/s systems have analogies in the passive input/state /output (i/s/o) systems theory and there exists intimately connections between these two theories, respective notions and results. In particular,this relate to the notions conservativeness, controllability, observability, simplicity, minimality, optimality,*-optimality, compression and dilation, losses internal and external channels and others. In an i/s/o system Σ iso =(S,X,U,Y) are considered a Hilbert state space X,(Krein or Hilbert) input and output spaces U and Y,resp.,and a may be multivaluate system operator S:X[+]U X + Y, that generate the sets of classical or generalized trajectories on the intervals (in particular,future,past and two-side) analogical as it was done for s/s system by its generating subspace V. 19

20 (3) For a such i/s/o system a quadratic form Q(u,y) = [ u y, G u y ] U + Y, G=G, G 1 εb(u + Y) (2) is considered (in control theory called supply rate ),that define respective Krein inner product on U[+]Y and this Krein space with this new inner product on it will be considered as signal space W of s/s.system Σ = (V, X, W), in which V={ z x u y : x u εdom(s), z y εs x u } 20

21 Starting system is called Q-passive (conservative) if constructed s/s system is passive (resp.,conservative). In special case, when U and Y are Hilbert spaces and G= I U 0 0 I Y this i/s/o system is called passive (conservative) scattering. If dimu=dimy and G= 0 Ψ Ψ 0 where Ψ is an unitary operator, it is a passive (conservative) impedance system. If U and Y are Krein spaces and G is defined as above for scattering system, it is a passive(conservative)transmission system. 21

22 In the scattering, impedance and transmission cases the in the constructed Krein signal space W it has natural ordered direct sum decomposition, W=U +Y,where U and Y are identificated with U {0} and {0} Y that in these cases are a fundamental, Lagrandian and orthogonal decompositions, respectively. Conversely, starting from arbitrary passive (conservative) s/s system Σ = V, X, W and considered arbitrary ordered direct sum decomposition of its signal space W=U +Y a respective Q-passive (conservative) i/s/o system (i/s/o representation of Σ) may be obtained, in particular scattering (for the fundamengtal decomposition),impedance(for a Lagrangian decomposition) and transmission(for an orthogonal decomposition). 22

23 Thus,there exists two-side connection between passive s/s and i/s/o systems. Moreover, to a passive s/s system correspond infinite many its passive i/s/o representations (scattering, or others) and if original system is conservative, or simple conservative, or controllable,or observable, or minimal,or energy preserving,with minimal losses,or,in this and only this case its i/s/o representation has same property, has same reachable and observable subspaces, has same internal or external losses channels. Above formulated in theorems1-4 results may be obtained from respective results for scattering i/s/o representations of considering s/s system with account that to any passive future behavior M + (W) corresponds W + W, the respective passive future behavior in the frequency domain,that is the Laplace transform of time domain passive future behavior and is a maximal nonnegative shift invariant subspace of the Krein space K + 2 (W) (the Laplace transform K 2 (R +, W), then W + (W) is the graph of the scattering matrix, a Schur class function s εs U, Y (contractive B(U,Y)- valued analytical in right half plane),where W = U + Yis a fundamental decomposition of W,and such s is the scattering matrix of respective scattering representation of s/s system with considered behavior. Conversely, results on the passive i/s/o systems may be obtained via respective results on s/s systems 23

24 In a passive scattering system Σ sc = (S, X, U, Y) the system operator S is single valued closed with dense domain in X U and the main operator A=P X S I X 0,that is the generator of a (evolution ) C o semi group T(t) of contractive operators in X. Theorem 6. Let Σ = (V, X, W) be a simple conservative s/s system, Let W=U[+]-Y be a fundamental decomposition of the Krein signal space W and let Σ sc = (S, X, U, Y) be respective scattering representation of Σ.Then 1.This representation is a simple conservative scattering system with evolution semi -group T(t). 2.If (D, V (t)) and (D +, V + (t)) are losses internal incoming and outgoing channals of Σ,then D + = {x o εx: T(t)x o = x o for every t εr + }, D = {x o εx: T t x o = x o for every tεr + }, 24 V + t = T(t)I D+, V t = T (t)i D.

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26 The evolution semigroup T(t) of considered above simple conservative scattering system is contractive coupling of the simple semi-unitary semi-groups V (t) and V + (t),that is a dissipative versia of generalized Lax-Phillips scattering scheme.for it is defined the notion of scattering suboperator s i,that is a contractive B(N, N + ) valued measurable imaginary axis function.in the terms of the properties of this function may be formulated criteries when original simple conservative system is controlable (N = {0}), observable (N + = {0}),minimal (both conditions satisfy),when all minimal passive s/s systems with given future passive behavior are unitary similar ( s i is boundary value of a Schur class function), when all them are similar ( the Hankel operator with symbol s i has closed range). These results follow from respective results for passive scattering systems. 26