arXiv:1309.0165v1 [cs.SY] 31 Aug 2013 oncint ieaueo iervriiiyadrelated and reversibility research. time time-o future on of literature pair to a u introduce connection we similarly, III and, Section processes In output respectively. proc mappings, generating anti-causal driving and of sample-paths between 1 spondence the from [10]) [5], ([1], particul theory In realization processes. stochastic stochastic corresponding of given the and a systems model these systems captur stochastic be time-opposite to shown was models time-opposite two whi characterize the necessarily (not to to general inputs order was noise-process in the introduced setting was process stochastic backwa giv or forward This direction, manner. time time-symmetric either a proce in ort in the running and of future process time, future and discrete and past past in between between white t dependence be as the to process studying (see assumed stochastic is smoothing a which [9]), model process (see to filtering is [8]), construction principal [7], [6], [5], [4], sa,i iceetm hs aetefr fasto differ of set a of form the take these discrete-time in usual, CESLnau etr T oa nttt fTechnology, of Institute Royal KTH Center, Linnaeus ACCESS eateto uoain hnhiJa ogUiest,S University, Tong Jiao Shanghai an Universi Automation, VR, Engineering, NSF, of Computer AFOSR, Department & from Electrical grants of by Department supported was research This nScinI eepanhwalfigo tt-yaisinto state-dynamics of lifting a how explain we II Section In deve to ideas of sets two these combine we note, present the In t between duality certain a [13]) (see context different a In stochasti in central is systems stochastic of reversal Time nti ae ecnie iceetm swl scontinuou as well as discrete-time consider we paper this In ntm-eesblt flna tcatcmodels stochastic linear of time-reversibility On ttoaypoesscnb iial iervre.B co realizatio By that stochastic duali in time-reversed. shown time-reversal a was similarly of it where be generalization work can moments, latter processes of this stationary theory In the solutions. in parametrize problems an filtering certain theory, with realization stochastic of development eeslo h iedrcini tcatcssesdriven systems stochastic in direction time the of Reversal rpo .Gogo n nesLindquist Anders and Georgiou T. Tryphon I S II. AEDNMC N ALL AND DYNAMICS TATE x ( t .I I. )= 1) + NTRODUCTION yo inst,Mnepls Minnesota, Minneapolis, Minnesota, of ty Abstract aga,Cia n etrfrIdsra n ple Mathe Applied and Industrial for Center and China, hanghai, theory. n tchl,Sweden, Stockholm, Ax mohn.Smlrieswr eeoe nconnection in developed were ideas Similar smoothing. d h SSF. the d ( t bnn hs w eso da epeethri a herein present we ideas of sets two these mbining + ) tcatcssesdie yabtaysecond-order arbitrary by driven systems stochastic susi hsc,adw niaedrcin for directions indicate we and , in issues e,adtecrepnec ewe h driving the between correspondence the and te), 7’ sn oe procedure. novel a using 970’s pst oes ial,i eto V edraw we IV, Section in Finally, models. pposite 1] 1] 1] n ytmietfiain The identification. system and [12]) [11], [10], iieti ehns ntecneto general of context the in mechanism this tilize results certain case special a as recover we ar, eotu falna ytmdie yanoise a by driven system linear a of output he - Bu ywieniehsbe eta hogotthe throughout central been has noise white by neequations ence yidcdb iervra a nrdcdto introduced was reversal time by induced ty di time. in rd ASEXTENSION PASS se,i poietm-ietos i causal via time-directions, opposite in esses, rcs.W td h eainhpbetween relationship the study We process. ouin ommn rbes nti new this In problems. moment to solutions elzto hoy(e,eg,[] 2,[3], [2], [1], e.g., (see, theory realization c s ti aua odcmoeteinterface the decompose to natural is it ss, -iesohsi iersaednmc.As state-dynamics. linear stochastic s-time ( ooa-nrmn ncniuu ie In time. continuous in hogonal-increment t db utbeda l-asdynamics. all-pass dual suitable by ed srs ossesrpeettoso the of representations systems to rise es ) etotm-ietosi oeiga modeling in time-directions two he nalps ytmalw ietcorre- direct allows system all-pass an [email protected] o eea rmwr hr two where framework general a lop [email protected] aisand matics (1) 1 2 where t ∈ Z, A ∈ Rn×n, B ∈ Rn×p, n, p ∈ N, A has all eigenvalues in the open unit disc D = {z ||z| < 1}, and u(t), x(t) are stationary vector-valued stochastic processes. The system of equations is assumed to be reachable, i.e., rank B, AB, ...An−1B = n, (2)   and non-trivial in the sense that B is full rank. In continuous-time, state-dynamics take the form of a system of stochastic differential equations dx(t)= Ax(t)dt + Bdu(t) (3) where, here, u(t), x(t) are stationary continuous-time vector-valued stochastic processes. Reachability (which in this case, is equivalent to controllability) of the pair (A, B) is also assumed throughout and the condition for this is identical to the one for discrete-time given above (as is well known). In continuous time, stability of the system of equations is equivalent to A having only eigenvalues with negative real part, and will be assumed throughout along with the condition that B has full rank. In either case, discrete-time or continuous-time, it is possible to define an output equation so that the overall system is all-pass. This is done next. The assumptions of stationarity and constant parameter matrices is made for simplicity of notation and brevity and can be easily removed.

A. All-pass extension in discrete-time

Consider the discrete-time Lyapunov equation P = AP A′ + BB′. (4) Since A has all eigenvalues inside the unit disc of the complex plane and (2) holds, (4) has as solution a matrix P which is positive definite. The state transformation

− 1 ξ = P 2 x, (5) and − 1 1 − 1 F = P 2 AP 2 , G = P 2 B, (6) brings (1) into ξ(t +1) = Fξ(t)+ Gu(t). (7)

′ ′ For this new system, the corresponding Lyapunov equation X = F XF + GG has In as solution, where In denotes the (n × n) identity matrix. This fact, namely, that ′ ′ In = F F + GG (8) implies that this [F,G] can be embedded as part of an orthogonal matrix F G U = , (9)  H J  ′ ′ i.e., such that UU = U U = In+p. Define the transfer function

−1 U(z) := H(zIn − F ) G + J (10) 3

corresponding to ξ(t +1) = Fξ(t)+ Gu(t) (11a) u¯(t)= Hξ(t)+ Ju(t). (11b) This is also the transfer function of x(t +1) = Ax(t)+ Bu(t) (12a) u¯(t)= B¯′x(t)+ Ju(t), (12b)

− 1 ′ where B¯ := P 2 H , since the two systems are related by a similarity transformation. Hence,

′ −1 U(z)= B¯ (zIn − A) B + J. (13) We claim that U(z) is an all-pass transfer function (with respect to the unit disc), i.e., that U(z) is a transfer function of a stable system (obvious) and that

−1 ′ −1 ′ U(z)U(z ) = U(z ) U(z)= Ip. (14)

′ The latter claim is immediate after we observe that, since U U = In+p,

′ ξ(t + 1) ξ(t) U = ,  u¯(t)   u(t)  and hence, ξ(t)= F ′ξ(t +1)+ H′u¯(t) (15a) u(t)= G′ξ(t +1)+ J ′u¯(t) (15b) or, equivalently, ′ − 1 ′ x(t)= P A P 1x(t +1)+ P 2 H u(t) (16a) u(t)= B′P −1x(t +1)+ J ′u¯(t). (16b) Setting x¯(t) := P −1x(t + 1), (17) (16) can be written x¯(t − 1) = A′x¯(t)+ B¯u¯(t) (18a) u(t)= B′x¯(t)+ J ′u¯(t) (18b) with transfer function

∗ ′ −1 ′ −1 ′ U(z) = B (z In − A ) B¯ + J . (19) Either of the above systems inverts the dynamical relation u → u¯ (in (12) or (11)).

u(t) u¯(t) ✲ U ✲

Fig. 1. Realization (12) in the forward time-direction. 4

u(t) u¯(t) ✛ U∗ ✛

Fig. 2. Realization (18) in the backward time-direction.

B. All-pass extension in continuous-time Consider the continuous-time Lyapunov equation AP + P A′ + BB′ =0. (20) Since A has all its eigenvalues in the left half of the complex plane and since (2) holds, (20) has as solution a positive definite matrix P . Once again, applying (5-6), the system in (3) becomes dξ(t)= Fξ(t)dt + Gdu(t). (21a) We now seek a completion by adding an output equation du¯(t)= Hξ(t)dt + Jdu(t) (21b) so that the transfer function −1 U(s) := H(sIn − F ) G + J (22) is all-pass (with respect to the imaginary axis), i.e., ′ ′ U(s)U(−s) = U(−s) U(s)= Ip. (23)

For this new system, the corresponding Lyapunov equation has as solution the identity matrix and hence, F + F ′ + GG′ =0. (24) Utilizing this relationship we note that −1 ′ ′ −1 (sIn − F ) GG (−sIn − F ) −1 ′ ′ −1 =(sIn − F ) (sIn − F − sIn − F )(−sIn − F ) −1 ′ −1 =(sIn − F ) +(−sIn − F ) , and we calculate that U(s)U(−s)′ −1 ′ ′ −1 ′ ′ =(H(sIn − F ) G + J)(G (−sIn − F ) H + J ) ′ −1 ′ ′ = JJ + H(sIn − F ) (GJ + H ) ′ ′ −1 ′ (JG + H)(−sIn − F ) H . For the product to equal the identity, ′ JJ = Ip H = −JG′. Thus, we may take

J = Ip H = −G′, 5 and the forward dynamics dξ(t)= Fξ(t)dt + Gdu(t) (25a) du¯(t)= −G′ξ(t)dt + du(t). (25b) Substituting F = −F ′ − GG′ from (24) into (25a) we obtain the reverse-time dynamics dξ(t)= −F ′ξ(t)dt + Gdu¯(t) (26a) du(t)= G′ξ(t)dt + du¯(t). (26b) Now defining x¯(t) := P −1x(t) (27) and using (5) and (6), (26) becomes dx¯(t)= −A′x¯(t)dt + Bd¯ u¯(t) (28a) du(t)= B′x¯(t)dt + du¯(t), (28b) with transfer function

∗ ′ ′ −1 U(s) = B (sIn + A ) B¯ + Ip, (29) where B¯ := P −1B. (30) Furthermore, the forward dynamics (25) can be expressed in the form dx(t)= Ax(t)dt + Bdu(t) (31a) du¯(t)= B¯′x(t)dt + du(t) (31b) with transfer function

′ ′ −1 U(s)= B¯ (sIn − A ) B + Ip. (32)

III. TIME-REVERSALOFLINEARSTOCHASTICSYSTEMS The development so far allows us to draw a connection between two linear stochastic systems having the same output and driven by a pair of arbitrary, but dual, stationary processes u(t) and u¯(t), one evolving forward in time and one evolving backward in time. When one of these two processes is (or, orthogonal increment process, in continuous-time), then so is the other. For this special case we recover results of [1] and [5] in stochastic realization theory.

A. Time-reversal of discrete-time stochastic systems Consider a stochastic linear system x(t +1) = Ax(t)+ Bu(t) (33a) y(t)= Cx(t)+ Du(t) (33b) with an m-dimensional output process y, and x, u, A, B are defined as in Section II-A. All processes are stationary and the system can be thought as evolving forward in time from the remote past (t = −∞). In particular, x(t + 1) is F u-measurable  y(t)  t 6

Z u for all t ∈ , where Ft is the σ-algebra generated by {u(s) | s ≤ t}. Next we construct a stochastic system x¯(t − 1) = A′x¯(t)+ B¯u¯(t) (34a) y(t)= C¯x¯(t)+ D¯u¯(t), (34b) which evolves backward in time from the remote future (t = ∞), and for which x¯(t − 1) is F¯u¯-measurable  y(t)  t Z ¯u¯ for all t ∈ , where Ft is the σ-algebra generated by {u¯(s) | s ≥ t}. The processes x,¯ x, u,u¯ relate as in the previous section. More specifically, as shown in Section II-A,

u u¯(t) is Ft -measurable while ¯u¯ u(t) is Ft -measurable for all t, as examplified in Figures 1 and 2. In fact, the all-pass extension (12) of (33a) yields u¯(t)= B¯′x(t)+ Ju(t) (35) It follows from (18b) that (35) can be inverted to yield u(t)= B′x¯(t)+ J ′u¯(t), (36) where x¯(t)= P −1x(t + 1), and that we have the reverse-time recursion x¯(t − 1) = A′x¯(t)+ B¯u¯(t). (37a) Then inserting (36) and x(t)= P x¯(t − 1) = P A′x¯(t)+ P B¯u¯(t) into (33b), we obtain y(t)= C¯x¯(t)+ D¯u¯(t), (37b) where D¯ := DJ ′ and C¯ := CP A′ + DB′. (38) Then, (37) is precisely what we wanted to establish. Moreover, the transfer functions

−1 W(z)= C(zIn − A) B + D (39) of (33) and

−1 ′ −1 W¯ (z)= C¯(z In − A ) B¯ + D¯ (40) of (34) satisfy W(z)= W¯ (z)U(z). (41) In the context of stochastic realization theory, discussed next, U(z) is called structural function ([3], [4]). 7

u(t) y(t) ✲ W ✲

Fig. 3. The forward stochastic system (33).

y(t) u¯(t) ✛ W¯ ✛

Fig. 4. The backward stochastic system (34)

1) Time-reversal of stochastic realizations.: Given an m-dimensional stationary process y, consider a minimal stochastic realization (33), evolving forward in time, where now u is a normalized white noise process, i.e., ′ E{u(t)u(s) } = Ipδt−s. Since U, given by (13), is all-pass, u¯ is also a normalized white noise process, i.e., ′ E{u¯(t)¯u(s) } = Ipδt−s. From the reverse-time recursion (34a) ∞ x¯(t)= (A′)k−(t+1)B¯u¯(k). kX=t+1 Since, u¯ is a white noise process, E{x¯(t)¯u(s)′} = 0 for all s ≤ t. Consequently, (34) is a backward stochastic realization in the sense of stochastic realization theory.

B. Time-reversal of continuous-time stochastic systems We now turn to the continuous-time case. Let dx = Axdt + Bdu (42a) dy = Cxdt + Ddu (42b) be a stochastic system with x, u, A, B as in Section II-B, evolving forward in time from the remote past (t = −∞). All processes have stationary increments and x(t) is F u-measurable  y(t)  t R u for all t ∈ , where Ft is the σ-algebra generated by {u(s) | s ≤ t}. The all-pass extension of Section II-B yields du¯ = du − B¯′xdt (43) as well as the reverse-time relation dx¯ = −A′xdt¯ + Bd¯ u¯ (44a) du = B′xdt¯ + du,¯ (44b) 8 where x¯(t)= P −1x(t). Inserting (44b) into

dy = CP xdt¯ + Ddu yields dy = C¯xdt¯ + Ddu,¯ where C¯ = CP + DB′. (45)

Thus, the reverse-time system is

dx¯ = −A′xdt¯ + Bd¯ u¯ (46a) dy = C¯xdt¯ + Ddu.¯ (46b)

From this, we deduce that x¯(t) is F¯u¯-measurable  y(t)  t for all t ∈ R. We also note that the transfer function

−1 W(s)= C(sIn − A) B + D of (42) and the transfer function

′ −1 W¯ (s)= C¯(sIn + A ) B¯ + D of (46) also satisfy W(s)= W¯ (s)U(s) as in discrete-time. 1) Time-reversal of stochastic realizations.: In continuous-time stochastic realization theory, (42) is a forward minimal stochastic realization of an m-dimensional process y with stationary increments provided u is a normalized orthogonal-increment process satisfying

′ E{du(t)du(t) } = Ipdt.

Since U(s) is all-pass, du¯ = du − B¯′xdt (47) also defines a stationary orthogonal-increment process u¯ such that

′ E{du¯(t)du¯(t) } = Ipdt.

It remains to show that (46) is a backward stochastic realization, that is, at each time t the past increments of u¯ are orthogonal to x¯(t). But this follows from the fact that ∞ ′ x¯(t)= e−A (t−s)Bd¯ u¯(s) Zt and u¯ has orthogonal increments. 9

IV. CONCLUDING REMARKS

The direction of time in physical laws and the fact that physical laws are symmetric with respect to time have occupied some of the most prominent minds in science and ([14], [15], [16]). These early consideration were motivated by no less an issue than that of the very nature of the quantum. Indeed, Erwin Schr¨odinger’s aim appears to have been to draw a classical analog to his famous equation. A large body of work followed. In particular, closer to our immediate interests, dual time-reversed models have been employed to model, in different time-directions, Brownian or Schr¨odinger bridges (see [17], [18]), a subject which is related to reciprocal processes ([19], [20], [21], [22]). The topic of time reversibility has also been central to thermodynamics, and in recent years studies have sought to elucidate its relation to systems theory (see [23], [24]). Possible connections between this body of work and our present paper will be the subject of future work. The thesis of the present work is that under mild assumptions on a , any model that consists of a linear stable driven by an appropriate input process can be reversed in time. In fact, a reverse-time dual system along with the corresponding input process can be obtained via an all-pass extension of the state equation. The correspondence between the two input processes can be expressed in terms of each other by a causal and an anti-causal map, respectively. The formalism of our paper can easily be extended to a non-stationary setting at a price of increased notational, but not conceptual, complexity. Informally, and in order to underscore the point, if u(t) is a non-stationary process and the linear system is time-varying, under suitable conditions, a reverse-time system and a process u¯(t) can be similarly constructed via a time-varying orthogonal transformation.

REFERENCES

[1] A. Lindquist and G. Picci, “On the stochastic realization problem,” SIAM J. Control Optim., vol. 17, no. 3, pp. 365–389, 1979. [2] ——, “Forward and backward models for Gaussian processes with stationary increments,” Stochastics, vol. 15, no. 1, pp. 1–50, 1985. [3] ——, “Realization theory for multivariate stationary Gaussian processes,” SIAM J. Control Optim., vol. 23, no. 6, pp. 809–857, 1985. [4] ——, “A geometric approach to modelling and estimation of linear stochastic systems,” J. Math. Systems Estim. Control, vol. 1, no. 3, pp. 241–333, 1991. [5] M. Pavon, “Stochastic realization and invariant directions of the matrix Riccati equation,” SIAM Journal on Control and Optimization, vol. 18, no. 2, pp. 155–180, 1980. [6] A. Lindquist and M. Pavon, “On the structure of state-space models for discrete-time stochastic vector processes,” IEEE Trans. Automat. Control, vol. 29, no. 5, pp. 418–432, 1984. [7] G. Michaletzky, J. Bokor, and P. V´arlaki, Representability of stochastic systems. Budapest: Akad´emiai Kiad´o, 1998. [8] G. Michaletzky and A. Ferrante, “Splitting subspaces and acausal spectral factors,” J. Math. Systems Estim. Control, vol. 5, no. 3, pp. 1–26, 1995. [9] A. Lindquist, “A new algorithm for optimal filtering of discrete-time stationary processes,” SIAM J. Control, vol. 12, pp. 736–746, 1974. [10] F. Badawi, A. Lindquist, and M. Pavon, “On the Mayne-Fraser smoothing formula and stochastic realization theory for nonstationary linear stochastic systems,” in Decision and Control including the Symposium on Adaptive Processes, 1979 18th IEEE Conference on, vol. 18. IEEE, 1979, pp. 505–510. [11] F. A. Badawi, A. Lindquist, and M. Pavon, “A stochastic realization approach to the smoothing problem,” IEEE Trans. Automat. Control, vol. 24, no. 6, pp. 878–888, 1979. [12] A. Ferrante and G. Picci, “Minimal realization and dynamic properties of optimal smoothers,” Automatic Control, IEEE Transactions on, vol. 45, no. 11, pp. 2028–2046, 2000. [13] T. T. Georgiou, “The Carath´eodory–Fej´er–Pisarenko decomposition and its multivariable counterpart,” Automatic Control, IEEE Transactions on, vol. 52, no. 2, pp. 212–228, 2007. [14] E. Schr¨odinger, Uber¨ die Umkehrung der Naturgesetze. Akad. d. Wissenschaften, 1931. [15] A. Kolmogorov, Selected Works of AN Kolmogorov: Probability theory and mathematical statistics. Springer, 1992, vol. 26. [16] A. Shiryayev, “On the reversibility of the statistical laws of nature,” in Selected Works of AN Kolmogorov. Springer, 1992, pp. 209–215. [17] M. Pavon and A. Wakolbinger, “On free energy, stochastic control, and Schr¨odinger processes,” in Modeling, Estimation and Control of Systems with Uncertainty. Springer, 1991, pp. 334–348. [18] P. Dai Pra and M. Pavon, “On the Markov processes of Schr¨odinger, the Feynman-Kac formula and stochastic control,” in Realization and Modelling in System Theory. Springer, 1990, pp. 497–504. 10

[19] B. Jamison, “Reciprocal processes,” Probability Theory and Related Fields, vol. 30, no. 1, pp. 65–86, 1974. [20] A. Krener, “Reciprocal processes and the stochastic realization problem for acausal systems,” in Modelling, Identification and Robust Control, C. I. Byrnes and A. Lindquist, Eds. Amsterdam: North-Holland, 1986, pp. 197–211. [21] B. C. Levy, R. Frezza, and A. J. Krener, “Modeling and estimation of discrete-time gaussian reciprocal processes,” Automatic Control, IEEE Transactions on, vol. 35, no. 9, pp. 1013–1023, 1990. [22] P. Dai Pra, “A stochastic control approach to reciprocal diffusion processes,” Applied mathematics and Optimization, vol. 23, no. 1, pp. 313–329, 1991. [23] W. M. Haddad, V. Chellaboina, and S. G. Nersesov, “Time-reversal symmetry, poincar´erecurrence, irreversibility, and the entropic arrow of time: From mechanics to system thermodynamics,” Nonlinear Analysis: Real World Applications, vol. 9, no. 2, pp. 250–271, 2008. [24] ——, Thermodynamics: A dynamical systems approach. Princeton University Press, 2009.