Feedback Control and the Arrow of Time
aa r X i v : . [ m a t h . D S ] A p r Feedback Control and the Arrow of Time
Tryphon T. Georgiou and Malcolm C. Smith
Abstract
The purpose of this paper is to highlight the central role that the time asymmetry of stabilityplays in feedback control. We show that this provides a new perspective on the use of doubly-infiniteor semi-infinite time axes for signal spaces in control theory. We then focus on the implicationof this time asymmetry in modeling uncertainty, regulation and robust control. We point out thatmodeling uncertainty and the ease of control depend critically on the direction of time. We alsodiscuss the relationship of this control-based time-arrow with the well known arrows of time inphysics.
I. I
NTRODUCTION
The origin and implications of the “arrow of time” is one of the deepest and leastunderstood subjects of physics. The “arrow” is an intrinsic part of the world as we knowit. Yet its emergence in thermodynamics and cosmology, from physical laws which areapparently impervious to it, remains a controversial subject [29]. At first sight, this subjectmay seem unconnected with the theory of feedback control. However, starting from thevery basic fact that our notion of stability in the sense of Lyapunov is time-asymmetric, weargue that the “arrow of time” does have important implications on modeling and uncertainty,robustness of stability, as well as on the topology for the study of the dynamics of feedbackinterconnections.The circle of ideas that gave rise to this paper began in a short note published by theauthors thirteen years ago [8]. There, it was pointed out that the doubly-infinite time axispresents some “intrinsic difficulties” for developing a suitable input-output systems theory—difficulties that are not present in the semi-infinite time axis setting. These difficulties arenot mere mathematical technicalities. Rather, they relate fundamentally to the consistency ofthe theory of stabilizability across different frameworks. Subsequently, a number of paperswere written which shed light on the problem [22], [23], [24], [15], [16], [17]. The presentpaper takes a fresh look and traces the origin of the “puzzle” to the arrow of Lyapunovstability, and then, explores the relevance of this arrow to the topology of dynamical systemsand feedback theory.The relationship of the modern theory of dynamical systems with classical physics andthermodynamics is a developing one. A classical contribution by Nyquist and Johnson [28],[18] is a derivation of the electromotive force due to thermal agitation in conductors. In [4]the issue of irreversibility is treated from the point of view of stochastic control theory. More
This work was partially supported by the National Science Foundation. T.T. Georgiou is with Department of Electricaland Computer Engineering, University of Minnesota, Minneapolis, MN 55455; [email protected]. M.C. Smith is withthe Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, U.K.; [email protected] ecently [13] has sought to formalize classical thermodynamics in the mathematical languageof modern dynamical systems (see also [5]). In [27] information flow and entropy have beenstudied in the context of the Kalman filter. In [31] it is shown that a linear macroscopicdissipative system can be approximated by a linear lossless microscopic system over arbitrarylong time intervals. Our point of view here is influenced by [29] and is somewhat differentto the above references in that our main goal is to highlight a time-asymmetry, point out itsimplications, and discuss its relationship to other well-known asymmetries.The present paper begins by providing a new explanation of the issues raised in [8]with regard to an input-output theory for the doubly-infinite time axis. In Section III weintroduce the time-conjugation operator and discuss the implications of the time-arrow inoptimal control problems. In Section IV we analyse the effect of the time-arrow on modellinguncertainty; we show that dynamical systems which are close in the usual sense, that acommon controller can stabilise and give similar closed-loop responses for either, may notbe close when the time-arrow is reversed. Then, in Section V, we further illuminate theinherent time-asymmetry in our ability to control a dynamical system with two specificexamples. These can be thought of as examples of time irreversible feedback phenomena(see Section V-B). In Section VI we briefly discuss the arrow of time in physics and itsrelation to the time-arrow of feedback stability. Finally, in Section VII we consider feedbackloops with small time delays and discuss the contrasting effects of delays and predictorsand the connection with the arrow of time.II. T
IME - ASYMMETRY AND STABILITY
A. Input-output and Lyapunov stability
We focus on finite-dimensional linear dynamical systems which, for the most part, areassumed to be time-invariant. The dimensions of input, state and output (column) vectors, aswell as the consistent sizes of transformation matrices in state-space models, are suppressedfor notational simplicity. The following result is basic and well-known, cf. [38, p. 52-53],[12, p. 82].
Proposition 1:
Let P be a linear time-invariant finite-dimensional system which iscontrollable and observable and is specified by ˙ x = Ax + Bu, (1) y = Cx + Du, (2) with an initial condition x (0) = 0 . Then y ∈ L [0 , ∞ ) for all u ∈ L [0 , ∞ ) if and only ifthe matrix A is Hurwitz. Moreover, if this condition holds, y is determined uniquely by ˆ y ( s ) = ( C ( sI − A ) − B + D )ˆ u ( s ) , where ˆ denotes the Laplace transform. Many variants and extensions of the result are familiar: signal spaces with differentnorms can also be used; there is a finite-gain property relating the L -norms of y and u ;even with x (0) = 0 the main equivalence in the proposition still holds. Here we wouldlike to highlight the fact that the result establishes an equivalence between stability definedin terms of the forced response and stability defined in terms of the free response , i.e. anquivalence between bounded-input/bounded-output (BIBO) stability and Lyapunov stabilityfor a system operating on the positive time-axis. Asymptotic stability in the sense ofLyapunov is obviously a time-asymmetric concept since convergence of the state vectoris required as t tends to PLUS infinity, starting from an arbitrary initial condition at t = 0 .In itself, BIBO stability does not appear to have this asymmetry, yet it is implicit in theformulation of Proposition 1.To further illustrate the point we can write down the following obvious corollary ofProposition 1, obtained by running time backwards from to −∞ . By changing the supportof the signal spaces from the positive half-line to the negative half-line stability definedthrough the forced response (BIBO stability) becomes equivalent to asymptotic stability inthe sense of Lyapunov for the reversed time-direction as t tends to MINUS infinity. Proposition 2:
Let P be a linear system as in Proposition 1 with x (0) = 0 . Then y ∈ L ( −∞ , for all u ∈ L ( −∞ , if and only if the matrix − A is Hurwitz. We now turn to the situation where inputs and outputs may have support on the doubly-infinite time-axis. In this case the following holds, e.g. see [44, p. 101].
Proposition 3:
Let P be a linear system as in Proposition 1. Then there exists y ∈L ( −∞ , ∞ ) for all u ∈ L ( −∞ , ∞ ) if and only if A has no imaginary-axis eigenvalues.Moreover, if this condition holds, y is determined uniquely by ˆ y ( s ) = ( C ( sI − A ) − B + D )ˆ u ( s ) . We remark that Proposition 3 is the natural generalisation of Proposition 1 when systemsare viewed as operators. A linear system in Proposition 1 becomes a multiplication operatoron the Fourier transformed spaces. The operator is bounded if and only if the “symbol”(the transfer-function) belongs to H ∞ , which under the controllability and observabilityassumption is equivalent to A being Hurwitz. On the double-axis a multiplication operatoron the Fourier transformed spaces is bounded if and only if the symbol belongs to L ∞ —which for rational symbols excludes only poles on the imaginary axis.In Proposition 3 there is no longer any relationship between a notion of BIBO stabilityand Lyapunov stability (in either time-direction). Clearly, both A and − A may fail tobe Hurwitz. Since only the existence of some y ∈ L ( −∞ , ∞ ) is required for a given u ∈ L ( −∞ , ∞ ) , and the free motion solutions of (1) are ignored, this is not surprising.Propositions 1 and 2, by contrast, establish a connection between BIBO stability and Lya-punov stability as t → + ∞ (respectively, t → −∞ ) without putting in explicit requirementson the free motion solutions.We now consider the feedback interconnection in the form of Fig. 1 where P and C arelinear systems. The existence of signals u i , y j ( i, j ∈ { , } ) in L [0 , ∞ ) which satisfy thefeedback equations for a given pair of external inputs u , y in L [0 , ∞ ) , for a given set ofinitial conditions, is a well-known and natural definition of stability in terms of the forcedresponse. From Proposition 1 stability in this sense is equivalent to asymptotic stability inthe sense of Lyapunov of the combined state-space (assuming minimal realizations for P and C and well-posedness). Again, BIBO stability inherits the required time-asymmetryfrom the asymmetry of the support interval [0 , ∞ ) . u y u y y ✲ + ♠ Σ ✲ P ❄ − ✻ − C ✛ ♠ Σ + ✛ Fig. 1. Standard feedback configuration.
It is apparent that the corresponding definition of BIBO stability for this feedbackinterconnection with L ( −∞ , ∞ ) signals, generalising Proposition 3, will not correspond toa sensible notion of closed-loop stability. Indeed, we can easily check that a system P withtransfer funtion P ( s ) = 1 / ( s − is “stabilised” by any of the controllers with C ( s ) = 2 , C ( s ) = 0 , or C ( s ) = − . / ( s + 1) . (In conventional terms the controllers give closed-looppoles which are in the open left-half plane (LHP), the open right-half plane (RHP), and inboth half planes, respectively.)We can summarize the points so far as follows. Stability is a time-asymmetric concept—the requirement of an asymptotic property as t tends to PLUS infinity defines a timearrow. If stability is defined by requiring bounded outputs in response to bounded inputsthen a time arrow is not obviously implied. However, for signal spaces with support on apositive (resp. negative) half-line, the definition turns out to imply a positive (resp. negative)time arrow. On the other hand, a bounded-input bounded-output definition of stability forsignals with support on the doubly-infinite time-axis does not define a preferred time arrow.Stable systems defined by bounded “multiplication operators” may be stable in the sense ofLyapunov in the positive time-direction, in the negative time-direction or in neither direction. B. The two-sided time axis and causality
The fact that the doubly-infinite time axis causes problems for the analysis of stabilityand of stabilisation was pointed out in [8]. The explanation given there is consistent withthat of Section II, but the overall argument was somewhat different. We now summarize thereasoning of [8].Two systems P i ( i = 1 , ) defined by convolution operators were considered: y ( t ) = Z ∞−∞ h i ( t − τ ) u ( τ ) dτ = h i ∗ u where h ( t ) = e t for t ≥ and zero otherwise, and h ( t ) = − e t for t ≤ and zerootherwise, respectively. Each system has (double-sided Laplace) transfer function equal to / ( s − , but with differing regions of convergence. The first system is unstable and causaland the second is stable and non-causal (in fact anticausal) according to the usual definitions.When viewed on L ( −∞ , ∞ ) , P is a bounded operator and hence is a stable system inan input-output sense. On the other hand, it was shown in [8] that P fails to be stabilisablen L ( −∞ , ∞ ) . This is a counterintuitive result since P is stabilisable in the ordinary wayon any positive half-line. The proof that P fails to be stabilisable on the doubly-infinitetime-axis reduces to the observation that the graph of P fails to be closed.It was also pointed out in [8] that the closure of the graph of P coincides with thegraph of P . Once the graph is closed there appears to be no problem with stabilisation.But in closing the graph “anti-causal” trajectories are brought in which are inconsistent withthe convolution representation of the system, so this was considered inadmissible.Another possible remedy discussed in [8] was to consider the underlying differentialequation representations rather than the convolution representations. In fact both systemsare defined by the same differential equation ˙ y = y + u. (3)More precisely, the trajectories of both P and P satisfy this equation. In terms of “flowof time” thinking, P appears to arise by solving this equation forwards in time while P is obtained by solving it backwards. This suggestion seems to make stronger the argumentto consider P and P to be the same system. But this was considered unnatural in [8] onthe grounds that it appears to abandon any notion of causality, or that it leaves the directionof time undefined.The discussion of Section II allows the difficulties pointed out in [8] to be explainedin a new way. Let us suppose we are willing to accept the closure of the graph of P which makes it “stabilizable” on the double-axis in a bounded-input/bounded-output sense.As explained, P and P can now be thought of as one and the same system defined by(3)—a state-space description as in (1-2) solved forwards or backwards as desired. Does theclosure of the graph resolve the difficulty pointed out in [8]? The answer is no, since thenotion of stability does not correspond to the usual notions. As is made clear by Proposition3, the feedback system may turn out to be stable in a conventional sense in the forward,backward or neither time-directions. C. The work of M¨akil¨a, Partington and Jacob
A number of interesting observations and contributions have followed from [8] whichwe would like to comment on here.The fact that a causal system on the double-axis can have a non-causal closure has ledto a study of “closability” and “causal closability” as questions in their own right. M¨akil¨a[25] has shown that the lack of causal closability for the example of [8] extends to general L p spaces on the double-axis. Jacob and Partington [17] give general characterisations ofthe graphs of time-invariant systems and derive necessary and sufficient conditions for theclosure of a closable system to be causal. In [22] M¨akil¨a and Partington consider weighted L -spaces on the double-axis and show that, when signals have very rapid decrease to zerotowards −∞ , causal convolution operators may be closed operators. (So there is no issueof causality being lost due to the operation of closure.)On the question of stabilization on the double time-axis, Jacob [15] has made aninteresting suggestion. We have seen already that closing the graph and applying the BIBOtability definition fails to recover the usual concept of stability. Jacob proposed that causalityof the closed-loop operators of the feedback system be added as an extra requirement. Jacobshowed that the resulting characterisation of stability agrees with the usual definitions forlinear time-invariant systems. In the context of the present paper we can re-interpret thisresult by saying that the causality condition forces the positive time-arrow into feedbacksystem stability. We can understand this as follows. In [17] it is shown that a closed lineartime-invariant system is causal on L ( −∞ , ∞ ) if and only if the corresponding transferfunction belongs to a certain Smirnov class. For finite dimensional systems this is equivalentto the transfer function having no right half-plane poles. Thus, in Proposition 3, if P isrequired to be causal, BIBO stability agrees with Lyapunov stability with the positive time-arrow.M¨akil¨a and Partington in [22] make an interesting observation on the possible extensionof Jacob’s idea to the time-varying situation. They consider a causal, convolution operatorderived from the underlying differential equation ˙ y ( t ) + a ( t ) y ( t ) = u ( t ) (4)where a ( t ) = − for t ≤ and a ( t ) = +1 for t > and point out that the closure of the L ( −∞ , ∞ ) graph of the convolution system is not the graph of an operator. Essentially thisboils down to the fact that there are free motion solutions y ( t ) = ce −| t | , u ( t ) = 0 , where c is a constant, which can be approximated arbitrarily closely by elements of the graph. Thisraises the question of whether the approach of Jacob can recover a theory of stabilizationwhich is consistent with the single-axis case. At the same time it is pointed out that thesystem is stabilizable in a Lyapunov sense by the feedback u ( t ) = − y ( t ) . (5)In the present context this example highlights the care that is needed in defining stability fortime-varying systems, even in the conventional sense. The open-loop system (4) is Lyapunovstable in the forward time-direction for any initial condition specified at any time (eitherpositive or negative), but not uniformly so. Incidentally, the same is true for stability in thebackwards time-direction. With the feedback (5) in force the system becomes uniformlystable in the sense of Lyapunov in the forward time-direction and unstable in reverse. Inthe perspective of the present paper, any method to force agreement between BIBO stabilityon the double-axis and conventional notions (such as requiring causality of the closed loopoperators) might be seen as tantamount to directly imposing the desired time arrow withinthe stability definition.In several papers (e.g., [24], [22], [23]) M¨akil¨a and Partington have advocated the useof a two-operator model for systems on the doubly infinite time-axis in the form Ay = Bu ,where A, B are causal, bounded operators, in contrast to a single-operator model y = P u ,where P is causal and possibly unbounded. Closed-loop stability is defined as the existenceof a causal, bounded inverse of the feedback system operator mapping system inputs toexogenous disturbances. Since this definition incorporates a causality requirement on theclosed-loop system there is evidently a close relationship between this idea and the approachof Jacob.II. T IME -A SYMMETRY AND O PTIMAL R EGULATION
This section focusses on the time-asymmetry of the definition of stability and its im-plications in the context of optimal regulation. Firstly, a time-conjugation operator will bedefined as well as the concepts of f-stability and b-stability. Then the finite-horizon quadraticregulator problem will be considered for a system running forwards in time and backwardsin time, and it will be shown that the optimal cost is generally different. The infinite-horizon(asymptotic) regulator will also be considered in the same way. It will be shown that theoptimal cost can be expressed in terms of the two extremal solutions of the appropriatealgebraic Riccati equation. The result shows that ease of optimal regulation depends on thetime-direction.
A. The time-conjugation operator, f-stability and b-stability
Let P denote a dynamical system described by the state-space equations in (1-2),initialized at time zero and running forwards in time. Let J denote the operation on P which corresponds to solving (1-2) backwards from t = 0 followed by a flip of the timeaxis (so the new system runs forward again). More specifically we set t = − t , so that ddt = − ddt , and then replace t by t which results in − ˙ x = Ax + Bu, with x (0) = x y = Cx + Du for the system J ( P ) . The effect on the transfer function is as follows: if P has transferfunction P ( s ) , then J ( P ) has transfer function P ( − s ) .Define the system P to be f-stable if A is Hurwitz, and define P to be b-stable if J ( P ) is f-stable, or equivalently, if − A is Hurwitz. It is immediately obvious that a lineartime-invariant system of the type (1-2) can never be both f-stable and b-stable. Similarly, acontroller which makes (1-2) f-stable cannot make it b-stable as well. B. The finite-horizon linear quadratic regulator
Let P be a linear time-invariant system which is controllable and observable and de-scribed by (1-2), as before, with D = 0 and x (0) = x . Consider the problem of theregulation of P with criterion J = Z T ( y ( t ) ′ Qy ( t ) + u ( t ) ′ Ru ( t )) dt + x ( T ) ′ Hx ( T ) . This has solution u ( t ) = − R − B ′ S ( t ) x ( t ) (6)where − ˙ S ( t ) = S ( t ) A + A ′ S ( t ) − S ( t ) BR − B ′ S ( t ) + C ′ QC (7)nd S ( T ) = H , with optimal cost J f , T = x ′ S (0) x [1]. When P runs backwards in timefrom x (0) = x with cost J = Z − T ( y ( t ) ′ Qy ( t ) + u ( t ) ′ Ru ( t )) dt + x ( − T ) ′ Hx ( − T ) we can check that the optimal control is still given by (6) where S ( t ) satisfies (7) with S ( − T ) = − H , and that the optimal cost is J b,T = − x ′ S (0) x . It can be readily verifiedthat S (0) (forward case) is in general different from − S (0) (backward case), and so theoptimal cost is different in the two cases, e.g., if A = B = C = Q = R = T = 1 and H = 10 , then S (0) = 2 . in the forward case and − S (0) = 0 . in the backwardcase. C. The infinite-horizon linear-quadratic regulator
Again let P be a linear time-invariant system which is controllable and observable anddescribed by (1-2) with D = 0 and x (0) = x . It is well-known [1] that J = Z ∞ ( y ( t ) ′ Qy ( t ) + u ( t ) ′ Ru ( t )) dt (8)has a minimum given by J f , ∞ = x ′ S + x where S + is the unique positive-definite solutionto the algebraic Riccati equation A ′ S + SA − SBR − B ′ S + C ′ QC = 0 . (9)It is also well-known that S + is the unique solution of (9) for which A − BR − B ′ S hasall its eigenvalues in the open LHP. In the language of the present paper we can say that S + is the unique solution of (9) which makes the system (1-2) f-stable with the controller u = − R − B ′ Sx .What happens if we require the minimisation of J = Z −∞ ( y ( t ) ′ Qy ( t ) + u ( t ) ′ Ru ( t )) dt (10)for (1-2) running backwards in time? This is the same as the conventional problem for thesystem J ( P ) . It is easy to see that the minimum is given by J b , ∞ = − x ′ S − x where S − is the unique negative-definite solution to (9). It is also well-known that S − is the uniquesolution of (9) for which A − BR − B ′ S has all its eigenvalues in the open RHP [44]. Inthe language of the present paper we can say that S − is the unique solution of (9) whichmakes the system (1-2) b-stable with the controller u = − R − B ′ Sx .In general J f , ∞ = x ′ S + x and J b , ∞ = − x ′ S − x are different. This shows that “difficultyof control” is time-asymmetric for the standard linear-quadratic regulator on the infinitehorizon. The difference can be significant, e.g. if A = 1 , B = ǫ , C = 1 , Q = 1 and R = 1 then S + = 2 /ǫ + 1 / O ( ǫ ) and S − = − / O ( ǫ ) for ǫ small.V. T IME -A SYMMETRY AND M ODELLING U NCERTAINTY
In this section we look at the topology for uncertainty in feedback control and how thisis affected by the time arrow. We will see that dynamical systems which are close in theusual sense, that a common controller can stabilise them and give a similar closed-loopbehaviour, may not be close if time is reversed.
A. The gap metric and robustness of stability
Zames and El-Sakkary [43] introduced a metric on dynamical systems for the purpose ofassessing robustness. This was based on the gap metric used in functional analysis to studyinvertibility of operators [19], [32]. Specifically, systems are considered to be operators on L [0 , ∞ ) with a graph which is a closed subspace of L [0 , ∞ ) . Consider two linear systems P i ( i = 1 , ) with transfer functions P i ( s ) = n i ( s ) ( m i ( s )) − where n i ( s ) and m i ( s ) are coprime polynomials or, more generally, right-coprime polyno-mial matrices. Let ( n i ( − s )) T n i ( s ) + ( m i ( − s )) T m i ( s ) = ( d i ( − s )) T d i ( s ) with det( d i ( s )) a Hurwitz polynomial and ( ) T representing matrix transpose—the exis-tence of such a polynomial (matrix) d i ( s ) is a standard result in the theory of canonicalfactorization [41]. Then, G P i , H := m i ( s )( d i ( s )) − n i ( s )( d i ( s )) − H := G i ( s ) H is (the Fourier transform of) the graph of P i , for i = 1 , . Thus, the graph symbol G i ( s ) generates the graph of P i as its range. Then the gap between P and P is defined to be δ H ( P , P ) := k Π G P , H − Π G P , H k where Π K denotes orthogonal projection onto a closedsubspace K .Let the feedback configuration of Fig. 1 be denoted by [ P , C ] , where P and C are linearsystems defined as operators on L [0 , ∞ ) which may possibly be unbounded. Define H P , C := (cid:18) IP (cid:19) ( I − PC ) − (cid:0) I − C (cid:1) to be the operator mapping (cid:0) u T y T (cid:1) T to (cid:0) u T y T (cid:1) T . The following are basic robustnessresults for gap metric uncertainty. Proposition 4: [9]
Assume that the closed-loop system [ P , C ] is f-stable. Then, [ P , C ] is f-stable for all P such that δ H ( P , P ) ≤ b if and only if b < b P , C where b P , C := k H P , C k − ∞ . roposition 5: [43] Assume that the closed-loop system [ P , C ] is f-stable. Then,the following are equivalent: (i) δ H ( P n , P ) → as n → ∞ . (ii) H P n , C is f-stable for sufficiently large n and k H P n , C − H P , C k ∞ → as n → ∞ . Proposition 5 was the primary justification for the claim in [43] that the gap metricdefines the “correct” topology for robustness of feedback systems. In the present context,it can be seen that the choice of a signal space with support on the positive half-line isessential in achieving an appropriate topology. To emphasize the point, if L [0 , ∞ ) werereplaced by L ( −∞ , then the above proposition would hold with f-stability replaced byb-stability.Let us consider the case where systems are defined on L ( −∞ , ∞ ) . Then we define δ L ( P , P ) := k Π G P , L − Π G P , L k where G P i , L := G i ( s ) L and L := L ( − j ∞ , j ∞ ) . With this definition, G P , L is always closed, but may contain“non-causal” input-output pairs (as pointed out in [8]—see also Section II-B). It is easy toconstruct examples to demonstrate that convergence of δ L ( P n , P ) to zero does not allowany closed-loop stability prediction, e.g., [ P , C ] f-stable does not imply [ P n , C ] f-stable forsufficiently large n .In [36] Vinnicombe introduced a new metric δ v ( · , · ) on dynamical systems which definesthe same topology as δ H ( · , · ) , and which satisfies the following inequality: δ L ( · , · ) ≤ δ v ( · , · ) ≤ δ H ( · , · ) . The v-gap between P and P is defined as follows: δ v ( P , P ) := δ L ( P , P ) if wno(det( G ( − s ) T G ( s ))) = 0 , otherwise, (11)where wno( g ( s )) denotes the winding number about the origin of g ( s ) , as s traces thestandard Nyquist D-contour [36], [37]. A simple expression for δ L ( · , · ) can be obtained usingleft fractional representations—let P i ( s ) = ( ˜ m i ( s )) − ˜ n i ( s ) be a left-coprime polynomialfraction, ˜ d i the Hurwitz polynomial matrices which satisfy ˜ n i ( s ) (˜ n i ( − s )) T + ˜ m i ( s ) ( ˜ m i ( − s )) T = ˜ d i ( s )( ˜ d i ( − s )) T , and define ˜ G i ( s ) := (cid:0) − ( ˜ d i ( s )) − ˜ n i ( s ) , ( ˜ d i ( s )) − ˜ m i ( s ) (cid:1) for i = 1 , . The graph of P i is the kernel of multiplication by ˜ G i ( s ) (in the respectivespace of signals H or L ). The L -gap can now be expressed as δ L ( P , P ) := k ˜ G ( s ) G ( s ) k ∞ . t turns out that Propositions 4 and 5 both hold with δ H replaced by δ v (see [36]). Since δ L = δ v when wno(det( G ( − s ) T G ( s ))) = 0 (12)holds, this condition effectively imposes a positive time-arrow on the double-axis graphwhich forces f-stability to be retained under small perturbations in δ v ( · , · ) . This is illustratedby the following result (which can be readily derived from [36, Theorem 4.2]; see also [10]). Proposition 6:
Let [ P , C ] be f-stable and suppose δ L ( P n , P ) → as n → ∞ . Then [ P n , C ] is f-stable for all sufficiently large n if and only if wno(det( G n ( − s ) T G ( s ))) = 0 for all sufficiently large n . B. The effect of the time-arrow on gap distances
We define a forward and a backward v-gap as follows, δ v,f ( P , P ) := δ v ( P , P ) δ v,b ( P , P ) := δ v ( J ( P ) , J ( P )) . It is straightforward to see that δ L ( P , P ) = δ L ( J ( P ) , J ( P )) , so any difference between δ v,f ( P , P ) and δ v,b ( P , P ) lies in the winding number conditionin (11). Let us examine this more closely. Note that det( G ( − s ) T G ( s )) = h ( s )det( d ( − s )) det( d ( s )) where h ( s ) := det( m ( − s ) T m ( s ) + n ( − s ) T n ( s )) . (13)If δ L ( P , P ) < then it can be shown that wno(det( G ( − s ) T G ( s ))) is well-defined [36],in which case h ( s ) admits a canonical factorization h ( s ) = h + ( s ) h − ( s ) (14)where h + ( s ) and h − ( − s ) are Hurwitz polynomials. Thus, wno(det( G ( − s ) T G ( s ))) = 0 ifand only if deg( h + ( s )) = deg(det( d ( s ))) , or equivalently deg( h − ( s )) = deg(det( d ( s ))) , It can be shown that the degree of det( ˆ d i ( s )) is equal to the McMillan degree of P i (e.g.using the uniqueness of normalised coprime factors over H ∞ up to a constant unitarytransformation and the corresponding state-space realisations [26], [35], [44]). Determiningthe graph symbol for J ( P i ) requires a canonical factorization ( n i ( s )) T n i ( − s ) + ( m i ( s )) T m i ( − s ) = ( ˆ d i ( − s )) T ˆ d i ( s ) ith det( ˆ d i ( s )) a Hurwitz polynomial. Again it can be shown that the degree of det( ˆ d i ( s )) is equal to the McMillan degree of P i . The corresponding winding number condition in δ v,b ( P , P ) can now be expressed as wno(det(( ˆ d ( − s ) − ) h ( − s )( ˆ d ( s ) − ))) = 0 which is equivalent to deg( h − ( s )) being equal to the McMillan degree of P . We thereforeobtain the following result. Proposition 7:
Let P i ( s ) ( i = 1 , ) be the rational transfer functions of linear time-invariant dynamical systems as above, with McMillan degrees µ i , and with h, h + , h − as in (13-14). Assume that δ L ( P , P ) < . The following are equivalent: a) δ v,f ( P , P ) < , b) deg( h + ( s )) = µ , c) deg( h − ( s )) = µ . The following are equivalent: a) δ v,b ( P , P ) < , b) deg( h − ( s )) = µ , c) deg( h + ( s )) = µ . The following are equivalent: a) δ v,f ( P , P ) = δ v,b ( P , P ) < , b) µ = µ = deg( h + ( s )) = deg( h − ( s )) . In the above proposition, expresses the zero winding number condition in (11) inan equivalent form, while does the same for δ v,b ( P , P ) . It is interesting that whenthe two conditions are combined as in the result is a very stringent requirement whichincludes the necessity that P ( s ) and P ( s ) have the same McMillan degree. This serves tohighlight the fact that “unmodelled dynamics” which may account for a small error in δ v,f (and which may be neglected in the design of a robust controller) will inevitably accountfor a substantial error in δ v,b . Example 8:
Consider two systems with different McMillan degrees, e.g. P ( s ) = 1 , P ( s ) = 1 /s . It can be computed that δ v,f ( P , P ) = 1 / √ . Proposition 7 then tells usimmediately that δ v,b ( P , P ) = 1 where P ( s ) = − /s . Similarly, if P ( s ) = 1 /s , thenProposition 7 tells us that both δ v,f ( P , P ) = δ v,b ( P , P ) = 1 since P ( s ) = P ( − s ) .V. T IME - ASYMMETRY AND ROBUST CONTROL
This section addresses the implications of the time-asymmetry in the theory of robustcontrol. In particular, we will also see that a system which is “easy” to control in onedirection of time may be far from easy to control in the opposite direction. . Optimal robustness and difficulty of control
In [11] it was shown that b P , C could be maximised over all stabilising C and that thisamounts to solving a Nehari problem [44]. This optimum value, which we denote by b opt , f ( P ) , can be interpreted as a measure of ease/difficulty of control, where a value near to meansthe plant is “easy to control” and a value near means the plant is “hard to control”.With the understanding that b opt , f ( P ) has the meaning of “ease of control” with respectto the forward time-arrow for stabilty, it is interesting to define b opt , b ( P ) := b opt , f ( J ( P )) , which represents “ease of control” with respect to the backwards time-arrow. Our mainpurpose in defining b opt , b ( P ) is to highlight the influence of the time-arrow in feedbackregulation.Let P be a controllable and observable system which is described by the state-spaceequations in (1-2), as before. Then, following [11], [9], b opt , f ( P ) = p − λ max ( Y + X + ) where Y + is the positive definite solution of the Riccati equation A Y + Y A ∗ − Y CR − C ∗ Y + B ( I − D ∗ R − D ∗ ) B ∗ = 0 (15)where A = A − BD ∗ R − C and R = I + DD ∗ , and X + is the corresponding solution tothe ( Y -dependent) Lyapunov equation ( A − Y C ∗ R − C ) ∗ X + X ( A − Y C ∗ R − C )+ C ∗ R − C = 0 (16)for Y = Y + . Similarly, it can be seen that b opt , b ( P ) = p − λ max ( Y − X − ) (17)where Y − is the negative definite solution of the Riccati equation (15) while X − is thecorresponding solution to (16) for Y = Y − .In the following two examples we will see situations where b opt , f ( · ) and b opt , b ( · ) arevery different. Example 9: Near pole-zero cancellations.
Consider P ( s ) = 1 + ǫs +1 . Letting − A = C = D = 1 and B = ǫ in equations (15)-(17)gives b opt , f ( P ) = s − p ǫ + ǫ / − − ǫ/ p ǫ + ǫ / . t follows that for small values of ǫ , b opt , f ( P ) = 1 − ǫ + O ( ǫ ) and hence, b opt , f ( P ) → as ǫ → . On the other hand, b opt , b ( P ) = s − p ǫ + ǫ / ǫ/ p ǫ + ǫ / . which leads to b opt , b ( P ) = 14 | ǫ | + O ( ǫ ) for small values of ǫ , and hence b opt , b ( P ) → as ǫ → . This is accounted for by the factthat P ( s ) has a near pole-zero cancellation in the LHP, which is innocuous for f-stabilisation,but highly challenging for b-stabilisation. The latter is equivalent to f-stabilisation of P ( − s ) ,which has a troublesome near pole-zero cancellation in the RHP. Example 10: Riding Bicycles.
A feedback stability problem in everyday experience is bicycle riding. An elementarymodel to study rider-bicycle stability is given in [2] which gives the following transferfunction from steering angle input to tilt angle: αV s + βVs − γ (18)where α, β, γ are positive constants and V is the forward speed. This model has one RHPpole, but the zero is in the LHP. As such, this plant is not too difficult to control.Let us consider what happens if we try to ride the bicycle backwards in time . Thiscorresponds to trying to stabilise the plant P ( − s ) forwards in time . The model still has oneRHP pole, but the zero is also in the RHP, which makes stabilisation much more difficult.Indeed if V β = √ γ the plant is technically not stabilisable. It is interesting to note that anexperimental bicycle with the steered wheel at the rear instead of the front has a transferfunction from steering angle input to tilt angle given by [2] (see also [21]) αV − s + βVs − γ . (19)This is exactly the transfer function for the conventional bicycle ridden backwards in time .Figure 2 shows the value of b opt , f and b opt , b versus V with parameter values α = 1 / , β = 2 and γ = 9 (which are deemed reasonably realistic). Recall that b opt , b is the same as b opt , f for the rear-wheel steered bicycle model (19) at the same V . It can be observed that b opt , b is less than b opt , f for any V . Also, b opt , b is very small for low V , indicating difficultyof control, and zero at V = 1 . m/s. For larger V , b opt , b increases, indicating that controlbecomes easier. These results are equivalent to the rear-wheeled steered bicycle being moredifficult to ride than the front-wheel steered one, but still being reasonably controllable athigher speeds [2]. opt,f vs. Vb opt,b vs. V PSfrag replacements V in m/s b o p t Fig. 2. b opt , f and b opt , b versus V for the bicycle model of (18) with α = 1 / , β = 2 , and γ = 9 . B. Time irreversible feedback phenomena
The concept of ease or difficulty of control gives a thought-provoking perspective onreversibility. Systems which in a limiting situation are very difficult to control (in the sensethat b opt , f ( P ) tends to zero) are unlikely to be observed in nature or technology. Nevertheless,such a system may be easy to control in the time-reversed direction (see Examples 9 and10). This is independent of the fact that the underlying differential equation can be integratedequally well in either time-direction. This is reminiscent of phenomena (such as a bottlefalling from the table and shattering into many pieces) that appear to be associated withan intrinsic direction of time even though classical physics would also allow the reversedmotion as a solution (see Section VI for a further discussion).We expand this point in the context of Example 10. The loss of stabilizability of the rear-wheeled steered bicycle at V = p γ/β has the following interesting consequence. Imaginea video of a rear-wheeled steered bicycle being ridden stably at this critical speed. Let usassume that it is possible to verify from the video the actual speed (e.g., by knowing theframe-rate and observing markings on the ground). An observer with a good grounding incontrol theory would be led to the inescapable conclusion that the video had been madewhen the said bicycle was actually being ridden backwards in space (i.e., with a negative V ) and then played backwards in time as well, giving the impression of a forward motion.VI. T HE ARROW OF TIME IN PHYSICS
The subject of the “arrow of time” is a well-known conundrum in physics. The secondlaw of thermodynamics states that the entropy of a system increases with time. It is thetime-asymmetry in this law which gives rise to the notion of the “thermodynamic arrow oftime”. The classical derivation of the second law in statistical mechanics due to Boltzmann isconnected with a famous puzzle known as Loschmidt’s paradox [40]. This essentially pointsout that the laws of mechanics used in the derivation of the second law are time-symmetric whereas the conclusion is not. Evidently the time-asymmetry creeps in through the statisticalssumptions. An illuminating discussion of this issue is given in [29]. Other arrows of timehave also been defined, for example (i) the “psychological arrow”—the direction in whichtime passes as perceived by a sentient being [14], [33], (ii) the “cosmological arrow”—the direction of time in which the universe is expanding. Hawking [14] argues that thethermodynamic and psychological arrows are always aligned with each other but these neednot always be aligned with the cosmological arrow (though they are at present).In this paper we have described the time-asymmetry in the definition of control systemsstability as a time-arrow. In the theory of dynamical systems there is also the notion ofpassivity, which again defines a time-arrow. For electrical circuits the time-arrow of passivitycan be seen in the behaviour of the resistor, in contrast to the inductor and capacitor whichare time-symmetric in their operation. If the electrical resistor were to operate backwardsin time one would observe a resistor gathering low-grade heat from the environment andcharging up a battery. This behaviour would be recognised as a violation of the secondlaw of thermodynamics (see [20, pages 260, 390-2]). In a similar way, an ideal lineardamper operating backwards in time extracts low-grade heat from the environment to createmechanical work, in violation of the second law. It seems that the arrow of time in passivesystems or circuits coincides with, or is the same as, the thermodynamic arrow.How does the arrow of time for control system stability relate to other time arrows? Itis highly unlikely that a control engineer who is designing a control system for a plant willgive even a moment’s thought to the preferred time arrow for control. Without expressingthe thought, the designer will seek decaying free motion solutions in the direction in whichtime is perceived to be passing. In this way the arrow of time for control could be saidto coincide with the psychological arrow. On the other hand, in biological systems, activecontrol is ubiquitous. It is less obvious that, for example, homeostasis in a cell is alignedwith the psychological arrow. Here we will be content to raise the question of whether thestability arrow for control systems in general can be directly related to the thermodynamicarrow, e.g. by considering information flow or the effect of internal energy sources.Finally, from a purely mathematical point of view, we observe that the arrow of time forcontrol systems stability appears identical with the arrow of time for passivity. This supportsthe conclusion that the arrow of time for control systems stability always coincides with thethermodynamic (and psychological) arrow.VII. F
EEDBACK LOOPS AND TIME DELAYS
Let D τ : x ( t ) x ( t − τ ) denote a time delay operator. It seems superfluous tosay that D τ is physically realisable for τ > . Indeed the delay is a common feature ofcommunication and control systems. For τ < , D τ is the ideal predictor which is notbelieved to be physically realisable as a “real-time” device. At first sight this “fact” appearsto be self-evident, but its subtlety is revealed on closer examination—indeed, a rigorousjustification appears not to be available at present. An insightful discussion of the issue of“causation” and its connection with the arrow of time is given in Price [29, Chapter 6]. Price’ssuggestion that the asymmetry of causation “is a projection of our own temporal asymmetryas agents in the world” [29, page 264] is similar to the view expressed by Bertrand Russell:“The law of causality, I believe, like much that passes muster among philosophers, is a relicf a bygone age, surviving, like the monarchy, only because it is erroneously supposed to dono harm”. This prevalence in physics and philosophy of an anthropocentric explanation ofcausation sits in opposition to the belief of the unrealizability of a “prediction machine” outof physical components and processes, and suggests that a deeper analysis of the questionis needed.In this paper we will not attempt to further debate the origin and explanation of causation.In the next section we will simply highlight the striking difference in behaviour of feedbackloops with small delays versus predictors and confirm the difference using the forward-timegap metric. A. Feedback stability, delays and predictors
Consider a feedback system which consists of an integrator in series with a time delayand negative unity feedback. The governing equation is ˙ x ( t ) + x ( t − τ ) = d ( t ) (20)where d ( t ) denotes an external disturbance. We set d ( t ) ≡ and consider the totality ofall free motion solutions of the system equations. If all solutions decay as t → + ∞ wesay the system is f-stable. This definition agrees with the one given in Section III-A forfinite-dimensional systems.For τ ≥ we can verify that (20) is f-stable. Taking Laplace transforms in (20) gives ˆ x ( s ) = 1 s + e − sτ ˆ d ( s ) . We can verify that all zeros of s + e − sτ = 0 are in the LHP so the system is f-stable.Now consider the case where τ < . Note that this corresponds to an integrator with apredictor in negative feedback, which we would not expect to be realizable in the forwardtime direction. In fact, s + e − sτ has infinitely many zeros in the RHP for any τ < andhence the system fails to be f-stable. It is evident that the system displays a discontinuityin the asymptotic (as t → ∞ ) behaviour of the free motion at the point τ = 0 .Let us now consider the closeness of the systems involved using the v-gap metric. Let P denote the integrator and P τ denote the integrator in series with D τ . Regarding these asoperators on L [0 , ∞ ) we have the graph: G P τ , H = (cid:18) ss +1 e − sτ s +1 (cid:19) H for τ ≥ . Then δ L ( P , P τ ) = k s ( s + 1) (1 − e − sτ ) k ∞ which tends to zero as τ → . Also G ( − s ) T G ( s ) = − s + e sτ − s + 1 o providing | τ | < π there are no crossings of the negative real axis of this function when s = jω . Hence, wno( G ( − s ) T G ( s )) = 0 , for τ sufficiently small. This implies δ v,f ( P , P τ ) = δ L ( P , P τ ) for τ ≥ and sufficiently small and δ v,f ( P , P τ ) → as τ → .Now consider the case of P τ with τ < . Again regarding P τ as an operator on L [0 , ∞ ) we have: G P τ = (cid:18) se sτ s +11 s +1 (cid:19) H and δ L ( P , P τ ) = k s ( s +1) ( e sτ − k ∞ which tends to zero as τ → . Also, G ( − s ) T G ( s ) = − s e − sτ + 1 − s + 1 , which behaves like e − sτ for large s , so the winding number of this function is not zero and δ v,f ( P , P τ ) = 1 for τ < .The above analysis with the gap agrees with the earlier conclusion on f-stability. For τ ≥ , f-stability was retained for sufficiently small τ , but lost for any τ < . Now we haveseen that, as long as τ ≥ , there is a small error in δ v,f , but for any τ < , δ v,f ( P , P τ ) = 1 .Finally, it is interesting to mention that the tolerance of feedback loops to small time-delays is guaranteed by a well-known sufficient condition that the high-frequency loop-gainof the feedback loop is smaller than one ([3], [6], [39], [42])—a condition routinely met inpractice. It is easy to check that robustness to an arbitrarily small “parasitic predictor” in theloop would be guaranteed theoretically by the loop-gain being greater than one at arbitrarilyhigh frequencies—a condition that appears impossible to achieve in a real feedback system.VIII. S YNOPSIS
1) Stability is a time-asymmetric concept. The requirement of an asymptotic property as t tends to PLUS infinity defines a time arrow.2) A stability definition which requires bounded outputs in response to bounded inputsdoes not obviously imply a time arrow. For signal spaces with support on a positive(resp. negative) half-line, the definition turns out to imply a positive (resp. negative)time arrow.3) A bounded-input bounded-output definition of stability for signals with support onthe doubly-infinite time-axis does not define a preferred time arrow. Stable systemsdefined by bounded multiplication operators may be stable in the sense of Lyapunovin the positive time direction, in the negative time direction or in neither direction.4) The fact that the closure of the graph of an unstable causal system may coincide withthe graph of a stable anti-causal system on the doubly-infinite time-axis need not bea fundamental obstacle in developing a usable control theory on the doubly-infinitetime-axis.5) Any method which modifies the BIBO definition of stability on the doubly-infinitetime-axis to agree with conventional stability notions could be interpreted as theimposition of a positive time-arrow.) A time-conjugation operator on systems was defined as well as the concepts of f-stability and b-stability.7) Both the finite-horizon and infinite-horizon quadratic regulators give a different optimalcost for a system running forwards in time and backwards in time. In the infinitehorizon case the optimal cost can be expressed in terms of the two extremal solutionsof the appropriate algebraic Riccati equation.8) The role of the positive time arrow in the gap metric measure of uncertainty fordynamical systems was highlighted. The usual H -gap metric inherits the positivetime arrow by virtue of systems being defined as operators on the positive half-line.The L -gap metric, which is well known to define an inappropriate topology for robustcontrol, does not have a preferred time-direction due to the underlying operators beingdefined on the double-axis. The v-gap metric may be interpreted as the L -gap withan imposed time-arrow.9) A time-conjugated v-gap metric was defined to measure closeness for robust b-stabilisation.It was seen that closeness of systems in the forward and backwards directions is astrong condition which includes the requirement of equal McMillan degrees.10) It was seen that ease or difficulty of control as measured by optimal robustness in thegap metric is a property that depends on the time-arrow.11) The situation of a plant which is easy to control in one time-direction but impossibleto control in the other shows that irreversibility can be intimately related to control.12) An engineering perspective of control suggests a close link between the control systemstability arrow and the psychological arrow. Unified mathematical frameworks forpassive circuits and feedback control suggest a close link between the control systemstability arrow and the thermodynamic arrow. The question was raised whether thestability arrow for control systems can be directly related to the thermodynamic arrow.13) The issue of the non-realizability of the pure predictor as a “real-time” device andthe connection with the arrow of time was highlighted as well as the difficulty of es-tablishing non-realizability rigorously. The strongly contrasting behaviour of feedbackloops in the presence of arbitrarily small time-delays or predictors was pointed out.IX. A CKNOWLEDGEMENT
We are grateful to Jan Willems for helpful comments on an earlier draft.R
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