Contraction and Observer Design on Cones

Transcription

1 Contraction an Observer Design on Cones Silvère Bonnabel, Alessanro Astolfi, an Roolphe Sepulchre Abstract We consier the problem of positive observer esign for positive systems efine on soli cones in Banach spaces. The esign is base on the Hilbert metric an convergence properties are analyze in the light of the Birkhoff theorem. Two main applications are iscusse : positive observers for systems efine in the positive orthant, an positive observers on the cone of positive semi-efinite matrices with a view on quantum systems. I. INTRODUCTION Positive systems arise in several areas, where the state variables represent quantities that o not have a physical meaning when they are negative see, for example, [8], [3], [7]. In the last ten years or so, the esign of positive observers for linear positive systems, i.e. observers such that the estimate state respects positivity, has attracte an ever growing attention. Inee, the requirement that the observer estimates be positive at any time seems esirable since it allows a physical interpretation of the estimation. Positive observers have been stuie for classes of linear systems an uner specific structural assumptions. In [] structural properties, incluing observability, of positive systems have been stuie, whereas observers for compartmental systems have been evelope []. In [] the positive observer esign problem has been ealt with using coorinates transformations an the theory of positive realization [], [], thus generalizing the results in [] an relaxing the conitions uner which positive observers exist. For a class of time-varying non-linear (or linear) systems, we avocate the fact that the use of special polar coorinates (a vector is parameterize by its norm, an an element of the unit sphere) as well as the use of a special metric on the unit sphere (more exactly the projective space), the Hilbert projective metric, simplifies the esign of observers an the convergence analysis. The key component of our approach is a theorem by Birkhoff, 97, that characterizes a class of positive mappings that are contractions in the projective space for the Hilbert metric. For those systems, a mere copy of the system provies a simple positive observer which converges exponentially in the projective space (at least). Our S. Bonnabel is with Centre e Robotique, MINES ParisTech, 77 Paris, France silvere.bonnabel@mines-paristech.fr A. Astolfi is with the Dept. of Electrical an Electronic Engineering, Imperial College Lonon, SW7 AZ, Lonon, UK an DISP, University of Roma Tor Vergata, 33 Rome, Italy a.astolfi@ic.ac.uk R. Sepulchre is with the Dept. of Electrical an Computer Engineering (Montefiore Institute, B8), University of Liège, Liège, Belgium. r.sepulchre@ulg.ac.be. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, an Optimization), fune by the Interuniversity Attraction Poles Programme, initiate by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors. approach can thus be relate to the one introuce by [9] where the convergence analysis of observers is base on the special choice of a contractive metric. The approach leas to potentially powerful alternatives to the usual methos, which are generally concerne with the conitions uner which a Luenberger observer is a positive system itself. More generally this paper eals with the general question of the esign of observers on soli cones in Banach spaces, the positive orthant being a particular case. The estimates are require to remain in the cone at any time. We will prove that the Hilbert projective an the Birkhoff Theorem allow to esign meaningful caniate observers with guarantee convergence properties for a large class of systems. The approach is vali in finite an infinite imension, an opens the way to the esign of simple observers which respect the unerlying structure of the problem, on several other cones than the positive orthant. In particular we apply the technique to the cone of positive semi-efinite matrices. The paper is organize as follows: in Section, the Hilbert metric an Birkhoff theorem are recalle. In Section 3, a class of observers on soli cones is propose. In Section, the results are applie to the positive orthant. In particular we introuce a system for which it is not possible to buil a linear convergent positive observer an for which we present a non-linear convergent observer. In Section we iscuss the esign of observers on the cone of positive semi-efinite matrices. II. BIRKHOFF-BUSHELL THEOREM In this section, we recall basic results presente in []. A soli cone K efine on a Banach space X, is a subset of X which is such that ) K, the interior of K, is not empty, ) K +K K, 3) λk K for all λ, an ) { K} K = {}. On such a space, a partial orer can be efine by the following relation : x y iff y x K. Let us now efine two important quantities. Let x,y K\{}. We let M(x/y) = inf{λ R + x λy} m(x/y) = sup{µ R + µy x} an M(x/y) = + if the set is empty. The notation is justifie by the fact that we always have m(x/y) y x M(x/y) y. Definition : The Hilbert projective metric in K\{} is efine by (x, y) = log(m(x/y)/m(x/y)). The metric can be calle projective, as we have (λx,µy) = (x,y) for all λ,µ >. Let A : K K be mapping:

2 A is sai positive if A : K K. A is monotone increasing if x y implies Ax Ay. A is sai to be homogeneous of egreep if for all λ > an x K we have A(λx) = λ p A(x). The projective iameter of a positive mapping A is efine by (A) = sup {(Ax,Ay) x,y K}. The contraction ratio is k(a) = inf{λ R + (Ax,Ay) λ(x,y)}. The Birkhoff theorem allows to characterize a class of systems that are contractions for the Hilbert metric. Theorem : [Birkhoff (97)] Let A be a monotone increasing mapping which is homogeneous of egree p in K. Then for all x,y K we have (Ax,Ay) p (x,y) If A is a positive linear mapping we have for all x,y K (Ax,Ay) (tanh( (A) ))(x,y) III. PROJECTIVE OBSERVER ON A CONE Consier the time-varying system on a soli cone K in a banach space X x k+ = A k (x k ) () y k = C k (x k ) where each A k is a positive map on K an C is a positive homogenous map of egree q. Then if the initial state x is in the cone, it remains in the cone for all times. We are concerne with the esign of observers whose estimate state ˆx remains in K for all times. A. A positive observer We start from the following ecomposition for all x K: x = rz, (r,z) R + (S K) wheresenotes the unit sphere inx an wherer is a scaling factor representing the norm of x, i.e. r k = x k. We have B. Convergence issues In this Section we stuy the asymptotic behavior of the z coorinate. The Birkhoff theorem implies that if A k is a positive homogeneous map of egree p, the Hilbert istance between z k an ẑ k oes not increase. As a result the observer oes not iverge (at least as long as the estimation of the z coorinate is concerne). Furthermore, when the observer s ynamics is a contraction, exponential convergence can be expecte for a large class of systems (see [9] for a general stuy of observers base on a contraction associate to a specific metric). The Birkhoff theorem allows to highlight some systems that are strict contractions: Proposition : Consier the system () an suppose there exists a finite horizon T such that either ) p ( < p < ) k N the operator A k+t A k+ A k is homogeneous of egree at most p ) Or R > k N A k+t A k+ A k is linear with projective iameter R Then observer (3) is such that (ẑ k,z k ) converges exponentially to zero. Proof: The proof is a straightforwar application of Birkhoff theorem over a finite horizon. IV. POSITIVE OBSERVERS FOR POSITIVE LINEAR SYSTEMS IN THE POSITIVE ORTHANT In this section we aress the particular case of linear systems on the positive orthantk = R n + = {x R n i x i }. The Hilbert metric on this cone is efine by: (x,y) = maxlog( x iy j ) i,j x j y i We consier the linear positive system in R n +. x k+ = A k x k +B k u k, y k = C k x k () where A k is positive, B k an C k are non-negative matrices. The positive observer (3) becomes z k+ = A kz k A k z k which is well efine as A k is a positive map. As the output map is suppose to be homogeneous of egree q we have y k = r q k C(z k) () ẑ K, ẑ k+ = A kẑ k +B k u k A k ẑ k +B k u k, ˆr k = y k C k (ẑ k ) In continuous time, the linear system becomes () Thus a simple caniate observer for the complete state is ẑ k+ = A kẑ k A k ẑ k, ˆr k = ( y k ) (3) /q C(ẑ k ) with ẑ K. The observer is well-efine as ẑ K an C are positive, an it elivers positive estimates ˆx k = ˆr k ẑ k, as ẑ remains in K. x = A(t)x+B(t)u(t), y = C(t)x (6) t with, A(t) Metzler, i.e. A ij (t) for i j, B(t) an C(t) non-negative, an observer () writes tẑ = ( ẑẑt )[A(t)ẑ +B(t)u(t)], ˆr = y(t) C(t)ẑ (7)

3 A. Convergence issues First of all, theorem 3. of [] proves that the contraction ratio of the mapx Ax+z withz is less than the one of A. Thus the aitional term B k u k helps convergence an nees not be consiere in the sequel. We have the following result extening Proposition to the full observer: Proposition : Consier the system () an suppose there exists a finite horizon T such that R > k N A k+t A k+ A k is linear with projective iameter R, Proposition implies the quantity (ẑ k,z k ) converges exponentially to zero. Moreover, if there exists ǫ,α,β > such that for k > the ball center ẑ k an raius ǫ satisfies S(ẑ k,ǫ) K, an α C k β, then observer () is such that the quantity ˆr k r k also converges exponentially to zero. Proof: Proposition implies that (ẑ k,z k ) tens exponentially to zero. So S(z k,ǫ/) K for k large enough. So the angle between z k an any line of C k is less than, say, θ < π/. It implies that r k ˆr k αcosθ C kẑ k C k z k αcosθ C k(ẑ k z k ) Moreover, the ifferential of the map S n K v (v,w) for w S n K satisfying S(w,ǫ/) K is uniformly boune from below in all irections on the subset {v S n K S(v,ǫ/) K}. So there exists γ > such that ẑ k z k γ (ẑ k,z k ) for k large enough. As a result, ẑ k z k converges exponentially to zero for k sufficiently large an so oes r k ˆrk. The previous result provies a convergence result that iffers from the usual linear exponential convergence. Inee the fact that ˆr k /r k converges exponentially to oes not imply the exponential convergence of the linear error ˆr k r k an ˆx k x k. Nevertheless, it coul be argue that the exponential convergence of the error ˆr k /r k is a meaningful alternative that acknowleges the nonlinear nature of the state-space. For instance, this error is invariant to scalings, which is meaningful from a physical point of view as scalings often correspon to a change of units. Such a state error is also naturally foun in the theory of symmetry-preserving observers [3]. Moreover, for boune trajectories, exponential convergence of the ratio ˆr k /r k oes imply exponential convergence of the linear error. Finally, it is of interest to observe that the proposition implies (non exponential) convergence for the following metric: p (ˆx k,x k ) = (ẑ k,z k ) + log(ˆr k /r k ) (8) The metric log(ˆr k /r k ) is calle the natural metric onr +, see e.g. [6]. The metric (8) is therefore a contractive metric for the observer. This result is reminiscent of the general work of [9]. B. -invariant case Birkhoff is a generalize version of the celebrate Perron theorem on an arbitrary cone by means of Hilbert s geometry. Inee, in the case where K is the positive orthant, the Perron theorem is a corollary of the Birkhoff theorem. Proposition can thus be formulate irectly with the help of Perron-Frobenius theory that eals with applications having a finite projective iameter. Proposition 3: Consier the time-invariant system x k+ = Ax k +Bu k, y k = Cx k Suppose that A is primitive, i.e. there exists a natural T N such that all the coorinates A T ij of the T -th power of A are strictly positive. Then observer () has the following properties: (ẑ k,z k ) an ˆr k rk converge exponentially to zero. p (ˆx k,x k ) where p is the metric (8). if the sequence r k is boune, the observer is exponentially convergent in the usual sense, i.e. ˆx k x k exponentially. Proof: Theorem an the Banach contraction mapping theorem (or the Perron-Frobenius theorem) imply there exists a vector v K (the fixe point) in K S such that both (ẑ k,v) an (z k,v) ten exponentially to zero (see [] for more etails). For k large enough, it implies that r k ˆr k Cv Cẑ k Cz k Cv C(ẑ k z k ) So there exists γ > such that r k /ˆr k γ ẑ k z k. On the tangent space to v/ v in S K the ifferential of the map w (v,w) is boune from below in all irections. As a result, ẑ k z k converges exponentially to zero for k sufficiently large an so oes r k ˆrk. C. An example We consier the positive continuous-time system of [] ẋ = 3 x, y = ( ) x 3 As all the off-iagonal coorinates of A are strictly positive, the associate iscrete-time map is primitive, an Proposition 3 applies for this system. The present paper allows to erive a convergent positive observer whereas [] proves that it is not possible to buil a convergent positive linear observer for this system. D. Positive observers an measurement noise Proposition proves that for a whole class of systems, one can buil a convergent positive observer with very weak assumptions on the output map. Inee, for those systems n coorinates (the z term) are estimate without the use of the output map. Thus the z estimate is never noisy, even when the measurement noise is very large. The remaining

4 -imensional term r is estimate via the following relation: ˆr k = y k / C k (ẑ k ). Note that any non-zero component of the output y k suffices to estimate the r coorinate, so that we will assume in the sequel the output is a scalar y k >. If the measurement noise is large, the estimation of observer () may not be easy to interpret as ˆr k can be as noisy as the output y k. For a better noise filtering, we propose the following moification: ˆr k = A k (ˆr k ẑ k )+B k u k +L r (y k /C k (ẑ k ) ˆr k ) First coorinate 8 6 Plot of the first coorinate of x an its estimate on r Secon coorinate. Plot of the secon coorinate of x an its estimate on z If L r is small enough the noise is efficiently filtere. Yet, L r must be large enough to ensure convergence. For instance if we consier the one-imensional time-invariant system: r = ar, y = cr t with a >, we see that convergence is guarantee as soon as L r > a. E. Numerical experiments Consier the continuous-time system []: ( ) ẋ = x, y = ( ) x (9) for which there exists no convergent linear Luenberger observer. Proposition 3 proves that observer () is positive an converges, as the off-iagonal terms of the matrix associate to the continuous time system are strictly positive. In the first numerical experiment we consier the noiseless system (see Fig ). We see that the estimates of the first an secon coorinates of the observer (), ˆx (t) an resp. ˆx (t) are always positive an that the error converges. In the secon experiment, a gaussian white noise with unit stanar eviation (more than % of the maximum value of the signal) was ae. Estimates of the following observer: tẑ = ( ẑẑt )[Aẑ +Bu(t)], () tˆr = y ˆr(ẑT Aẑ)+L r ( C(ẑ) ˆr) with L r = 3 s are presente on Figure. We see that the noise is efficiently filtere an the observer is still positive an convergent. In both experiments the initial conitions are: x() = (,/) T, ˆx() = (/,) T. As a final remark, note that, looking at the figures we see the error ˆr/r seems much more aapte than the usual error ˆx x when the norm of x iverges (i.e. r ). Inee, the interesting transient behavior of the estimation error is crushe by the plot scale as the coorinates of x grow exponentially. For example if we plot ˆx x over a 6-secons horizon, the initial errors are barely visible on the plot. V. OBSERVERS IN THE CONE OF POSITIVE SEMI-DEFINITE MATRICES Positive semi-efinite matrices appear in various contexts of applie mathematics an engineering. They appear as variables (convex programming, LMI, Lyapunov equation), Fig.. Results of observer (7) for system (9) with noiseless measurement. Top left: x (t) (plain line) an ˆx (t) (ashe line). They are equal by efinition of the observer since the output is x. Top right: x (t) (plain line) an ˆx (t) (ashe line). Bottom left: ˆr/r. Bottom right: ẑ z. First coorinate Plot of the first coorinate of x an its estimate on r Secon coorinate.. Plot of the secon coorinate of x an its estimate... 3 on z... 3 Fig.. Results of observer () for system (9) with noisy measurement (white noise with unit stanar eviation). Top left: Measurement y(t) (plain line) an ˆx (t) (ashe line). The noise is efficiently filter (see top right graphics) an ˆx is thus maske by y. Top right: x (t) (plain line) an ˆx (t) (ashe line). Bottom left: ˆr/r. Bottom right: ẑ z. an as covariance matrices (statistics, signal processing, Kalman filtering), iffusion tensors (biomeical imaging), an kernels in machine learning. The stuy of the cone of positive semi-efinite hermitian matrices has thus receive ever growing attention in the last years, an we propose to apply the theory eveloppe in this paper on this cone. It writes the Hilbert metric is K = {X C n n X = X, X } (X,Y) = log( λ max(xy ) λ min (XY ) ) Consiering a linear system efine on K X k+ = A k (X k )+B k (u), y k = C k (X k ) the whole analysis eveloppe in Section II can be applie. We are now going to iscuss a particular possible omain of application : esign of quantum filters.

5 A. Quantum filtering as a positive observer problem An interesting linear system on K is the evolution of the ensity matrix characterizing a state in a quantum channel. Inee consier a quantum channel associate to the Kraus map (which is a positive linear map on K) K(ρ) = n M µ ρm µ µ= where ρ is the ensity matrix, i.e. an hermitian semi-efinite positive matrix of trace one, escribing the input state, K(ρ) is the output state, an m M µm µ = I. The evolution is escribe by the following iscrete-time system ρ k+ = M µk (ρ k ) := Tr(M µk ρ k M µ k ) M µ k ρ k M µ k where ρ k is the quantum state at time t k an µ k = {,..,m} is a ranom variable such that µ k = j with probability Tr(M j ρ k M j ). The process preserves the trace an it is a concatenation of a linear map an a renormalization. The problem of quantum filtering is the following: consier a realization associate to the Kraus map efine above, an assume that at each step the jump µ k {,..,n} is etecte, but the initial state ρ is not known. It is typically an observer problem. The following observer is known as a quantum filter : ˆρ k+ = M µk (ˆρ k ) () an it is a mere copy of the ynamics which takes into account the jump information µ k. Observer () suits in the framework eveloppe in Section. Moreover, as the true processρ k must be of trace, there is no nee to estimate the scaling r k of Section. The quantum filters can be analyze as positive observers in the light of the theory eveloppe in this paper. In particular we have the following sufficient conition for exponential convergence: Proposition : If for any µ n the map M µ has a finite projective iameter, the quantum filter () converges exponentially, i.e. (ˆρ k,ρ k ) exponentially. Note that a similar result has alreay been unerline recently by one of the authors in [] for applications in consensus. Proposition proves that the Hilbert istance is a goo metric to analyze convergence of quantum filters, as the Kraus maps are contractions for the Hilbert metric. Inee they are linear maps so Theorem proves the the contraction ratio oes not excee. The Hilbert metric may thus prove to be a useful alternative to the celebrate trace norm, i.e. (ρ,ρ ) = Tr( ρ ρ ) for the stuy of quantum filters. See [] an references therein for more etails. The theory was applie to two cones: the positive orthant of the eucliean space an the cone of hermitian positive efinite matrices. In the positive orthant, the theory allows to buil very simple convergent positive observers for a large class of systems. For instance, we prove exponential convergence of the observer for the linear system t x = Ax + Bu, y = Cx as soon as the off-iagonal terms of A are strictly positive, an the construction of the observer is trivial. In the cone of positive efinite matrices, the application of Birkhoff theorem provies a framework to esign an analyze quantum filters. Another application on the cone of semi-efinite positive matrices woul be to stuy the convergence of the istribute Kalman filter with the tools introuce in this article. This is left for future research. As a concluing remark, note that the metho eveloppe in this paper may seem magical as the observer esign becomes trivial. However the price to pay is that the convergence spee is not freely chosen as in Luenberger observer esign for observable linear systems where the eigenvalues of the close-loop system can be freely assigne. REFERENCES [] J. Back an A. Astolfi. 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On metric analysis an observers for nonlinear systems. Automatica, 3(6): , 998. [] Y. Ohta, H. Maea, an S. Koama. Reachability, observability an realizability of continuous-time positive systems. SIAM Journal of Control an Optimization, ():7 8, 98. [] P. Rouchon. Fielity is a sub-martingale for any quantum filter. Technical report, Submitte, Available on Arxiv.. [] S. Sepulchre, A. Sarlette, an P. Rouchon. Consensus in noncommutative spaces. In 9th IEEE Conference on Decision an Control, pages ,. [3] H.L. Smith. Monotone Dynamical Systems-An introuction to the theory of competitive an cooperative systems. American mathematical society, Province, RI, 99. [] J.M. van en Hof. Positive linear observers for linear compartmental systems. SIAM Journal of Control an Optimization, 36():9 68, 998. VI. CONCLUSION In this paper, we propose a new esign metho for positive observers on soli cones. The convergence analysis is base on Birkhoff theorem. It allows to buil convergent positive observers for a large class of systems, which are homogeneous an not necessarily linear.