Uniform observability of linear time-varying systems and application to robotics problems

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Many methods have been proposed to estimate the state of a nonlinear dynamical system from uncomplete measurements. This paper concerns an approach that consists in lifting the estimation problem into a higher-dimensional state-space so as to transform an original nonlinear problem into a linear problem. Although the associated linear system is usually time-varying, one can then rely on Kalman’s linear filtering theory to achieve strong convergence and optimality properties. In this paper, we first present a technical result on the uniform observability of linear time-varying systems. Then, we illustrate through a problem arising in robotics how this result and the lifting method evoked above lead to explicit observability conditions and linear observers.

Uniform observability of linear time-varying systems and application to robotics problems

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application/pdf Uniform observability of linear time-varying systems and application to robotics problems Pascal Morin, Alexandre Eudes, Glauco Scandaroli
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Uniform observability of linear time-varying systems and application to robotics problems
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Uniform observability of linear time-varying systems and application to robotics problems⋆ Pascal Morin1 , Alexandre Eudes2 , and Glauco Scandaroli1 1 ISIR, Sorbonne Universités, UPMC Univ. Paris 06, CNRS UMR 7222, Paris, France morin@isir.upmc.fr, scandaroli@upmc.fr, 2 ONERA - The French Aerospace Lab, Palaiseau, France alexandre.eudes@onera.fr Abstract. Many methods have been proposed to estimate the state of a nonlinear dynamical system from uncomplete measurements. This paper concerns an approach that consists in lifting the estimation problem into a higher-dimensional state-space so as to transform an original nonlinear problem into a linear problem. Although the associated linear system is usually time-varying, one can then rely on Kalman’s linear filtering theory to achieve strong convergence and optimality properties. In this paper, we first present a technical result on the uniform observability of linear time-varying systems. Then, we illustrate through a problem arising in robotics how this result and the lifting method evoked above lead to explicit observability conditions and linear observers. Keywords: observability, observer design, filtering Introduction The general problem of observability and observer design concerns the recon- struction of the state x of a dynamical system ẋ = f(x, u, t) from the knowledge of the input u and an output function y = h(x, u, t). There is a vast control litterature on this topic for both linear [12, 11] and nonlinear systems [9]. For linear systems, i.e. f(x, u, t) = A(t)x + B(t)u, h(x, u, t) = C(t)x + D(t)u, ob- servability is independent of the input u and is directly related to the system’s Grammian, which only depends on the matrices A and C. By contrast, observ- ability of a nonlinear system usually depends on the input. This dependence is a source of difficulty for both observability analysis and observer design. As a matter of fact, the main observability characterization result for nonlinear sys- tems [9] only ensures a ”weak” form of observability, i.e. existence of control inputs for which the system is observable. Neither the characterization of these ”good inputs”, nor the characterization of the associated observability property (e.g. uniform versus non-uniform) is provided in [9]. Concerning observer design, in many application fields involving nonlinear systems state estimation relies on Extended Kalman Filters (EKFs), which often perform well but also, sometimes, yield divergent estimation errors or/and unconsistent results. One of the main ⋆ These results were obtained while all authors were with ISIR. This work was sup- ported by the ”Chaire d’excellence en Robotique RTE-UPMC”. progress achieved in recent years on the topic of nonlinear observers concerns systems with symmetries [2]. When both the system’s dynamics and output func- tions are invariant under a transformation group, so-called ”invariant observers” can be built with improved convergence properties w.r.t. EKFs. In particular, if the system’s state space is a Lie group and the system’s dynamics and out- put functions are invariant w.r.t. the Lie group operation, observers with error dynamics independent of the trajectory can be obtained [3]. W.r.t. EKFs this implies stronger convergence results, as demonstrated for several applications [15, 4]. As a remaining difficulty, the error dynamics is still nonlinear. This can make global or semi-global convergence properties difficult to achieve. This paper concerns a different approach, which consists in lifting the estimation problem into a higher-dimensional state-space so as to transform the original nonlinear estimation problem into a linear estimation problem in higher-dimension. Like for invariant observers, the objective is still to simplify the observability analysis and observer design. In this case the goal is fully achieved thanks to the strong observability and observer design results for linear time-varying systems. The paper is organized as follows. The main technical result of this paper is given in Section 1. Then, a robotics application example is treated in Section 2. 1 Observability of linear time-varying systems Consider a general linear time-varying (LTV) system  ẋ = A(t)x + B(t)u y = C(t)x (1) There exist different types of observability properties for LTV systems, like e.g., differential, instantaneous, or uniform observability (see e.g. [6, Ch. 5] for more details). Here we focus on uniform observability, which ensures that the state estimation process is well-conditionned and can be solved via the design of ex- ponentially stable observers. The following assumption will be used. Assumption 1 The matrix-valued functions A, B, and C of the LTV system (1) are continuous and bounded on [0, +∞). Definition 1 A LTV system (1) satisfying Assumption 1 is uniformly observ- able if there exist τ, δ > 0 such that ∀t ≥ 0, 0 < δI ≤ W(t, t + τ) ∆ = Z t+τ t Ψ(s, t)T CT (s)C(s)Ψ(s, t) ds (2) with Ψ(s, t) the state transition matrix of ẋ = A(t)x and I the identity matrix. The matrix W is called observability Grammian of System (1). From this definition uniform observability is independent of B. Thus, we say without distinction that System (1) or the pair (A, C) is uniformly observable. Note also, as a consequence of Assumption 1, that W(t, t + τ) is upper bounded by some δ̄I for any t ≥ 0. The following theorem recalls two properties of uniformly observable systems. The first property follows from [1, Lemma 3] and the duality principle (see, e.g. [6, Th. 5-10]). This principle, together with [10, Th. 3], imply the second property. Theorem 1 For a LTV system (1) satisfying Assumption 1 the following prop- erties hold. 1. The pair (A, C) is uniformly observable iff the pair (A−LC, C) is uniformly observable, with L(.) any bounded matrix-valued time-function. 2. If the pair (A, C) is uniformly observable, then for any a > 0 there exists a bounded matrix La(t) such that the linear observer ˙ x̂ = A(t)x̂ + La(t)(y − C(t)x̂) is uniformly exponentially stable with convergence rate given by a, i.e. there exists ca > 0 such that kx̂(t) − x(t)k ≤ cae−a(t−t0) kx̂(t0) − x(t0)k for any t ≥ t0 and any x(0), x̂(0). Main technical result: Checking uniform observability of a LTV system can be difficult since calculation of the Grammian requires integration of the solutions of ẋ = A(t)x. It is well known that observability properties of LTV systems are related to properties of the observability space O(t) defined by ([6, Ch. 5]): O(t) ∆ =    N0(t) N1(t) . . .    , N0 ∆ = C, Nk+1 ∆ = NkA + Ṅk for k = 1, . . . (3) For example, instantaneous observability at t is guaranteed if Rank(On−1(t)) = n. For general LTV systems, however, uniform observability cannot be charac- terized in term of rank conditions only. We propose below a sufficient condition for uniform observability. Proposition 1 Consider a LTV system (1) satisfying Assumption 1. Assume that there exists a positive integer K such that: 1. The k-th order derivative of A (resp. C) is well defined and bounded on [0, +∞) up to k = K (resp. up to k = K + 1). 2. There exist a n × n matrix M composed of row vectors of N0, . . . , NK, and two scalars δ̄, τ̄ > 0 such that ∀t ≥ 0, 0 < δ̄ ≤ Z t+τ̄ t |det(M(s))| ds (4) with det(M) the determinant of M. Then, System (1) is uniformly observable. The proof of this result is given in the appendix. As illustrated in the following section, this result leads to very explicit uniform observability conditions. Remark: In [5] a sufficient condition for uniform complete observability (a prop- erty equivalent to uniform observability under Assumption 1) is provided. There are similarities between that condition and (4) but the latter is less demanding as it only requires positivity ”in average” while the positivity condition in [5] must hold for any time-instant. 2 Application to a robotics estimation problem A classical robotics problem consists in recovering the motion of a robot from measurements given by a vision system. For compacity reasons, monocular vi- sion can be preferred to stereo vision but then, full 3D motion estimation cannot be performed due to depth ambiguity. To remedy this difficuty, vision data can be fused with measurements provided by an IMU (Inertial Measurement Unit). This is called visuo-inertial fusion. In this section, we describe how the problem of monocular visuo-inertial fusion is commonly posed as a nonlinear estimation problem, and we show how it can be transformed by lifting into a linear esti- mation problem in higher dimension. The material presented in this section is based on [7] to which we refer the reader for more details. The monocular visuo-inertial problem Consider two images IA and IB of a planar scene taken by a monocular camera. Each image I∗ (∗ ∈ {A, B}) is taken from a specific pose of the camera and we denote by F∗(∗ ∈ {A, B}) an associated camera frame with origin corresponding to the optical center of the camera and third basis vector aligned with the optical axis. We also denote by d∗ and n∗ respectively the distance from the origin of F∗ to the planar scene and the normal to the scene expressed in F∗. Let R denote the rotation matrix from FB to FA and p ∈ R3 the coordinate vector of the origin of FB expressed in FA. The problem here considered consists in estimating R and p. From IA and IB one can compute (see, e.g., [13]) the so-called ”homography matrix” H = RT − 1 dA RT pnT A (5) Considering also a (strapped-down) IMU, we obtain as additional measurements ω, the angular velocity vector of the sensor w.r.t. the inertial frame expressed in body frame, and as, the so-called specific acceleration. Assuming that FA is an inertial frame and FB is the body frame3 , ω and as are defined by: Ṙ = RS(ω) , p̈ = gA + Ras (6) where gA denotes the gravitational acceleration field expressed in FA and S(x) is the skew-symmetric matrix associated with the cross product by x, i.e. S(x)y = x × y with × the cross product. Note that as can also be defined by the relation v̇ = −S(ω)v + as + gB (7) where v = RT ṗ denotes the velocity of FB w.r.t. FA expressed in FB, and gB = RT gA is the gravitational acceleration field expressed in FB. Visuo-inertial fusion: Diverse estimation algorithms have been proposed. In [16] the state is defined as x = (R, p, v, nA, dA), with measurement y = (H, ω, as). Since H is a nonlinear function of x, the estimation problem is nonlinear and an EKF is used. In [14, 8] the state is defined as x = (H̄, M, nA) where H̄ = 3 For simplicity we assume that the camera frame and IMU frame coincide. det(H)− 1 3 H and M = v nT A dA , with measurement y = (H̄, ω, as). The measurement then becomes a linear function of x, but the dynamics of x is nonlinear. Using the fact that H̄ belongs to the Special Linear group SL(3), nonlinear observers with convergence guarantees are proposed in [14, 8], but under restrictive motion assumptions. As an alternative solution, define the state as x = (H, M, ns, Q) with H defined by (5) and M = vnT s , ns = nA dA , Q = gBnT s . Since nA and dA are constant quantities, one verifies from (6) and (7) that:  Ḣ = −S(ω)H − M , ṅs = 0 Ṁ = −S(ω)M + Q + asnT s , Q̇ = −S(ω)Q (8) Since ω and as are known time-functions, the above system is a linear time- varying system in x. In other words, the estimation problem has been trans- formed into a linear estimation problem by lifting to a higher-dimensional state space. One verifies (see [7] for details) that R and p can be extracted from x. From this point, one can make use of existing tools of linear estimation theory. Proposition 1 provides the following characterization of uniform observability in term of the IMU data. It was initially obtained in [7]. Proposition 2 System (8) with measurement y = (H, ω, as) is uniformly ob- servable if i) ω and as are continuous and bounded on [0, +∞), and their first, second, and third-order time-derivatives are well defined and bounded on [0, +∞); ii) there exists two scalars δ, σ > 0 such that ∀t ≥ 0, 0 < δ ≤ Z t+σ t kȧs(τ) + ω(τ) × as(τ)k dτ (9) 3 Conclusion We have provided a technical result on uniform observability of linear time- varying system and we have illustrated the application of this result to an es- timation problem. We have also shown through this problem how an original nonlinear estimation problem could be transformed into a linear estimation prob- lem through lifting of the state space. The main open problem is to characterize systems for which such a lifting exists. Appendix: Proof of Proposition 1: We must show the existence of constants τ, δ > 0 such that (2) is satisfied. The inequality in (2) is equivalent to xT W(t, t+ τ)x ≥ δkxk2 for any vector x ∈ D = {x ∈ Rn : kxk = 1}. Thus, the proof consists in showing the existence of constants τ, δ > 0 such that ∀t ≥ 0, 0 < δ ≤ inf x∈D Z t+τ t kC(s)Ψ(s, t)xk2 ds We proceed by contradiction. Assume that ∀τ > 0, ∀δ > 0, ∃t(τ, δ) : inf x∈D Z t(τ,δ)+τ t(τ,δ) kC(s)Ψ(s, t(τ, δ))xk2 ds < δ (10) Take τ = τ̄ with τ̄ the constant in (4), and consider the sequence (δp = 1/p). Thus, for any p ∈ N, there exists tp such that inf x∈D Z tp+τ̄ tp kC(s)Ψ(s, tp)xk2 ds < 1 p so that there exists xp ∈ D such that Z tp+τ̄ tp kC(s)Ψ(s, tp)xpk2 ds < 1 p (11) Since D is compact, a sub-sequence of the sequence (xp) converges to some x̄ ∈ D. From Assumption 1, A is bounded on [0, +∞). Therefore, ∀x ∈ Rn , ∀t ≤ s, e−(s−t)kAk∞ kxk ≤ kΨ(s, t)xk ≤ e(s−t)kAk∞ kxk (12) with kAk∞ = supt≥0 kA(t)k. Since C is also bounded (from Assumption 1) and the interval of integration in (11) is of fixed length τ̄, it follows that lim p→+∞ Z tp+τ̄ tp kC(s)Ψ(s, tp)x̄k2 ds = 0 By a change of integration variable, this equation can be written as lim p→+∞ Z τ̄ 0 kfp(s)k2 ds = 0 (13) with fp(t) = C(t + tp)Ψ(t + tp, tp)x̄. It is well known, and easy to verify, that f(k) p (t) = Nk(t + tp)Ψ(t + tp, tp)x̄ (14) with f (k) p the k-th order derivative of fp and Nk defined by (3). The existence of f (k) p , for any k = 0, · · · , K + 1, follows by Assumption 1 of Proposition 1. The end of the proof relies on the following lemma, proved further. Lemma 1 Assume that (10) is satisfied. Then, ∀k = 0, . . . , K, lim p→+∞ Z τ̄ 0 kf(k) p (s)k2 ds = 0 (15) Since the matrix M in (4) is composed of row vectors of N0, . . . , NK, it follows from (14) that Z τ̄ 0 kM(s + t, p)Ψ(s + tp, tp)x̄k2 ds ≤ K X k=0 Z τ̄ 0 kf(k) p (s)k2 ds Therefore, from Lemma 1, lim p→+∞ Z tp+τ̄ tp kM(s)Ψ(s, tp)x̄k2 ds = lim p→+∞ Z τ̄ 0 kM(s + tp)Ψ(s + tp, tp)x̄k2 ds = 0 (16) Then, for any ξ ∈ Rn kM(s)ξk2 = ξT MT (s)M(s)ξ ≥ kξk2 min i λi(MT (s)M(s)) = kξk2 λ1(MT (s)M(s)) (17) with λ1(MT (s)M(s)) ≤ · · · ≤ λn(MT (s)M(s)) the eigenvalues of MT (s)M(s) in increasing order. Furthermore, since M is bounded on [0, +∞) (as a consequence of Assumption 1 and the definition of M), there exists a constant c > 0 such that maxi λi(MT (s)M(s)) ≤ c for all s. Thus λ1(MT (s)M(s)) = det(MT (s)M(s)) Q j>1 λj(MT (s)M(s)) ≥ det(MT (s)M(s)) cn−1 ≥ (det(M(s)))2 cn−1 It follows from this inequality, (12), (17), and the fact that kx̄k = 1 that ∀p ∈ N, Z tp+τ̄ tp kM(s)Ψ(s, tp)x̄k2 ds ≥ c̄ Z tp+τ̄ tp (det(M(s)))2 ds (18) with c̄ = e−2τ̄kAk∞ /cn−1 > 0. Furthermore, Schwarz inequality implies that Z tp+τ̄ tp |det(M(s))| ds ≤ Z tp+τ̄ tp 1 ds !1/2 Z tp+τ̄ tp (det(M(s)))2 ds !1/2 Thus, it follows from (4) and (18) that ∀p ∈ N, Z tp+τ̄ tp kM(s)Ψ(s, tp)x̄k2 ds ≥ c̄δ̄2 /τ̄ > 0 which contradicts (16). To complete the proof, we must prove Lemma 1. Proof of Lemma 1: We proceed by induction. By assumption (10) is satisfied, which implies that (13) holds true. Thus, (15) holds true for k = 0. Assuming now that (15) holds true for k = 0, . . . , k̄ < K, we show that it holds true for k = k̄ +1 too. From Assumption 1 of Proposition 1, recall that ∀j = 1, . . . K +1, f (j) p is well defined and bounded on [0, τ̄], uniformly w.r.t. p. We claim that f (k̄) p (0) tends to zero as p tends to +∞. Assume on the contrary that f (k̄) p (0) does not tend to zero. Then, ∃ε > 0 and a subsequence (f (k̄) pj ) of (f (k̄) p ) such that kf (k̄) pj (0)k > ε , ∀j ∈ N. Since kf (k̄+1) pj (0)k is bounded uniformly w.r.t. j (because f (k̄+1) p is bounded on [0, τ̄] uniformly w.r.t. p), ∃t′ > 0 such that ∀j ∈ N, ∀t ∈ [0, t′ ], kf (k̄) pj (t)k > ε/2. By (15), this contradicts the induction hypothesis. Therefore, f (k̄) p (0) tends to zero as p tends to +∞. By a similar argument, one can show that f (k̄) p (τ̄) tends to zero as p tends to +∞. Now, Z τ̄ 0 kf(k̄+1) p (s)k2 ds = n X i=1 Z τ̄ 0  f (k̄+1) p,i (s) 2 ds = − n X i=1 Z τ̄ 0 f (k̄) p,i (s)f (k̄+2) p,i (s) ds + n X i=1 h f (k̄) p,i (s)f (k̄+1) p,i (s) iτ̄ 0 ≤ n X i=1 Z τ̄ 0  f (k̄) p,i (s) 2 ds 1/2 Z τ̄ 0  f (k̄+2) p,i (s) 2 ds 1/2 + n X i=1 h f (k̄) p,i (s)f (k̄+1) p,i (s) iτ̄ 0 Each term Z τ̄ 0  f (k̄) p,i (s) 2 ds 1/2 Z τ̄ 0  f (k̄+2) p,i (s) 2 ds 1/2 in the first sum tends to zero as p tends to infinity due to (15) for k = k̄ and the fact that f (k̄+2) p is bounded uniformly w.r.t. p. Boundary terms in the second sum also tend to zero as p tends to infinity since f (k̄) p (0) and f (k̄) p (τ̄) tend to zero, and f (k̄+1) p is bounded. As a result, (15) is satisfied for k = k̄ + 1. References 1. B.D.O. Anderson and J.B. Moore. New results in linear system stability. SIAM Journal on Control, 7(3):398–414, 1969. 2. S. Bonnabel, P. Martin, and P. Rouchon. Symmetry-preserving observers. IEEE Trans. on Automatic Control, 53(11):2514–2526, 2008. 3. S. Bonnabel, P. 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