Identification of Analytic Nonlinear Dynamical Systems with Non-asymptotic Guarantees

Authors: negin  musavi
geir  dullerud
yingying  li
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 21:11:30

Abstract: This paper focuses on the system identification of an important class of nonlinear systems: linearly parameterized nonlinear systems, which enjoys wide applications in robotics and other mechanical systems. We consider two system identification methods: least-squares estimation (LSE), which is a point estimation method; and set-membership estimation (SME), which estimates an uncertainty set that contains the true parameters. We provide non-asymptotic convergence rates for LSE and SME under  i.i.d. control inputs and control policies with i.i.d. random perturbations, both of which are considered as non-active-exploration inputs. Compared with the counter-example based on piecewise-affine systems in the literature, the success of non-active exploration in our setting relies on a key assumption on the system dynamics: we require the system functions to be real-analytic. Our results, together with the piecewise-affine counter-example, reveal the importance of differentiability in nonlinear system identification through non-active exploration. Lastly, we numerically compare our theoretical bounds with the empirical performance of LSE and SME on a pendulum example and a quadrotor example.

Main paper: 
 introduction: 
 Learning control-dynamical systems with statistical methodology has received significant attention in the past decade (Sarker et al., 2023; Li et al., 2023b; Chen and Hazan, 2021; Simchowitz and Foster, 2020; Wagenmaker and Jamieson, 2020; Simchowitz et al., 2018; Dean et al., 2018; Abbasi-Yadkori and Szepesvári, 2011; Li et al., 2021b) . In particular, the estimation of linear dynamical systems, e.g. x t+1 = A * x t +B * u t +w t , is relatively well-studied: it has been shown that non-active exploration by i.i.d. noises on control inputs u t and system disturbances w t are already enough for accurate system identification, and least square estimation (LSE) can achieve the optimal estimation convergence rate (Simchowitz and Foster, 2020; Simchowitz et al., 2018) .However, nonlinear control systems are ubiquitous in real-world applications, e.g. robotics (Siciliano et al., 2010; Alaimo et al., 2013) , power systems (Simpson-Porco et al., 2016) , transportation (Kong et al., 2015) , etc. Motivated by this, there has been a lot of attention on learning nonlinear systems recently. One natural and popular direction to study nonlinear system identification is on learning linearly parameterized nonlinear systems as defined below, which is a straightforward extension from the standard linear systems (Mania et al., 2022; Khosravi, 2023; Foster et al., 2020) x t+1 = θ * ϕ(x t , u t ) + w t where θ * is a vector of unknown parameters and ϕ(x t , u t ) is a known vector of nonlinear features.On the one hand, some classes of these systems are shown to enjoy similar benefits of linear systems. For example, bilinear systems can also be estimated by LSE under non-active exploration with i.i.d. noises (Sattar et al., 2022) , as well as linear systems with randomly perturbed nonlinear policies (Li et al., 2023b) .On the other hand, it is also known that non-active exploration is insufficient for general linearly parameterized nonlinear systems. In particular, (Mania et al., 2022) provides a counter example showing that non-active exploration is insufficient to learn accurate models under piece-wise affine feature functions. This motivates a sequence of follow-up work on the design of active exploration for nonlinear system estimation, which is largely motivated by the non-smooth feature functions such as ReLu in neural networks (Mania et al., 2022; Kowshik et al., 2021; Khosravi, 2023) .However, there is a big gap between bilinear systems, which is infinitely differentiable, and the counter example by non-smooth systems. A natural question is: to what extent can non-active exploration still work for linearly parameterized nonlinear systems?Contributions. One major contribution of this paper is showing that LSE with non-active i.i.d. noises can efficiently learn any linearly parameterized nonlinear systems with real-analytic feature functions and provide a non-asymptotic convergence rate Notice that real-analytic feature functions are common in physical systems. For example, polynomial systems satisfy this requirement and have wide applications in power systems (Simpson-Porco et al., 2016) , fluid dynamics (Noack et al., 2003) , etc. Further, trigonometric functions also satisfy the real-analytic property so a large range of robotics and mechanical systems also satisfy this requirement (Siciliano et al., 2010; Alaimo et al., 2013) .A side product of our LSE convergence rate analysis is the convergence rate for another commonly used uncertainty quantification method in control: set membership estimation (SME).Numerically, we test our theoretical results in pendulum and quadrotor systems. Simulations show that LSE and SME can indeed efficiently explore the system and converge to the true parameter under non-active exploration noises.Technically, the key step in our proof is establishing the block-martingale-small-ball condition (BMSB) for general analytic feature functions, which greatly generalizes the bilinear feature function in Sattar et al. (2022) . Our result is built on an intuition inspired by the counter example in (Mania et al., 2022) : the counter example in (Mania et al., 2022) requires that some feature function is zero in a certain region, so nothing can be learned about its parameter if the states stay in this region. However, analytic functions cannot be a constant zero in a positive-measure region unless it is a constant zero everywhere. Therefore, the counter example does not work, and non-active exploration around any states can provide some useful information. Our proof formalizes this intuition by utilizing the Paley-Zygmund Petrov inequality (Petrov, 2007) .Inspired by neural network parameterization, nonlinear systems of the form x t+1 = ϕ(A * x t ) + w t is also studied in the literature, where ϕ(•) is a known nonlinear link function and A * is unknown. The least square cost is no longer quadratic or even convex in this case and various optimization methods have been proposed to learn this type of systems (Kowshik et al., 2021; Sattar et al., 2022; Foster et al., 2020) .Another related line of research focuses on nonlinear regression with dependent data (Ziemann and Tu, 2022; Ziemann et al., 2023 Ziemann et al., , 2024)) , 1 which can be applied to nonlinear system identification. The nonlinear regression in (Ziemann and Tu, 2022; Ziemann et al., 2023 Ziemann et al., , 2024) ) is based on nonparametric LSE and its variants, and their convergence rates under different scenarios have been analyzed. It is interesting to note that this line of work usually assumes certain persistent excitation assumptions,2 whereas our paper demonstrates that persistent excitation holds by establishing the BMSB condition for linearly parameterized and real-analytic nonlinear control systems.Uncertainty set estimation is crucial for robust control under model uncertainties Lu and Cannon (2023) ; Lorenzen et al. (2019) ; Li et al. (2021a) . SME is a widely adopted uncertainty set estimation method in robust adaptive control (Lorenzen et al., 2019; Lu and Cannon, 2023; Bertsekas, 1971; Bai et al., 1995) . Recently, there is an emerging interest in analyzing SME's convergence and convergence rate for dynamical systems (Li et al., 2024; Lu et al., 2019; Xu and Li, 2024) , because previous analysis focus more on the linear regression problem (e.g. (Akçay, 2004; Bai et al., 1998) ). There are also recent applications of SME to online control Yu et al. (2023) , power systems Yeh et al. (2024), and computer vision Gao et al. (2024) ; Tang et al. (2024) .Notation. The set of non-negative real numbers is denoted by R ≥0 . The notation ⌈•⌉ stands for the ceiling function. For a real vector z ∈ R n , ∥z∥ 2 represents its ℓ 2 norm, ∥z∥ ∞ represents its ℓ ∞ norm, and z i represents its i-th component with i = 1 • • • n. The set of real symmetric matrices is denoted by S n . For a real matrix Z, Z ⊺ represents its transpose, ∥Z∥ 2 its maximum singular value, ∥Z∥ F its Frobenius norm, σ min (Z) its minimum singular value, vec(Z) its vectorization obtained by stacking its columns, and for a real square matrix Z, tr(Z) represents its trace. For a real symmetric matrix Z, Z ≻ 0 and Z ⪰ 0 indicate that Z is positive definite and positive semi-definite, respectively. For a measurable set E ⊂ R n , λ n (E) represents its Lebesgue measure in R n and E c represents its complement in R n . The notation ∅ stands for the empty set. For a set T of matrices θ ∈ R n×m , diam(T ) denotes its diameter and it is defined as diam(T ) = sup θ,θ ′ ∈T ∥θ -θ ′ ∥ F . For z i ∈ R with i = 1, • • • , ℓ, the notation diag(z 1 , • • • , z ℓ ) denotes a matrix in R ℓ×ℓ with diagonal entries of z i .This paper uses truncated-Gaussian(0, σ w , [-w max , w max ]) to refer to the truncated Gaussian distribution generated by Gaussian distribution with zero mean and σ 2 w variance with truncated range [-w max , w max ]. The same applies to multi-variate truncated Gaussian distributions.Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?Answer: [Yes] Justification: For details refer to Sections 2, 3 and 4 Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 
problem formulation and preliminaries: 
 This paper studies the system identification/estimation of linearly parameterized nonlinear systems:EQUATIONwhere x t ∈ R nx , u t ∈ R nu , and w t ∈ R nx denote the state, control input, and system disturbance respectively; θ * ∈ R nx×n ϕ denotes the unknown parameters to be estimated, and ϕ(•) denotes a vector of known nonlinear feature/basis functions, i.e., ϕ(•) = (ϕ 1 (•), • • • , ϕ n ϕ (•)) ⊺, where ϕ i (•) : R nx+nu → R. Without loss of generality, we consider zero initial condition, i.e., x 0 = 0, and linearly independent feature functions, that is,n ϕ i=1 c i ϕ i (x t , u t ) = 0 implies that c i = 0 for all i. 3The linearly parameterized nonlinear system (1) is a natural generalization of linear control systems x t+1 = A * x t + B * u t + w t and has wide applications in, e.g. robotics (Siciliano et al., 2010; Alaimo et al., 2013) , power systems (Simpson-Porco et al., 2016) , transportation (Kong et al., 2015) , etc. Therefore, there has been a lot of research on learning this type of systems (1) by utilizing the methodology and insights from linear system estimation. For example, it is common to estimate a linearly parameterized nonlinear system by least square estimation (LSE), which enjoys desirable performance in linear systems.In particular, LSE for (1) is reviewed belowθT = arg min θ T -1 t=0 x t+1 -θϕ(x t , u t ) 2 2 .(2)For linear systems, LSE enjoys the following good property: LSE can achieve the optimal rate of convergence with i.i.d. noises w t and i.i.d. control inputs u t under proper conditions ( (Simchowitz et al., 2018) ). This good property has been generalized to some linearly parameterized nonlinear systems, such as bilinear systems, and linear systems with nonlinear control policies. Unfortunately, general linearly parameterized nonlinear systems do not enjoy this good property of linear systems, meaning i.i.d. random inputs may not provide enough exploration for non-smooth feature functions ϕ(•). Therefore, a sequence of follow-up work focuses on the design of active exploration methods.However, due to the simplicity of implementation, i.i.d. random inputs remain a popular method in empirical research of system identification and enjoy satisfactory performance sometimes, despite the lack of theoretical guarantees. Therefore, this paper aims to establish more general conditions that allow provable convergence of nonlinear system estimation under i.i.d. random inputs.In the rest of this paper, we will show that with certain smoothness and continuous conditions, i.i.d. random inputs are sufficient for estimation of (1), which recovers the good property of linear systems.In the following, we formally describe the smoothness and continuity conditions that enables efficient exploration of (1) by i.i.d. random inputs.Assumption 1 (Analytic feature functions). All the components of feature vector ϕ(•) are real analytic functions in R nx+nu ,4 i.e., for every 1 ≤ i ≤ n ϕ , ϕ i (x, u) is an infinitely differentiable function such that the Taylor expansion at every (x, ū) converges point-wisely to the ϕ i (x, u) in a neighborhood of (x, ū).Analytic functions include polynomial functions and trigonometric functions, which are important components of many physical systems in real-world applications, e.g. power systems, robotics, transportation systems, etc. In particular, we provide two illustrative examples below.Example 1 (Pendulum). Many multi-link robotic manipulators can be understood as interconnected pendulum dynamics. The motion equations of a single pendulum, consisting of a mass m suspended from a weightless rod of length l fixed at a pivot with no friction, can be expressed as:α = - g l sin(α) + u ml 2 + w,where α represents the angle of the rod relative to the vertical axis, g is the gravity constant, u is the torque input, and w is the disturbance applied to this system. After discretization the system dynamics can be rewritten in the structure of (1) with the feature vector consisting of expressions involving sin(α) and u, all of which are analytic functions. The matrix of unknown parameters contains terms of the pendulum's mass and the rod's length.Example 2 (Quadrotor (Alaimo et al., 2013) ). Let p ∈ R 3 and v ∈ R 3 represent the center of mass position and velocity of the quadrotor in the inertial frame, respectively; let ω ∈ R 3 denote its angular velocity in the body-fixed frame, and q ∈ R 4 denote the quaternion vector. The quadrotor's equations of motion can then be expressed as:d dt    p v q ω    =    v -ge z + 1 m Qf u 1 2 Ωq I -1 (τ u -ω × Iω)    + w,where g is the gravity constant, m is its total mass, I = diag(I xx , I yy , I zz ) its inertia matrix with respect to the body-fixed frame, f u ∈ R the total thrust, τ u ∈ R 3 the total moment in the body-fixed frame, e z = (0, 0, 1) ⊺ , Q =   q 2 0 + q 2 1 -q 2 2 -q 2 3 2(q 1 q 2 -q 0 q 3 ) 2(q 0 q 2 -q 1 q 3 ) 2(q 1 q 2 -q 0 q 3 ) q 2 0 -q 2 1 + q 2 2 -q 2 3 2(q 2 q 3 -q 0 q 1 ) 2(q 1 q 3 -q 0 q 2 ) 2(q 0 q 1 -q 2 q 3 ) q 2 0 -q 2 1 -q 2 2 + q 2 3   , andΩ =    0 -ω 1 -ω 2 -ω 3 ω 1 0 ω 3 -ω 2 ω 2 -ω 3 0 ω 1 ω 3 ω 2 -ω 1 0   .Similar to the pendulum example, after discretization the system dynamics can be rewritten in the structure of (1) with the feature vector consisting of cubic polynomials in states and inputs, which are real-analytic. The unknown parameters contain terms of the mass and inertial moments of the quadrotor.Next, we introduce the assumption on the random inputs, which relies on the following definition.Definition 1 (Semi-continuous distribution). We define a probability distribution P as semi-continuous if there does not exist a set E with Lebesgue measure zero such that P(E) = 1.The semi-continuous distribution is a weaker requirement than continuous distributions. In particular, any continuous distributions, or a mixture distribution with one component as a continuous distribution can satisfy the requirement of semi-continuity. The semi-continuity can also be interpreted by the Lebesgue Decomposition Theorem (Chapter 6 of (Halmos, 2013)) as discussed below.Remark 1 (Connection with Lebesgue Decomposition Theorem). Definition 1 can be interpreted by the Lebesgue Decomposition Theorem, which suggests that any probability distribution can be decomposed into a purely atomic component and a non-atomic component (see more details in Halmos ( 2013)). A semi-continuous distribution as defined in Definition 1 requires the distribution's non-atomic component to be nonzero.In the following, we provide the assumptions on w t and u t using the semi-continuity definition.Assumption 2 (Bounded i.i.d. and semi-continuous disturbance). w t is i.i.d. following a semicontinuous distribution with zero mean and a positive definite covariance matrix Σ w ⪰ σ 2 w I nx ≻ 0 and a bounded support, i.e. ∥w t ∥ ∞ ≤ w max almost surely for all t.The i.i.d. assumption is common in the literature of system identification for linear and nonlinear systems. As for the bounded assumption on w t , though being stronger than the sub-Gaussian assumption on w t in the literature of linear system estimation, it is a common assumption in the literature of nonlinear system estimation (Mania et al., 2022; Shi et al., 2021; Kim and Lavaei, 2024) . Further, in many physical applications, noises are usually bounded, e.g. the wind disturbances in quadrotor systems are bounded, the renewable energy injections in power systems are also bounded, etc.The semi-continuity assumption may seem restrictive, since it rules out the discrete distributions. However, the disturbances in many realistic systems can satisfy the semi-continuity because realistic noises are usually generated from a mixture distribution where at least one component is continuous, e.g. the wind disturbances and renewable generations are continuous.As for the control inputs u t , we first impose the same assumption as Assumption 2 for simplicity. Later in Section 3, we will also discuss the relaxation of this assumption to include control policies. Assumption 3 (Bounded i.i.d. and semi-continuous inputs). u t is i.i.d. following a semi-continuous distribution with zero mean and a positive definite covariance Σ u ⪰ σ 2 u I nx ≻ 0 and bounded support, i.e. ∥u t ∥ ∞ ≤ u max almost surely for all t.Lastly, we introduce our stability assumption based on the input-to-state stability definition below. Definition 2 (Locally input-to-state stability (LISS)). Consider the general nonlinear system x t+1 = f (x t , d t ) with x t ∈ R nx , d t ∈ R n d , f being a continuous function such that f (0, 0) = 0. This system is called locally input-to-state stable (LISS) if there exist constants ρ x > 0, ρ > 0 and functions γ ∈ K, β ∈ KL such that for all x 0 ∈ {x 0 ∈ R nx : ∥x 0 ∥ 2 ≤ ρ x } and any input d t ∈ {d ∈ R n d : sup t ∥d t ∥ ∞ ≤ ρ} , it holds that ∥x t ∥ 2 ≤ β ∥x 0 ∥ 2 , t + γ sup t ∥d t ∥ ∞ for all t ≥ 0.5 Assumption 4 (LISS system). System (1) is LISS with parameters ρ x and ρ such that ρ x ≥ ∥x 0 ∥ 2 and ρ ≥ max(w max , u max ), respectively. Assumption 4 is imposed, together with the bounded disturbances and inputs in Assumptions 2 and 3, to guarantee bounded states during the control dynamics (for instance, see the proof of Theorem 1in Appendix A). Notably, many studies on learning-based nonlinear control require certain boundedness on the states for theoretical analysis Sattar and Oymak (2022) ; Foster et al. (2020) ; Li et al. (2023a) .In addition, it is interesting to note that this paper only requires local stability of the dynamics, whereas several learning-based nonlinear control papers assume certain global properties, such as global exponential stability in (Foster et al., 2020) , global exponential incremental stability in (Sattar and Oymak, 2022; Li et al., 2023a; Lin et al., 2024) , or global Lipschitz smoothness in (Lee et al., 2024). 6 This difference in the dynamics assumption reflects a trade-off with the disturbance assumptions: we assume a stronger assumption on the boundedness of disturbances and a weaker assumption on local stability, whereas much of the literature considers (sub)Gaussian distributions (which can be unbounded) but requires stronger global properties for dynamics. Since this paper is largely motivated by physical systems, which typically encounter bounded disturbances/inputs and generally only satisfy local stability (Slotine and Li, 1991) , we address this trade-off through our current set of assumptions, leaving it as an exciting future direction to consider relaxing these assumptions.Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes] Justification: We discuss the limitations of this work specifically in Section 5.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
main results: 
 In this section, we provide the estimation error bounds of LSE for linearly parameterized nonlinear systems under i.i.d. random inputs. The estimation error bounds rely on the establishment of probabilistic persistent excitation, which will be introduced in the first subsection. Later, we also generalize the results to include control policies and discuss the convergence rate of another popular uncertainty quantification method in the control literature, set membership estimation, whose formal definition is deferred to the corresponding subsection.It is well-known that persistent excitation (PE) is a crucial condition for successful system identification (Narendra and Annaswamy, 1987) . In the following, we introduce the persistent excitation condition for our linearly parameterized nonlinear systems. Definition 3 (Persistent excitation (Skantze et al., 2000; Sastry and Bodson, 2011) ). System (1) is persistently excited if there exist s > 0 and m ≥ 1 such that for any t 0 ≥ 0, we have1 m t0+m-1 t=t0 ϕ x t , u t ϕ ⊺ x t , u t ⪰ s 2 I n ϕ .In the stochastic setting, PE is closely related with a block-martingale small-ball (BMSB) condition proposed in Simchowitz et al. (2018) , which can be viewed as a probabilistic version of PE. Definition 4 (BMSB (Simchowitz et al., 2018) ). Let {F t } t≥1 denote a filtration and let {y t } t≥1 be an {F t } t≥1 -adapted random process taking values in R ny . We say {y t } t≥1 satisfies the (k, Γ sb , p)block martingale small-ball (BMSB) condition for a positive integer k, a Γ sb ≻ 0, and a p ∈[0, 1], if for any fixed v ∈ R ny such that ∥v∥ 2 = 1, the process {y t } t≥1 satisfies 1 k k i=1 P |v ⊺ y t+i | ≥ √ v ⊺ Γ sb v | F t ≥ p almost surely for any t ≥ 1.One major contribution of this paper is formally establishing the BMSB condition for linearly parameterized nonlinear systems with real-analytic feature functions.In the following, we first investigate the open-loop system with i.i.d. inputs and later extend the results to the closed-loop systems with inputs u t = π(x t ) + η t , where η t represents the noise and π : R nx → R nu denotes a control policy. The following theorem considers the open-loop systems.Theorem 1 (BMSB for open-loop systems). Let u t = η t and consider the filtrationF t = F(w 0 , • • • , w t-1 , x 0 , • • • , x t , η 0 , • • • , η t ).Suppose Assumptions 1, 2, 3, 4 hold, then there exist s ϕ > 0 and p ϕ ∈ (0, 1) such that the {F t } t≥1 -adapted process ϕ x t , u t t≥1 satisfies the 1, s 2 ϕ I n ϕ , p ϕ -BMSB condition.Proof Sketch. Intuitively, BMSB requires that any linear combination of feature functions remains positive with a non-vanishing probability. Notice that a linear combination of real-analytic functions is itself real-analytic, and the zeros of an analytic function have measure zero. These facts allow us to show that the probability of a linear combination of linearly independent feature functions equaling zero is less than one, as long as the noises follow semi-continuous distributions, by the connection of the Lebesgue measure and the probability measure in Definition 1.In more detail, the proof leverages a variant of the Paley-Zygmund argument (Petrov, 2007) , which provides a lower bound for the tail properties of positive random variables. Specifically, it states that the probability of a positive random variable being small depends on the ratio of its even moments.We apply this result to the random variable |v T ϕ x t+1 , u t+1 | F t | with ∥v∥ 2 = 1 and aim to show that the lower bound is non-trivial for any direction v with ∥v∥ 2 = 1 and any filtration F t , t ≥ 0. We then use results from measure theory to demonstrate the existence of such a non-trivial lower bound. This is done by showing that the Lebesgue measure of the set where |v T ϕ x t+1 , u t+1 | = 0 is zero, and thus the even moments of |v T ϕ x t+1 , u t+1 | F t | are non-zero, provided that the noise and disturbance distributions are semi-continuous. For further details, please refer to Appendix A.It is worth pointing out that Theorem 1 only establishes the existence of the constants (s ϕ , p ϕ ), and deriving explicit formulas of these constants are left for future work. In particular, it can be challenging to derive a generic formula for all linearly parameterized nonlinear systems, but an exciting direction is to study reasonable sub-classes of systems and construct their corresponding formulas of the constants (s ϕ , p ϕ ).We are now prepared to present the non-asymptotic convergence rate for the LSE in learning the unknown parameters of the system (1).Theorem 2 (LSE's convergence rate for open-loop systems). Consider the dynamical system described in (1) with i.i.d. inputs u t = η t and assume that Assumptions 1, 2, 3, 4 are satisfied. Let s ϕ and p ϕ be as defined in Theorem 1, and define bϕ = sup t≥0 E ∥ϕ(z t )∥ 2 2 . For a fixed δ ∈ (0, 1) andT ≥ 1, if T satisfies the condition T ≥ 10 p ϕ log 1 δ + 2n ϕ log 10 p ϕ + n ϕ log bϕ δs 2 ϕ ,then LSE's estimation θT satisfies the following error bound with probability at least 1 -3δ.θT -θ * 2 ≤ 90σ w p ϕ n x + log 1 δ + n ϕ log 10 p ϕ + n ϕ log bϕ δs 2 ϕ T s 2 ϕ .The proof relies on Theorem 1 and Theorem 2.4 in (Simchowitz et al., 2018) . The complete proof is provided in Appendix B.1.Theorem 2 demonstrates that LSE converges to the true parameters under random control inputs and random disturbances (non-active exploration) at a rate of 1 √ T for linearly parameterized nonlinear systems. This is consistent with the convergence rates of LSE for linear systems in terms of T .Regarding the dimension dependence in the convergence rate, the explicit dependence is√ n x + n ϕ ,where n x and n ϕ refer to the dimensions of the state and the feature vector, respectively. Besides, it is worth mentioning that other parameters, such as s ϕ , p ϕ , bϕ , may implicitly depend on the dimensions as well. For some special systems, such as bilinear systems, it has been shown that these constants are independent of the dimensions (Sattar et al., 2022) . It is left as future work to explore other nonlinear systems' implicit dimension dependence.Next, we can generalize the i.i.d. inputs u t to include control policies, i.e., u t = π(x t ) + η t , where η t satisfies Assumption 3 and π(x t ) is analytic.Corollary 1 (LSE's convergence rate for closed-loop systems). Consider inputs u t = π(x t ) + η t , where π(•) is real-analytic, η t satisfies Assumption 3, and the closed-loop system x t+1 = θ * ϕ(x t , π(x t ) + η t ) + w t satisfies Assumption 4 for both w t and η t . Then, the same convergence rate in Theorem 2 holds.The proof is provided in Appendix B.2.Set membership estimation (SME) is another popular method for uncertainty quantification in control system estimation (Lu and Cannon, 2023; Bertsekas, 1971; Li et al., 2024) . Unlike LSE, SME is a set-estimator and directly estimates the uncertainty set. Since the analysis of SME also relies on the probabilistic persistent excitation analysis, we can also establish the convergence rate of SME for linearly parameterized nonlinear systems under i.i.d. noises in the following. In particular, SME estimates the uncertainty set asEQUATIONwhere W is a bounded set such that w t ∈ W for all t ≥ 0.The convergence of SME relies on an additional assumption as shown below: the tightness of the bound W on w t 's support. This tightness assumption is commonly considered in SME's literature (Li et al., 2024; Lu et al., 2019; Akçay, 2004) . Further, (Li et al., 2024) discusses the relaxation of this assumption by learning a tight bound at the same time of learning the uncertainty set of θ * for linear systems. Similar tricks can be applied to nonlinear systems, but this paper only considers the vanilla case of SME for simplicity.Assumption 5 (Tight bound on disturbances). Assume for any ϵ > 0, there exists q w (ϵ) > 0, such that for any 1 ≤ j ≤ n and t ≥ 0, we have P(w j t + w max ≤ ϵ) ≥ q w (ϵ) > 0, P(w max -w j t ≤ ϵ) ≥ q w (ϵ) > 0.Assumption 5 requires that w t can visit set W's boundary arbitrarily closely with a positive probability. For example, for a one-dimensional w t bounded by -w max ≤ w t ≤ w max , Assumption 5 requires that there is a positive probability that w t is close to w max and -w max , i.e., for any ϵ > 0, we have P(w max -ϵ ≤ w t ≤ w max ) > 0 and P(-w max ≤ w t ≤ -w max + ϵ) > 0.Next, we state a non-asymptotic bound on the diameter of the uncertainty set estimated by SME. Theorem 3 (SME's diameter bound for open-loop systems). Consider system (1) with i.i.d. inputs u t = η t . Suppose Assumptions 1, 2, 3, 4 are satisfied. Consider s ϕ and p ϕ defined in Theorem 1 and let b ϕ = sup t≥0 ∥ϕ(z t )∥ 2 . For any m ≥ 0 and δ ∈ (0, 1), when T > m, we haveP diam(Θ T ) > δ ≤ T m Õ n 2.5 ϕ a n ϕ 2 exp(-a 3 m) + Õ n 2.5 x n 2.5 ϕ a nxn ϕ 4 1 -q w a 1 δ 4 √ n x T m, where a 1 =s ϕ p ϕ 4 , a 2 = 64b 2 ϕ s 2 ϕ p 2 ϕ , a 3 = p 2 ϕ 8 , a 4 = 16b ϕ √ nx s ϕ p ϕ . The constants hidden in Õ are provided in the Appendix C.1.The proof of Theorem 3 relies on Theorem 1 in this paper and Theorem 1 from (Li et al., 2024) . The detailed proof is provided in Appendix C.1.Theorem 3 establishes an upper bound on the "failure" probability of SME, i.e., the probability that the uncertainty set's diameter exceeds δ. To ensure the failure probability is less than 1, one can select m = O(log(T )) and choose a sufficiently large T such that T ≥ m = O(log(T )). If w t is more likely to visit the boundaries of the set W (meaning a larger q(ℓ)), SME is less likely to estimate an uncertainty set with a diameter greater than δ.To provide more intuitions on the diameter bound in Theorem 3, we consider q w (ℓ) = c w ℓ for some c w > 0. Note that several common distributions, including the uniform distribution and the truncated Gaussian distribution, satisfy this property on q w (ℓ) (see Appendix C.2 for explicit formulas of c w ). With q w (ℓ) = c w ℓ, we can provide a convergence rate of SME in terms of T in the following. Corollary 2 (SME's convergence rate when q w (ℓ) = c w ℓ ). For any ϵ > 0, letm ≥ O log T ϵ + n ϕ log 8b ϕ s ϕ p ϕ p 2 ϕ .If w t 's distribution satisfies q w (ℓ) = c w ℓ for all ℓ > 0, then with probability at least 1 -2ϵ, we have:diam(Θ T ) ≤ O √ n x log 1 ϵ + n 1.5 x n ϕ log b ϕ nx s ϕ p ϕ c w s ϕ p ϕ T ,where the constants hidden in O( Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Answer: [Yes] Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?Answer: [Yes] Justification: We provide the codes as part of our supplementary material. The reader can also refer to our source available at https://github.com/NeginMusavi/ real-analytic-nonlinear-sys-id.Guidelines:• The answer NA means that paper does not include experiments requiring code. • Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. 
numerical experiments: 
 In this section, we evaluate the performance of LSE in estimating the unknown parameter θ * and SME in estimating the uncertainty set for the unknown parameters using the pendulum and quadrotor examples outlined in Section 2. We compare the empirical convergence rates of LSE and SME with the theoretical rates in Theorem 2 and Corollary 2. In each case, the input u t is composed of a control policy and i.i.d. noise, such that u t = π(x t ) + η t . For our experiments, we employ noise and disturbances drawn from uniform and truncated-Gaussian distributions. To compute theoretical rates, we numerically estimate parameters such as s ϕ and p ϕ (see Appendix E). Further details can be found in our source code.7 The details for these scenarios are outlined below:• Pendulum 1: In the pendulum example described in Section 2, the control input is u t = -k αt + η t . This scenario includes two unknown parameters: θ 1 = 1 l and θ 2 = 1 ml 2 . • Quadrotor 2: For the quadrotor example in Section 2, the control input is defined as u t = π(x t ) + η t , where π(x t ) follows the controller proposed by Alaimo et al. (2013) . The quadrotor system involves 13 states and 4 inputs, with the unknown parameter matrix θ * containing 7 parameters, including the mass m and specific elements of the inertia matrix I.Further details on controller gains and unknown parameters are provided in Appendix D. ), which is consistent with the results from Corollary 2 and with the related results for linear systems in (Li et al., 2024) . A similar result can be observed for the quadrotor example, in Figures 2c and 2d . Additionally, Figure 3 shows the uncertainty sets estimated by SME for the two unknown parameters, labeled θ 1 and θ 2 , in the pendulum example, along with the diameters of these sets as trajectory length grows. We observe that these sets contract as trajectory length increases, with the true values of the unknown parameters lying within the estimated uncertainty sets. The illustration of uncertainty sets for the quadrotor example is provided in Appendix D.2.  
concluding remarks: 
 Conclusion. This study examines the probabilistic persistent excitation in a class of nonlinear systems influenced by i.i.d. noise and stochastic disturbances, with the stipulation that their distributions do not concentrate on sets of Lebesgue measure zero. Based on this we then present an explicit bound on the convergence rate of SME estimations and LSE estimations for this class of dynamical systems. Additionally, numerical experiments in the context of robotics are provided to illustrate both methods.Limitations. One limitation of this work is that our analysis relies on a specific class of i.i.d. noises and stochastic disturbances, where the probability distribution is not concentrated on sets of Lebesgue measure zero. While this is a sufficient condition, it is possible that the BMSB conditions are satisfied under other circumstances. Another limitation is that, though we provide sufficient conditions for the existence of parameters satisfying the BMSB condition, the explicit dependence is not detailed here. Lastly, imperfect observations are not considered here.Future Work. Our future work includes several promising directions, e.g., to explore cases that do not satisfy our semi-continuity assumption, such as discrete noises, and to investigate the explicit dependence of the BMSB parameter on system attributes, such as state, input, and feature dimensions, etc. Furthermore, extending this work to imperfect state observations is an important next step. Finally, a potential direction is to provide a non-asymptotic analysis of the volumes of uncertainty sets estimated by SME uncertainty sets, as opposed to the current focus on their diameters.• Appendix A provides a proof of Theorem 1.• Appendix B provides proofs of Theorem 2 and Corollary 1.• Appendix C presents a proof of Theorem 3 and Corollaries 2 and 3.• Appendix D provides more details of the simulation settings.• Appendix E discusses the numerical estimation of the BMSB parameters (s ϕ , p ϕ ) in Theorem 1.• The NeurIPS Paper Checklist is provided after the appendices.A Proof Theorem 1Proof. Given that u t = η t and satisfies the conditions in Assumption 3, u t is bounded, meaning u t ∈ U, where U is a compact set. Moreover, since the system is LISS, there exist functions γ ∈ K and β ∈ KL, such that for all t ≥ 0, the following holds:x t ∈ X = x ∈ R n : ∥x∥ 2 ≤ β(ρ x , 0) + γ(ρ)with parameters ρ x ≥ ∥x 0 ∥ 2 and ρ ≥ max(w max , u max ). Let Z = X × U, then z t ∈ Z for all t ≥ 0.The set Z is a compact subset of R nx+nu .To show that the {F t } t≥1 -adapted process {ϕ(z t )} t≥1 satisfies the BMSB condition, it is sufficient to demonstrate that there exist s ϕ > 0 and p ϕ ∈ (0, 1) such that for all t ≥ 0 and for any v ∈ R n ϕ with ∥v∥ 2 = 1, the following holds:EQUATIONTo establish this, we apply the Paley-Zygmund inequality, which gives a lower bound on the tail probability of a non-negative random variable:Lemma 1. (Paley-Zygmund (Petrov, 2007)) Let x be a non-negative random variable. Then for any r ∈ (0, 1), the following holds:P x > r E[x 2 ] ≥ (1 -r 2 ) 2 E[x 2 ] 2 E[x 4 ].Based on this result, for any r ∈ (0, 1), we have:P v ⊺ ϕ(z t+1 ) > r E v ⊺ ϕ(z t+1 ) 2 F t F t ≥ (1 -r 2 ) 2 E v ⊺ ϕ(z t+1 ) 2 F t 2 E v ⊺ ϕ(z t+1 ) 4 F t . (5) Let V = {v ∈ R n ϕ : ∥v∥ 2 = 1}.To show that the BMSB condition holds, it is sufficient to establish the following two points:• inf Ft, t≥0 inf v∈V E v ⊺ ϕ(z t+1 ) 2 F t > 0,• and supFt, t≥0 sup v∈V E v ⊺ ϕ(z t+1 ) 4 F t < ∞.These conditions ensure that the {F t } t≥1 -adapted process {ϕ(z t )} t≥1 satisfies the BMSB condition with some constants s ϕ > 0 and p ϕ ∈ (0, 1). We will divide the proof into two parts:Step 1. Showing that infFt, t≥0 inf v∈V E v ⊺ ϕ(z t+1 ) 2 F t > 0:We begin by noting the following:inf Ft, t≥0 inf v∈V E v ⊺ ϕ(z t+1 ) 2 F t = inf Ft, t≥0 inf v∈V E v ⊺ ϕ x t+1 , u t+1 2 F t = inf Ft, t≥0 inf v∈V E v ⊺ ϕ θ * ϕ(z t ) + w t , u t+1 2 F t .Since z t ∈ F t while w t , u t+1 ̸ ∈ F t , we can treat z t as a constant and w t , u t+1 = η t+1 as random variables. From the continuity of features ϕ(•), we can conclude that:inf Ft, t≥0 inf v∈V E v ⊺ ϕ(z t+1 ) 2 F t = inf z∈Z inf v∈V E v ⊺ ϕ θ * ϕ(z) + w =: h(z,w) , η 2 ,where w, η are independent random variables, as assumed in Assumptions 2 and 3. Now, letN z v = (w, η) ∈ W × U : v ⊺ ϕ h(z, w), η = 0 , and we have:E v ⊺ ϕ h(z, w), η 2 = E v ⊺ ϕ h(z, w), η 2 1 v ⊺ ϕ h(z, w), η = 0 =0 + E v ⊺ ϕ h(z, w), η 2 1 v ⊺ ϕ h(z, w), η ̸ = 0 = E v ⊺ ϕ h(z, w), η 2 (w, η) ̸ ∈ N z v P (w, η) ̸ ∈ N z v = E v ⊺ ϕ h(z, w), η 2 (w, η) ̸ ∈ N z v 1 -P (w, η) ∈ N z v .Therefore, we have:inf z∈Z inf v∈V E v ⊺ ϕ h(z, w), η 2 = inf z∈Z inf v∈V E v ⊺ ϕ h(z, w), η 2 (w, η) ̸ ∈ N z v × 1 -sup z∈Z sup v∈V P (w, η) ∈ N z v . (6) It is evident that if N z v = ∅, then inf z∈Z inf v∈V E v ⊺ ϕ h(z, w), η 2 ̸ = 0, and sup z∈Z sup v∈V P (w, η) ∈ N z v = 0, leading to inf z∈Z inf v∈V E v ⊺ ϕ h(z, w), η 2 > 0.Now we proceed with the case where N z v ̸ = ∅. For this, we can use the following lemma concerning the zero set of real-analytic functions in terms of Lebesgue measure.Lemma 2 (The zero set of real-analytic functions (Crȃciun and Ghoshdastidar, 2024) ). The set of zeros of a non-trivial real-analytic function f : R n → R has a Lebesgue measure zero in R n . This is a known result and can be proved using the identity theorem along with Fubini's theorem. For further information on this topic, see sources such as (Krantz and Parks, 2002; Bogachev and Ruas, 2007) .h(z, w) = θ * ϕ(z) + w. Notice that h(•, •) is real-analytic. Now consider v ⊺ ϕ h(z, w), η = n ϕ i=1 v i ϕ i h(z, w), η ,where ϕ i h(z, w), η are linearly independent. Hence, the sum n ϕ i=1 v i ϕ i h(z, w), η ̸ ≡ 0 for any v ∈ V. This implies that v ⊺ ϕ h(z, w), η is real-analytic and non-zero. Consequently, by Lemma 2, λ nx+nu (N z v ) = 0 for any v ∈ V. Under Assumptions 3 and 2, there cannot exist a set E ⊂ Z of Lebesgue measure zero in R nx+nu for which the P (w, η) ∈ E = 1. Taking this into account, along with the fact that λ nx+nu (N z v ) = 0 and that the sets V and Z are closed sets (implying they include all their limit points), we can conclude thatsup z∈Z sup v∈V P (w, η) ∈ N z v ̸ = 1. Moreover, since λ nx+nu (N z v ) = 0 and λ nx+nu (W × U) ̸ = 0, it follows that (N z v ) c ̸ = ∅. This implies inf z∈Z inf v∈V E v ⊺ ϕ h(z, w), η 2 (w, η) ̸ ∈ N z v ̸ = 0.Substituting these results into (6), we obtain:inf Ft, t≥0 inf v∈V E v ⊺ ϕ(z t+1 ) 2 F t > 0.Step 2. Showing that supFt, t≥0sup v∈V E v ⊺ ϕ(z t+1 ) 4 F t < ∞:Since z t ∈ Z for t ≥ 0, and considering that the noise and disturbances are bounded while the features are real-analytic, it follows that z t+1 |F t is a bounded random variable. Consequently, v ⊺ ϕ(z t+1 )|F t is also bounded. Given that both Z and V are compact sets-meaning they contain all their limit points-and that any random variable with bounded support has finite moments, we conclude that supFt, t≥0 sup v∈V E v ⊺ ϕ(z t+1 ) 4 F t < ∞.We finalize the proof by combining the results from Step 1 and Step 2.Proof. The proof hinges on the following key meta-theorem about the LSE convergence rate:Theorem 4 (LSE meta-theorem (Simchowitz et al., 2018) ). Fix δ ∈ (0, 1), T ≥ 1, and 0 ≺ Γ sb ≺ Γ. Consider a random process {(y t , x t )} t≥1 ∈ (R ny × R nx ) T , and a filtration {F t } t≥1 . Suppose the following conditions hold:• x t = θ * y t + w t , where w t |F t is a zero mean σ 2 w -sub-Gaussian, • {y t } t≥1 is an {F t } t≥1 -adapted random process satisfying the (k, Γ sb , p)-BMSB condition, • P T t=1 y t y ⊺ t ̸ ⪯ T Γ ≤ δ.If the trajectory length T satisfiesT ≥ 10k p 2 log 1 δ + log det ΓΓ -1 sb + 2n y log 10 p ,then with probability at least 1 -3δ, LSE estimation error is bounded by:θT -θ * 2 ≤ 90σ w p n x + log 1 δ + log det ΓΓ -1 sb + n y log 10 p T σ min (Γ sb ) .Since w t satisfies the Assumption 2), and w t / ∈ F t , then σw|F t is sub-Gaussian with parameter σw. Additionally, system (1) is linear in unknown parameters θ * . From Theorem 1, the {F t } t≥1 -adapted process {ϕ(z t )} t≥1 satisfies the (1, s 2 ϕ I n ϕ , p ϕ )-BMSB condition for some s ϕ > 0 and p ϕ ∈ (0, 1]. To complete the proof, it is left to show that for any δ ∈ (0, 1), there exists a Γ ≻ s 2 ϕ I n ϕ , such thatP T t=1 ϕ(z t )ϕ ⊺ (z t ) ⪯̸ T Γ ≤ δ.To see this, note that for bϕ = sup t≥0 E ∥ϕ(z t )∥ 2 2 , we have:P T t=1 ϕ(z t )ϕ ⊺ (z t ) ⪯̸ bϕ T δ I n ϕ = P λ max T t=1 ϕ(z t )ϕ ⊺ (z t ) ≻ λ max bϕ T δ I n ϕ = P T t=1 ϕ(z t )ϕ ⊺ (z t ) 2 ≻ bϕ T δ ≤ δE T t=1 ϕ(z t )ϕ ⊺ (z t ) 2 bϕ T .In addition, we have:E T t=1 ϕ(z t )ϕ ⊺ (z t ) 2 ≤ T t=1 E ϕ(z t )ϕ ⊺ (z t ) 2 ≤ T t=1 E ∥ϕ(z t )∥ 2 2 ≤ T sup t≥0 E ∥ϕ(z t )∥ 2 2 ,which implies that:EQUATIONwhere Γ = bϕ δ I n ϕ . since z t ∈ Z for all t ≥ 0, with Z being a compact set (due to the system's LISS property and features ϕ(•) being real-analytic), such a bounded bϕ exists, completing the proof.Proof. We start the proof by stating the following lemma which is extension of Theorem 1 to the case with u t = π(x t ) + η t .Lemma 3. Let u t = π(x t )+η t , π(•) is real-analytic, η t satisfies Assumption 3. Consider the filtrationF t = F(w 0 , • • • , w t-1 , x 0 , • • • , x t , π(x 0 ), • • • , π(x t ), η 0 , • • • , η t ).Suppose Assumptions 1, 2 hold and that the closed-loop system x t+1 = θ * ϕ(x t , π(x t ) + η t ) + w t satisfies Assumption 4 for both w t and η t . Then there exist s ϕ > 0, and p ϕ ∈ (0, 1) such that the {F t } t≥1 -adapted process ϕ x t , u t t≥1 satisfies the 1, s 2 ϕ I n ϕ , p ϕ -BMSB condition.Proof of Lemma 3. The proof closely follows the steps of Theorem 1. First, observe that u t = π(x t ) + η t . Since η t satisfies the conditions outlined in Assumption 3, and the closed-loop system x t+1 = θ * ϕ(x t , π(x t ) + η t ) + w t adheres to Assumption 4 with respect to both w t and η t , there exist functions γ ∈ K and β ∈ KL such that for all t ≥ 0 the following holds:x t ∈ X = x ∈ R n : ∥x∥ 2 ≤ β(ρ x , 0) + γ(ρ)with parameters ρ x ≥ ∥x 0 ∥ 2 and ρ ≥ max(w max , u max ). Let Z = X × U, where U is a compact set containing u t for all t ≥ 0. Thus, z t ∈ Z for all t ≥ 0. The set Z is a compact subset of R nx+nu . The remaining part of the proof, specifically to show that supFt, t≥0 sup v∈V E v ⊺ ϕ(z t+1 ) 4 F t < ∞follows similarly to the proof in Theorem 1.Ft, t≥0inf v∈V E v ⊺ ϕ(z t+1 ) 2 F t > 0.This can be shown as follows:inf Ft, t≥0 inf v∈V E v ⊺ ϕ(z t+1 ) 2 F t = inf Ft, t≥0 inf v∈V E v ⊺ ϕ x t+1 , u t+1 2 F t = inf Ft, t≥0 inf v∈V E v ⊺ ϕ x t+1 , π(x t+1 ) + η t+1 2 F t = inf Ft, t≥0 inf v∈V E v ⊺ ϕ θ * ϕ(z t ) + w t , π θ * ϕ(z t ) + w t + η t+1 2 F t .Since z t ∈ F t and w t , η t+1 ̸ ∈ F t then z t is treated as constant while w t , η t+1 are considered random variables. Then, by the continuity of ϕ we conclude thatinf Ft, t≥0 inf v∈V E v ⊺ ϕ(z t+1 ) 2 F t = inf z∈Z inf v∈V E v ⊺ ϕ θ * ϕ(z) + w =: h(z,w) , π θ * ϕ(z) + w =: h(z,w) + η 2 = inf z∈Z inf v∈V E v ⊺ ϕ h(z, w), π h(z, w) + η 2where w and η are independent random variables constrained as described in Assumptions 2 and 3. By lettingN z v = (w, η) ∈ W × U : v ⊺ ϕ h(z, w), π h(z, w) + η = 0 , similar to (6) we have:inf z∈Z inf v∈V E v ⊺ ϕ h(z, w),π h(z, w) + η 2 = inf z∈Z inf v∈V E v ⊺ ϕ h(z, w), π h(z, w) + η 2 (w, u) ̸ ∈ N z v × 1 -sup z∈Z sup v∈V P (w, η) ∈ N z v .We aim to show that λ nx+nu (N z v ) = 0 by applying Lemma 2. Note that ϕ(•), h(•), and π(•) are realanalytic. To use the results of Lemma 2, we need to establish that v ⊺ ϕ h(z, w), π h(z, w) + η is non-zero for any v ∈ V. First, observe that:v ⊺ ϕ h(z, w), π(h(z, w)) + η = n ϕ i=1 v i ϕ i h(z, w), π(h(z, w)) + η) .• All components of π h(z, w) are linearly independent with any component of h(z, w).• At least one component of π h(z, w) is linearly dependent with one or more components of h(z, w).In both cases, due to the additive nature of η, all the functions ϕ i h(z, w), π(h(z, w)) + η) with i = 1, • • • , n ϕ are linearly independent, ensuring that v ⊺ ϕ h(z, w), π(h(z, w)) + η ̸ ≡ 0 for any v ∈ V. The remainder of the proof follows similarly to the argument in Theorem 1.Using this Lemma, Theorem 4, and reasoning similar to that in the proof of Theorem 2, the proof can be completed.C Proofs for Theorem 3, Corollary 2, and Corollary 3C.1 Proof of Theorem 3Proof. The proof follows from applying the following meta-theorem on the convergence rate of SME.Theorem 5 (SME meta-theorem (Li et al., 2024) ). Consider a general time series model with linear responses as follows:x t = θ * y t + w t , t ≥ 0.Also, define the filtrationF t = {w 0 , • • • , w t , y 0 , • • • , y t }.Assume the following conditions are met:• w t are i.i.d. with variance σ 2 w I nx , and box-constrained, i.e., w t ∈ W = {w ∈ R nx : ∥w∥ ∞ ≤ w max } for some w max > 0.• {y t } t≥1 is an {F t } t≥1 -adapted random process satisfying the (k, s 2 y I ny , p y )-BMSB condition.• There exists b y > 0 such that ∥y t ∥ 2 ≤ b y almost surely for all t ≥ 0.• For any ℓ > 0, there exists q w (ℓ) > 0, such that for any 1 ≤ j ≤ n and t ≥ 0, we have P(w j t + w max ≤ ℓ) ≥ q w (ℓ) > 0, P(w max -w j t ≤ ℓ) ≥ q w (ℓ) > 0.Then for any m ≥ 1 and any δ ∈ (0, 1), when T > m, the diameter of the uncertainty setΘ T = T -1 t=0 θ : x t -θy t ∈ W satisfies: P diam(Θ T ) > δ ≤ 544 T m n 2.5 y log(a 2 n y )a ny 2 exp(-a 3 m) + 544n 2.5 x n 2.5 y log(a 4 n x n y )a nxny 4 1 -q w a 1 δ 4 √ n x ⌈T /m⌉ , where a 1 = sypy 4 , a 2 = 64b 2 y s 2 y p 2 y , a 3 = p 2 y 8 , a 4 = max 4by √ nx a1, 1 .Observe that system (1) is linear in the unknown parameters θ * , and we can prove Theorem 3 by showing that the {F t } t≥1 -adapted process {ϕ(z t )} t≥1 and w t meet the conditions of the metatheorem. By Assumptions 2 and 5, and since w t ̸ ∈ F t , the noise w t fulfills all the requirements of the meta-theorem. Moreover, according to Theorem 1, the {F t } t≥1 -adapted process ϕ(z t ) t≥1 satisfies the (1, s 2 ϕ I n ϕ , p ϕ )-BMSB condition for some s ϕ > 0 and p ϕ ∈ (0, 1]. Lastly, since the system is LISS, we have z t ∈ Z for all t ≥ 0, where Z is the compact set defined in the proof of Theorem 1. Therefore, there exists a constant b ϕ > 0 such that sup t≥0 ∥ϕ(z t )∥ 2 ≤ b ϕ , completing the proof of the theorem.Explicitly, this means that for any m ≥ 1, for any δ ∈ (0, 1), when T > m, we have:EQUATIONwherea 1 = s ϕ p ϕ 4 , a 2 = 64b 2 ϕ s 2 ϕ p 2 ϕ , a 3 = p 2 ϕ 8 , a 4 = 16b ϕ √ nx s ϕ p ϕ .Let us provide two example distributions, truncated Gaussian and uniform, along with their corresponding q w (•) (from (Li et al., 2024) ):• If w t follows a uniform distribution on [-w max , w max ] nx , then q w (ℓ) = c w ℓ with c w = 1 2wmax . • If w t follows a truncated-Gaussian distribution on [-w max , w max ] nx , generated by a Gaussian distribution with zero mean and covariance matrix σ 2 w I nx , then q w (ℓ) = c w ℓ withc w = 1 min( √ 2πσw,2wmax) exp( -w 2 max 2σ 2 w ).Now, fix ϵ ∈ (0, 1). We want to show that ifEQUATIONthen for all T ≥ m, we haveEQUATIONwith probability at least 1 -2ϵ.Let the two terms in right hand-side of (8) be denoted by "term 1" and "term 2". We proceed with the proof in two steps as follows:Step 1: showing that with choice of m in (9), term 1 ≤ ϵ :With this choice of m, it is straightforward to see that544T n 2.5 ϕ log(a 2 n ϕ )a n ϕ2 exp(-a 3 m) ≤ ϵ, and thus term 1 ≤ ϵ.Step 2: letting term 2 = ϵ and showing that δ satisfies the inequality in (10):Assuming without loss of generality that T m is an integer, note that term 2 = ϵ implies:q w a 1 δ 4 √ n x = 1 - ϵ 544n 2.5x n 2.5 ϕ log(a 4 n x n ϕ )anxn ϕ 4 m T ≤ -log ϵ 544n 2.5 x n 2.5 ϕ log(a 4 n x n ϕ )a nxn ϕ 4 m T = - m T log ϵ 544n 2.5 x n 2.5 ϕ log(a 4 n x n ϕ )a nxn ϕ 4 = m T log 1 ϵ + log(a 4 )n x n ϕ + 2.5 log(n x n ϕ ) + log log(a 4 n x n ϕ ) + log(544) . If q w a1δ 4 √ nx = c w a1δ 4√ nx for some constant c w > 0, then:δ ≤ 4 √ n x m c w a 1 T log 1 ϵ + log(a 4 )n x n ϕ + 2.5 log(n x n ϕ ) + log log(a 4 n x n ϕ ) + log(544) ≤ 16 √ n x m c w s ϕ p ϕ T O log 1 ϵ + n x n ϕ log 16b ϕ √ n x s ϕ p ϕ ≤ O √ n x log 1 ϵ + n 1.5 x n ϕ log b ϕ nx s ϕ p ϕ c w s ϕ p ϕ T .Combining these two steps, we conclude that, with probability at least 1 -2ϵ,diam(Θ T ) ≤ O √ n x log 1 ϵ + n 1.5 x n ϕ log b ϕ nx s ϕ p ϕ c w s ϕ p ϕ T .Figure 4 displays the uncertainty set estimated by SME for the seven unknown parameters in the quadrotor example for various trajectory lengths, with η t and w t being i.i.d. samples from truncated-Gaussian distributions. The uncertainty sets are observed to shrink as the trajectory length increases, consistent with our theoretical results. Note that the ground truth value is contained within all the uncertainty sets.E Numerical Estimation of BMSB Parameters (s ϕ , p ϕ )We compare the empirical rates of LSE and SME with their theoretical counterparts in Section 4. The theoretical results presented in Theorem 2 and Corollary 2 rely on the parameters b ϕ , bϕ , and the BMSB parameters (s ϕ , p ϕ ). However, the explicit relationship of these parameters with system, noise, and disturbance characteristics such as n x , n u , n ϕ , σ u , and σ w is not known and we will address this in our future work. Consequently, we estimate these parameters numerically and utilize these estimates to calculate the theoretical rates discussed in Section 4. While b ϕ and bϕ are straightforward to estimate, special attention is required to estimate the BMSB parameters. This section is dedicated to describing this estimation process.For this, consider a system of the form (1). For this system, our goal is to estimate s ϕ and p ϕ , wherep ϕ = inf Ft,t≥0 inf ∥v∥2=1 P |v T ϕ(z t+1 )| ≥ s ϕ F t numerically. First, observe that ϕ(z t+1 ) | F t = ϕ(θ T * ϕ(z t ) + w t , u t+1 ), where z t ∈ F t . This implies that ϕ(z t+1 ) | F t is a random variable influenced by w t and u t+1 . We proceed by fixing s ϕ = s (for some s > 0) and empirically estimate p ϕ . To accomplish this, we first select a time horizon T and generate several trajectories of length T for the system. Let D T represent the set of these trajectories, while D t denotes a subset containing all trajectories up to t ≤ T . Additionally, we create multiple vectors v ∈ R n ϕ such that ∥v∥ 2 = 1; we refer to this collection as V. These vectors are randomly sampled from a Gaussian distribution and subsequently normalized.We then estimate p as:p = min t∈[T ] min z∈D t min v∈ V P |v T ϕ(θ T * ϕ(z) + w t , u t+1 )| ≥ s .As T increases, along with the number of trajectories and vectors v, the minimum estimates will more accurately reflect the infimums. In this context, p represents the minimum of P |v T ϕ(θ T * ϕ(z) + w t , u t+1 )| ≥ s across all combinations in [T ] × D T × V. For each combination in this set, we estimate the probability P |v T ϕ(θ T * ϕ(z) + w t , u t+1 )| ≥ s using Monte Carlo simulations. This process entails generating multiple random samples based on the distributions of w t and u t+1 , verifying whether each sample satisfies the condition |v T ϕ(z)| ≥ s, and tallying how many samples meet this criterion. We repeat this procedure for various values of s until we identify a pair of (s, p) such that p ∈ (0, 1). The estimated probability is calculated as the ratio of the count of successful samples to the total number of samples. According to the law of large numbers, this ratio converges to the true probability as the number of samples increases. For our estimations, we select T = 50, | V| = 1000, and |D T | = 20. 
experimental setting/details: 
 Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?Answer: [Yes] Justification: We provide the details in Section 4 and Appendix D.Guidelines:• The answer NA means that the paper does not include experiments.• The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material. 
experiment statistical significance: 
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institutional review board (irb) approvals or equivalent for research with human subjects: 
 Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?Answer: [NA] Justification: This is a theoretical paper.Guidelines:• The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.• Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. • We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. • For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.yt = f * (xt) + wt is considered, where xt and yt correlate with the historical data.For example,(Ziemann and Tu, 2022) assumes hyper-contractivity, and(Ziemann et al., 2024) assumes the empirical covariance of the {xt} t≥0 process is invertible with high probability (Corollary3.2).If the features are not independent, they can be converted to independent ones since the features are known.This assumption can be relaxed to locally analytic functions in a large enough bounded set.A function γ : R ≥0 → R ≥0 is a K function if it is continuous, strictly increasing and γ(0) = 0. A function β : R ≥0 × R ≥0 → R ≥0 is a KL function if, for each fixed t ≥ 0, the function β(•, t) is a K function, and for each fixed s ≥ 0, the function β(s, •) is decreasing and β(s, t) → 0 as t → ∞.Global Lipschitz smoothness may exclude system dynamics with higher-order polynomials.https://github.com/NeginMusavi/real-analytic-nonlinear-sys-id 
 
 Appendix:  


Decision: This paper extends important results from linear and bi-linear system identification to a broader class of nonlinear systems by providing non-asymptotic guarantees for both least-squares estimators (LSE) and set-membership estimators (SME) within a framework that incorporates analytic functions. The work is rigorous, original, and well-presented, with reviewers particularly appreciating the clarity and significance of the theoretical contributions. While some concerns were raised regarding the restrictive nature of certain assumptions and the conservative nature of the theoretical bounds, the overall impact and novelty of the work still hold. Therefore, I recommend acceptance.

Reviews: review: paper_summary: paper_summary: The authors extend the work of Simchowitz et al. (2018) [linear] and Sattar et al. (2022) [bi-linear] to linear in the parameters but analytic nonlinear features showing that PE of inputs still results in PE of the states which for general nonlinear systems is not true. From this result LSE results like those originally in Simchowitz et al. (2018) are recovered for this new class of systems and SME results like those in Li et al. (2023a) are recovered for this broader class of systems.
strength_weakness: strength_weakness: - the biggest weakness of the paper is that very little time is spent on the analysis in the main text. yes their is a proof sketch for the main theorem, but it would be nice to get more intuition from the main text.
questions: questions: - I would much prefer the simulations in the appendix and the inclusion of the entirety of the proof of the main theorem in the main text with clear exposition of how the properties of analytic functions can be exploited to prove PE of u results in PE of x for these systems. This paper is not as bad as others where the main text is only an advertisement of the main result, but it would be a much tighter and more useful paper if this was done. This reviewer may be an outlier though as it seems the trend in these conferences is to do just the opposite of what I have suggested.
limitations: limitations: limitations are discussed
paper_summary: paper_summary: The authors study the problem of system identification from a trajectory generated by an unknown linearly parametrized nonlinear system whose nonlinearity is an analytic function. Specifically, they theoretically analyze to estimators: the least squares estimator, and the set membership estimator. Both estimators, while being widely used in practice, do not have theoretical guarantees in many of the settings in which they are applied. The authors give such guarantees, and also conduct numerical experiments on certain nonlinear systems.
strength_weakness: strength_weakness: While the theoretical arguments presented seem sound, I am not sure that the extension of system identification results to the case of linearly parametrized smooth nonlinear systems is sufficiently significant from a theoreitcal perspective.
questions: questions: If instead of assuming that the nonlinearity is analytic, one assumed directly that there does not exist an open set on which it vanishes identically, would the proof carry through?
limitations: limitations: The authors adequately adress the limitations of their work.
paper_summary: paper_summary: The manuscript provides theoretical guarantees for nonlinear system identification using non-active i.i.d. noises, extending from linear systems to linearly parametrized nonlinear systems with analytic feature functions.  The findings demonstrate that non-active i.i.d. noises are capable of efficiently learning these systems with a non-asymptotic convergence rate.
strength_weakness: strength_weakness: Although the authors present an interesting idea, the manuscript could be improved by addressing the following concerns: 1. It is suggested to discuss the potential limitations of the non-asymptotic convergence rate in practical applications. 2. The analysis of the numerical experiments lacks depth. Could the authors further explore the convergence characteristics and the advantages and disadvantages of using i.i.d. noise for excitation? 3. The proof sketches could benefit from additional clarifications to enhance readability. For example, in line 234, the meaning of $\delta$ and $\bar{b}_\phi$ could be re-mentioned.
questions: questions: Can the authors please clarify the implications of the assumptions regarding bounded noises in more practical scenarios? It would be highly beneficial to delve deeper into the impact of these theoretical assumptions on real-world applications.
limitations: limitations: It is recommended to further explore the potential limitations of this  theoretical guarantees in practical applications. For instance, whether non-asymptotic convergence implies reduced data collection efficiency for some high-dimensional systems, and whether it is still preferable to actively design experiments in certain scenarios.
paper_summary: paper_summary: This paper studies the problem of identifying an observable stochastic nonlinear dynamical system, in case the transition function is linearly parametrized and the noise is additive. The authors assume that the feature functions are analytic, and both the inputs and the noises are i.i.d., bounded, semi-continuous and have nonvanishing variances in each coordinate direction. It is also assumed that the system is locally input-to-state stable. Two kinds of estimation methods are studied: the classical least-squares estimate (LSE), which provides point estimates, and a set membership method, which provides region estimates. Finite sample bounds for the performance of LSE are proved, based on a block-martingale-small-ball condition, for both the open-loop and the close-loop cases. The sample complexity of the set membership method is studied under an additional assumption (tight bound on disturbance) and only for open-loop (i.i.d.) inputs. Finally, the authors present some numerical evaluations of LSE and set membership methods on pendulum and quadrotor examples to empirically illustrate their convergence rates.
strength_weakness: strength_weakness: - Many of the assumptions are restrictive, such as bounded noises and inputs. Moreover, the inputs should also be i.i.d. (or should have an additive i.i.d. exploration noise, for the closed-loop case) which is unrealistic.
- The error bounds contain terms which are unknown in practice, such as s_\phi.
- Corollary 1: the controller for the closed-loop case also contains an additive noise term that satisfies the assumptions for the open-loop inputs, and the system with the controller satisfies the stability assumption, in which case the statement becomes a simple consequence of Theorem 1. A case without additional noises on the inputs would have been much more interesting.
- The figures do not show the actual bounds deducted by the paper, they only illustrate the empirical performance of LSE and set membership identification, which is a bit pointless, as they are classical, well-studied methods. I understand that the bounds of the paper are theoretical in nature, so they are conservative and mostly just give the convergence rate, still showing them (for example, on a logarithmic scale) would have been informative.
- The title could be misleading, as the word "analytic" should refer to the feature vectors, but in the current title the term "analytic system identification" could also be understood in a way that the obtained solution is analytic.
questions: questions: - What is the intuitive meaning of Assumption 5 (tight bound on disturbance)? This should be explained in the paper.
limitations: limitations: There is a section dedicated to the limitations of the work, which is a good thing. On the other hand, this section did not mention some key limitations, such as bounded noises, bounded and i.i.d. inputs, as well was fully observable states.

Comments: title: Official Comment by Reviewer wxwx
Author-reviewer discussion
Thank you for your feedback!
comment: Thank you very much for reading our responses and raising your score! We are glad to hear that you can accept most of our arguments! Regarding the bounded noises assumption, we will incorporate the discussions above into our revised manuscript, and for future work, we will continue exploring the convergence rate under weaker assumptions on noises yet stronger assumptions on system dynamics as considered in the literature mentioned in our responses. Regarding the unknown $s_\phi, p_\phi$, we will add the numerical estimation methods and the plots of these values to our revised manuscript, as well as the explicit formulas of these variables in special cases as illustrating examples. We hope these revisions can alleviate your concerns! Thank you again for your positive feedback!
Thank you very much for reading our responses and your positive feedback! We are glad that we have addressed most of your concerns!
Thank you for the detailed responses and the additional simulation. The responses have addressed most of my concerns. I believe my initial rating still reflects my overall assessment, so I will leave it as is.
Thank you for your comprehensive answers and for your additional experiments. I can accept most of your arguments and even though I still think that the boundedness assumption is a weakness of the paper and having typically unknown quantities in the bounds is also an issue, I have raised my rating based on the other points.
Dear reviewers,
Thank you for your insightful feedback during the initial review. Kindly ensure that you review the author's rebuttal and, if necessary, engage in scientific conversation with them (the discussion period ends on Aug 13th). If the authors have satisfactorily addressed your concerns, please consider adjusting your scores. This is also an opportune moment to review feedback from other reviewers. Even if you find that your score remains unaltered, I would appreciate it if you could acknowledge reviewing the author's responses.
Once again, I truly appreciate your time and attention.
Best regards,
Area Chair
Thank you very much for your feedback! Do you have remaining concerns or questions? We are more than happy to have further discussions!
I would like to thank the authors for their detailed response. I have decided to leave my score unchanged.