Estimation Problems and the Modulating Function Method: The Algebra of Modulating Functions

arXiv:2605.12244 · eess.SY, cs.SY, math.RA · Submitted 2026-05-12 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "Estimation Problems and the Modulating Function Method".

Dev: State and parameter estimation, along with fault detection, are three crucial estimation problems within the control systems community.

Rosa: First, who's behind it and why it matters.

Title: Rosa: So we're looking at this paper today titled "Estimation Problems and the Modulating Function Method: The Algebra of Modulating Functions," and the authors are Davi G. Accioli a and Jerome Jouffroy a, which sounds pretty academic. I was curious about what kind of estimation problems they're tackling with this approach since it seems to cover several areas like state estimation, parameter estimation, and even fault detection.

Dev: It does sound broad, Rosa; it's trying to unify three different kinds of control problems—state and parameter estimation, fault detection, and even distributed or fractional systems—under one framework using modulating functions. I wonder how unified they actually manage to make that work technically without the complexity getting out of hand.

Taro: From my angle as an autonomy researcher, if this method is truly unified across these different system types, it could be really powerful when we're dealing with complex mobile platforms where sensors fail or dynamics change unexpectedly in real-time. I'm interested in seeing how flexible this unification actually is when the physical situation gets messy.

Rosa: Exactly, Taro; that flexibility is what interests me most—can this method handle the kind of unpredictable behavior we see out in the field, and if so, for how long can we expect it to be reliable outside a controlled lab environment?

Dev: That's a critical question for me. If it relies on specific mathematical structures like these modulating functions, its real-world performance hinges entirely on the stability of those underlying operators and how sensitive the filter characteristics are to noise in that environment.

Taro: I think if they manage to build systems robust enough to handle misbehavior, then this could be a huge step for autonomous systems operating in uncontrolled environments where perfect model knowledge is impossible.

Rosa: That makes sense; it moves the focus from just lab validation to general applicability, and I'm hoping they show some data on how well it performs when you take it into the field.

Dev: The paper starts by setting up this unifying concept, focusing on how a single modulating function dictates the filter characteristics for different estimation tasks like state estimation or parameter identification.

Paper discussion segment 1: Rosa: Moving on, Dev, we've talked about the title and authors of this paper today, and now I want to get a better idea of what the core summary actually lays out regarding the Modulating Function Method. What are they telling us is the main contribution?

Dev: Well, they explain that at its heart is this modulating function, which basically evaluates to zero at the boundaries based on a certain order of derivatives. The key point is that by just picking these functions right, you directly control what kind of filter characteristics you end up with for your system estimation or detection tasks.

Taro: So, it's about having a direct mapping between the function chosen and the desired filter behavior, which simplifies the design process considerably compared to trying to build filters from scratch. That sounds like a nice simplification for complex problems.

Rosa: It really does sound elegant; they propose constructing new families of modulating functions by formalizing their algebraic properties, showing that you can combine or multiply existing ones in predictable ways to generate even more useful functions, like the logarithmic and non-analytic families they introduce.

Dev: And they also provide a straightforward algorithm for generating any type of modulating function from any sufficiently smooth function, which is pretty practical because it lets us build custom filters without needing to reinvent the wheel every single time we need a new one.

Taro: If you can generate new functions systematically, that opens up a lot of possibilities for creating specialized observers or detectors tailored exactly to the specific dynamics of our system, rather than relying on generic solutions.

Rosa: That's the exciting part; it’s not just about using existing tools but about having the mathematical machinery to create entirely new tools tailored to our specific needs, which is what makes this paper so interesting from a research standpoint.

Dev: The authors formalize definitions for different types of modulating functions, like total modulating functions denoted as phi T, left modulating functions denoted as phi L, and right modulating functions denoted as phi R.

Paper discussion segment 2: Rosa: Okay, so we understand the basic idea now—the construction and algebraic properties of these functions—but what's really exciting is how they suggest improvements or new avenues for using this method? What are the practical enhancements they propose?

Dev: They focus heavily on how to utilize the newly introduced Total Modulating Function vector space, showing that you can construct orthonormal total modulating functions specifically for parameter estimation problems, which is a major step because it circumvents those matrix inversion issues mentioned earlier.

Taro: Avoiding matrix inversion issues is huge; that's a major hurdle in many state and parameter estimation schemes, especially when you need to estimate multiple parameters at once, and having an orthonormal set makes the math much cleaner and more stable for those kinds of tasks.

Rosa: And they also detail how to use left and right modulating functions to estimate state derivatives or output derivatives, which opens up new avenues for estimation beyond just static parameter identification, like estimating how fast a system is changing.

Dev: That's interesting because it suggests we can move beyond simple static parameter fitting into more dynamic estimation tasks when dealing with system behavior.

Taro: Moving into derivative estimation brings in the real-time aspect of autonomy; if you can estimate derivatives reliably, then the AI could react much faster to unexpected environmental changes, which is crucial for maneuvering safely.

Rosa: That's interesting because it suggests we can move beyond simple static parameter fitting into more dynamic estimation tasks when dealing with system behavior.

Dev: They also show that by exploiting concepts from linear operator theory, it's possible to define the modulation operator, which has a concise notation and is used in integral operators.

Conclusion: Rosa: So wrapping up our discussion on "Estimation Problems and the Modulating Function Method: The Algebra of Modulating Functions," the main takeaway is that this paper provides a unified framework for state, parameter estimation, and fault detection using modulating functions.

Dev: I think we've covered how they formalize these functions, show how to build new ones systematically, and highlighted the improvement of using orthonormal Total Modulating Functions to bypass matrix inversion issues in parameter estimation.

Taro: From my view, the real implication is that this algebraic foundation gives us a way to handle complex system identification problems in a more structured manner when dealing with unpredictable real-world conditions where standard methods fall short.

Rosa: I think we've covered how they formalize these functions, show how to build new ones systematically, and highlighted the improvement of using orthonormal Total Modulating Functions to bypass matrix inversion issues in parameter estimation.

Dev: I agree, and the fact that all these modulating functions can be calculated offline means we don't need any heavy online auxiliary systems for the estimation loop, which is a big win for latency and reliability.

Taro: It really gives us a systematic way to approach model expansion, which is something that's going to be useful as autonomous systems get more sophisticated and need better ways to handle uncertainty in their operational domains.

Rosa: Absolutely; this whole paper shows how exploiting the algebraic properties of these modulating functions lets us create estimators that are fundamentally more stable and computationally lighter than what we used to implement.

Dev: Well, I think this paper solidifies the idea that pre-calculating these functions offline saves significant computational cost during runtime for parameter estimation.

Rosa: Fantastic discussion; it was fascinating exploring the structure behind this approach in "Estimation Problems and the Modulating Function Method: The Algebra of Modulating Functions." I'm looking forward to seeing how this method translates from theory into practical field-tested solutions.

SDU Mechatronics, University of Southern Denmark

eess.SY, cs.SY, math.RA

Submitted: 2026-05-12

Updated: 2026-09-25

Comments: 11 pages, 5 figures

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 82/100

The gist: State and parameter estimation, along with fault detection, are three crucial estimation problems within the control systems community.

Terminology

Core — 6× in this paper · explained in 2 episodes
Conference on Decision and Control — 2× in this paper
Hilbert Space — 2× in this paper · explained in 2 episodes
System Identification — 2× in this paper · explained in 3 episodes
Unified Framework — 1× in this paper · explained in 5 episodes

Summary

State and parameter estimation, along with fault detection, are three crucial estimation problems within the control systems community. Although different approaches have been proposed for each type of problem, the modulating function method proposes a more unified approach to all three problem classes, being used for state and parameter estimation of lumped systems, fault detection, and estimation of distributed and fractional systems. At the core of the method is the modulating function: a function that evaluates to 0 at the left or right boundaries up to a certain order of derivatives. By selecting the modulating functions, one directly determines the filter characteristics, and, for that reason, different function families have been proposed over the years. Nevertheless, many families of modulating functions are given in a rather similar mathematical structure. In light of these structures, this paper formally discusses the algebraic properties of modulating functions, and after formalizing the closedness and group properties of modulating functions, a simple algorithm to construct new modulating functions is proposed, discussed, and illustrated with the construction of the newly introduced logarithmic modulating function families and 3 non-analytic modulating function families. Moreover, the fact that total modulating functions form a vector space and an algebra is exploited to construct orthonormal modulating functions, which are then used for the parameter estimation of a boat’s roll dynamics, effectively avoiding matrix inversion issues.

The paper formally discusses the algebraic properties of modulating functions. It introduces definitions for different types of modulating functions:

if φ(i) (0) = φ(i) (T) = 0 ∀i, then it is called a total modulating function (TMF), denoted φT;

if φ(i) (0) = 0 ∀i and ∃i s.t. φ(i) (T) 6= 0, then it is called a left modulating function (LMF), denoted φL;

if φ(i) (T) = 0 ∀i and ∃i s.t. φ(i) (0) 6= 0, then it is called a right modulating function (RMF), denoted φR.

The paper also introduces the modulation operator:

"Definition 2 Let φ: R[0, T] 7→ R be a modulating function of suﬀicient order q ≥ i ∈ N. Then, a modulation operator is given by Mi [y(t)](t0, t1):= ∫ t1 (−1)i φ(i) (τ − t0)y(τ)dτ, for user-defined t0, t1 ∈ R, t1 > t0, and T:= t1 − t0."

Theorem 5 establishes the estimation framework:

"Given a system described by (5), modulation operator (2), and linearly independent modulating functions not fully orthogonal to the measured signals y(t) ∈ R and u(t) ∈ R, the system parameters can be estimated as θ̂ = W−1 z,"

where:

z:= [1 MnT [y],..., 2n MnT [y]],

and the matrix W is defined by:

W:= −Wy Wu, (9)-(11)

Lemma 4 shows how the modulation operator transfers derivatives to the modulating function:

"Let φ ∈ C i ([0, T]). Then, applying modulation operator (2) to the derivative of a signal transfers the derivative to the modulating function, i.e. M[y] = M [y] + P, where P:= −i−1 ∑ (−1)j φ(j) (T)y(i−1−j) (t1) j=0 −(−1) φ j (j)."

The paper details the construction of new modulating functions through algebraic properties:

"Theorem 7 Let ΦL, ΦR, and ΦT respectively be the sets of all left, right, and total modulating functions, and Φ be the set of modulating functions, i.e. Φ = [ΦL, ΦR, φT]. Then, it follows that (24) the product of any two TMFs is a TMF of order at least 1... The set of TMFs ΦT includes the trivial solution φT (τ) = 0 and is closed under addition since 1 φT (0) + 2 φT (0) = 0 + 0 = 0.... ΦT forms an abelian group under scalar addition, a vector space, and an associative and commutative algebra."

"Corollary 10 Let [1 q L, 1 q R] be the order of the suﬀiciently smooth modulating function φ(τ) on the left and right boundaries; and [2 q L, 2 q R] be the order of the suﬀiciently smooth modulating function φ(τ) on the left and right boundaries, respectively. Then, it follows that Proposition 8 Given any two modulating functions, the following properties are satisfied:... Combining the two modulating functions using a product, i.e. 3 φ(τ) = 1 φ(τ) 2 φ(τ), increases the order of the left and right boundary conditions to 3 q L = 1 q L + 2 q L and 3 q R = [1 q R + 2 q L];... Combining the two modulating functions using addition, i.e. 3 φ(τ) = 1 φ(τ) + 2 φ(τ), changes the order to [3 q L ≥ min(1 q L, 2 q L) and [3 q R ≥ min(1 q R, 2 q R)]."

The paper illustrates the construction of new families of modulating functions using Theorem 11:

"Theorem 11 Given any suﬀiciently smooth scalar function g: R[0, T] 7→ R, a family of modulating functions can be obtained as φ(τ) = F (τ) (g(τ) − g(0)) 1 (g(τ) − g(T)) 2, where the suﬀiciently smooth function F (τ) ∈ R, as well as the coeﬀicients T ∈ R>0 and q1, q2 ∈ N."

Examples of constructed families include:

4.1 Analytic Function Example... a left modulating function of any order q can be obtained by applying Theorem 11, giving rise to the logarithmic LMF family φL (τ) = (ln(cτ + 1))q.

4.2 Non-analytic Function Example... a family of logarithmic RMFs can be constructed using Theorem 11 as φR (τ) = (ln(cτ + 1) − ln(cT + 1))q.

The paper also constructs analytic and non-analytic modulating functions, including:

"4.3 Orthogonal Total Modulating Functions... By following the Gram-Schmidt process performed in the inner product space ΦT ⊂ L2 [0, T] for the candidates 1 ψT = t3 (T − t)4, together with 2 ψT, 3 ψT, and 4 ψT given in (53)-(55).... you obtain the 4 orthonormal TMFs shown in Figure 4, guaranteeing linearly independence of the total modulating functions. Moreover, due to Corollary 10, all modulating functions are of order qi ≥ 3."

The application example involves parameter estimation for a boat’s roll dynamics described by the nonlinear system:

ϕ̈ + a1 ϕ̇ + a0 ϕ + anl ϕ3 = b0 u,

The estimation utilizes the four orthonormal TMFs illustrated in Section 4.3, and the results show that using 5 orthonormal TMFs improves the norm of the RMS relative estimation error from 0.2956 to 0.0998, an improvement of over 66%."

The paper concludes by summarizing that by exploiting the algebraic properties of modulating functions, it is possible to effectively avoid the known matrix inversion issue related to this approach, and there is no need to implement an auxiliary system, as the modulating functions are all calculated offline. The final conclusion states that an algorithm to obtain any type of modulating function of any order using any suﬀiciently smooth function is obtained, along with a procedure to construct new modulating functions based on existing ones. It also notes that total modulating functions also form an abelian group under addition, a vector space, and an associative and commutative algebra.

The paper provides references including: Accioli & Jouffroy (2025), Asiri et al. (2021), Belkhatir & Laleg-Kirati (2018), Baurichter et al. (2025), Fischer et al. (2021, 2025), Friedrich et al. (Fokken & Reger, 2025), Grubb (2008), Hestenes (1999), Ionesi et al. (Ramezani & Jouffroy, 2019), Jouffroy & Reger (2015), Korder et al. (Noack & Reger, 2022), Li et al. (Boem et al., Pin & Parisini, 2020), Loeb & Cahen (1965), M. Shinbrot (1954), Mahony et al. (Hamel & Pflimlin, 2008).

The paper explicitly states that "the true model parameters were then selected as a0 = 1.33, a1 = 0.64, anl = 2.43, and b0 = 6.4 · 10−6, being obtained from the canting keel parameters given in Ramezani, Chaudhuri, Jerome Jouffroy, Baurichter, & Mattrup Hansen, 2025. It also notes that the simulation used the 4 orthonormal modulating functions discussed in Section 4.3 along with the fifth MF obtained from (58) and the Gram-Schmidt process. It concludes by stating that the results proposed in this paper not only extend the repertoire of modulating functions but also show how, by exploiting the algebraic properties of modulating functions, it is possible to effectively avoid the known matrix inversion issue related to this approach. Moreover, there is no need to implement an auxiliary system, as the modulating functions are all calculated offline, effectively avoiding higher computational costs during implementation."

The summary is: The paper formally discusses the algebraic properties of modulating functions. It introduces definitions for different types of modulating functions: if φ(i) (0) = φ(i) (T) = 0 ∀i, then it is called a total modulating function (TMF), denoted φT; if φ(i) (0) = 0 ∀i and ∃i s.t. φ(i) (T) 6= 0, then it is called a left modulating function (LMF), denoted φL; if φ(i) (T) = 0 ∀i and ∃i s.t. φ(0) 6= 0, then it is called a right modulating function (RMF). It introduces the modulation operator: "Definition 2 Let φ: R[0, T] 7→ R be a modulating function of suﬀicient order q ≥ i ∈ N. Then, a modulation operator is given by Mi [y(t)](t0, t1):= ∫ t1 (−1)i φ(i) (τ − t0)y(τ)dτ, for user-defined t0, t1 ∈ R, t1 > t0, and T:= t1 − t0. It establishes the estimation framework: Given a system described by (5), modulation operator (2), and linearly independent modulating functions not fully orthogonal to the measured signals y(t) ∈ R and u(t) ∈ R, the system parameters can be estimated as θ̂ = W−1 z, where z:= [1 MnT [y],..., 2n MnT [y]], and W is defined by W:= −Wy Wu. Lemma 4 shows how the modulation operator transfers derivatives to the modulating function: Let φ ∈ C i ([0, T]). Then, applying modulation operator (2) to the derivative of a signal transfers the derivative to the modulating function, i.e. M[y] = M [y] + P, where P:= −i−1 ∑ (−1)j φ(j) (T)y(i−1−j) (t1) j=0 −(−1) φ j (j). The paper details the construction of new modulating functions through algebraic properties, stating: Theorem 7 Let ΦL, ΦR, and ΦT respectively be the sets of all left, right, and total modulating functions, and Φ be the set of modulating functions, i.e. Φ = [ΦL, φR, φT]. Then, it follows that (24) the product of any two TMFs is a TMF of order at least 1. It also provides Theorem 11 for constructing new MFs: Theorem 11 Given any suﬀiciently smooth scalar function g: R[0, T] 7→ R, a family of modulating functions can be obtained as φ(τ) = F (τ) (g(τ) − g(0)) 1 (g(τ) − g(T)) 2. It also constructs new families including: 4.1 Analytic Function Example and 4.2 Non-analytic Function Example. Furthermore, it constructs orthonormal TMFs using the Gram-Schmidt process: 4.3 Orthogonal Total Modulating Functions." The application example involves parameter estimation for a boat’s roll dynamics, where the use of 5 orthonormal TMFs improves the norm of the RMS relative estimation error from 0.2956 to 0.0998, an improvement of over 66%. Finally, it concludes that the results proposed in this paper not only extend the repertoire of modulating functions but also show how, by exploiting the algebraic properties of modulating functions, it is possible to effectively avoid the known matrix inversion issue related to this approach. Moreover, there is no need to implement an auxiliary system, as the modulating functions are all calculated offline. The final conclusion states that an algorithm to obtain any type of modulating function of any order using any suﬀiciently smooth function is obtained, along with a procedure to construct new modulating functions based on existing ones. It also notes that total modulating functions also form an abelian group under addition, a vector space, and an associative and commutative algebra."

The paper explicitly states that "the true model parameters were then selected as a0 = 1.33, a1 = 0.64, anl = 2.43, and b0 = 6.4 · 10−6, being obtained from the canting keel parameters given in Ramezani, Chaudhuri, Jerome Jouffroy, Baurichter, & Mattrup Hansen, 2025. It also notes that the simulation used the 4 orthonormal modulating functions discussed in Section 4.3 along with the fifth MF obtained from (58) and the Gram-Schmidt process. It concludes by stating that the results proposed in this paper not only extend the repertoire of modulating functions but also show how, by exploiting the algebraic properties of modulating functions, it is possible to effectively avoid the known matrix inversion issue related to this approach. Moreover, there is no need to implement an auxiliary system, as the modulating functions are all calculated offline. The final conclusion states that an algorithm to obtain any type of modulating function of any order using any suﬀiciently smooth function is obtained, along with a procedure to construct new modulating functions based on existing ones. It also notes that total modulating functions also form an abelian group under addition, a vector space, and an associative and commutative algebra."

The paper provides references including: Accioli & Jouffroy (2025), Asiri et al. (2021), Belkhatir & Laleg-Kirati (2018), Baurichter et al. 2025, Fischer et al. (Fokken & Reger, 2025), Grubb (2008), Hestenes (1999), Ionesi et al. (Ramezani & Jouffroy, 2019), Jouffroy & Reger (2015), Korder et al. (Noack & Reger, 2022), Li et al. (Boem et al., Pin & Parisini, 2020), Loeb & Cahen (1965), M. Shinbrot (1954), Mahony et al. (Hamel & Pflimlin, 2008).

The paper explicitly states that "the true model parameters were then selected as a0 = 1.33, a1 = 0.64, anl = 2.43, and b0 = 6.4 · 10−6, being obtained from the canting keel parameters given in Ramezani, Chaudhuri, Jerome Jouffroy, Baurichter & Mattrup Hansen, 2025. It also notes that the simulation used the 4 orthonormal modulating functions discussed in Section 4.3 along with the fifth MF obtained from (58) and the Gram-Schmidt process. It concludes by stating that the results proposed in this paper not only extend the repertoire of modulating functions but also show how, by exploiting the algebraic properties of modulating functions, it is possible to effectively avoid the known matrix inversion issue related to this approach. Moreover, there is no need to implement an auxiliary system, as the modulating functions are all calculated offline. The final conclusion states that an algorithm to obtain any type of modulating function of any order using any suﬀiciently smooth function is obtained, along with a procedure to construct new modulating functions based on existing ones. It also notes that total modulating functions also form an abelian group under addition, a vector space, and an associative and commutative algebra."

The paper explicitly states that "the true model parameters were then selected as a0 = 1.33, a1 = 0.64, anl = 2.43, and b0 = 6.4 · 10−6, being obtained from the canting keel parameters given in Ramezani, Chaudhuri, Jerome Jouffroy, Baurichter & Mattrup Hansen. It also notes that the simulation used the 4 orthonormal modulating functions discussed in Section 4.3 along with the fifth MF obtained from (58) and the Gram-Schmidt process. It concludes by stating that the results proposed in this paper not only extend the repertoire of modulating functions but also show how, by exploiting the algebraic properties of modulating functions, it is possible to effectively avoid the known matrix inversion issue related to this approach. Moreover, there is no need to implement an auxiliary system, as the modulating functions are all calculated offline. The final conclusion states that an algorithm to obtain any type of modulating function of any order using any suﬀiciently smooth function is obtained, along with a procedure to construct new modulating functions based on existing ones. It also notes that total modulating functions also form an abelian group under addition, a vector space, and an associative and commutative algebra."

Improvements for AI systems

Here are the specific improvements to an AI system based on the Modulating Function Method (MFM) as described in the paper, along with what those improved systems can achieve:

The core improvement lies in replacing traditional, computationally expensive matrix inversion methods with a structurally sound algebraic framework based on modulating functions.

[[Improvement: Parameter Estimation via Orthonormal Total Modulating Functions (TMFs)]]
[[Specific Functionality: Robust and Efficient Parameter Identification]]
[[Mechanism: Gram-Schmidt Orthogonalization in the TMF Vector Space]]

The improved AI system can perform the following specific tasks:

[[Task 1: High-Accuracy Roll Dynamics Estimation for Marine/Robotic Systems]]
[[Specific Capability: Estimating 4 or more physical parameters (e.g., inertia, damping, restoring forces) of a boat's roll dynamics simultaneously in fixed time]]
[[Benefit: Avoiding the Sharp Peak Instability Issue]]
[[Benefit: Significant Reduction in Relative Estimation Error (up to 66% improvement shown in simulations) compared to previous methods (e.g., Co & Ungarala, 1997)]]
[[Task 2: Real-Time Parameter Tracking and Fault Detection]]
[[Specific Capability: Utilizing Left Modulating Functions (LMFs) and Right Modulating Functions (RMFs) to estimate state derivatives or output derivatives]]
[[Benefit: Eliminating the need for online, computationally intensive auxiliary systems, as all modulating functions are calculated offline. This reduces computational load significantly compared to Schmid & Roppenecker (2011) approaches.]]
[[Task 3: Model Expansion and Repertoire Generation]]
[[Specific Capability: Dynamically constructing new, linearly independent modulating functions by adding and multiplying existing ones (e.g., combining sine, polynomial, and hyperbolic families) using simple arithmetic operations (addition/multiplication), thereby expanding the estimation repertoire without complex system identification procedures.]]

In summary, the improved AI system will be a highly efficient parameter estimator capable of:

Estimating multiple physical parameters simultaneously in a fixed time frame for nonlinear systems (like boat roll dynamics).
Maintaining stability and accuracy by leveraging an orthonormal basis of total modulating functions, thus mitigating the common problem of non-invertible matrices.
Achieving superior estimation performance while drastically reducing computational overhead by relying on pre-calculated, offline modulating functions instead of online calculations.

Abstract

The modulating function method is an algebraic estimation framework that has been used for state and parameter estimation, as well as fault detection, of linear and some nonlinear systems. At the core of the method is the modulating function: a function that evaluates to 0 at the left or right boundaries up to a certain order of derivatives, being classified as left, right, or total modulating functions. Despite being an algebraic framework, there is no paper that analyzes the algebraic properties of the three types of modulating functions and the consequences of these properties. In light of this gap, this paper discusses the algebraic properties of modulating functions and, after formalizing their closedness and group properties, simple algorithms to construct new modulating functions are proposed, discussed, and illustrated. Moreover, the fact that total modulating functions form an algebra and a vector space is exploited to construct orthonormal modulating functions, which are then used to significantly improve the well-known matrix conditioning issue of the modulating function method, illustrated with the parameter estimation of a boat's nonlinear roll dynamics.