Simultaneous state estimation and control for nonlinear systems subject to bounded disturbances

arXiv:1906.10043 · eess.SY, cs.SY · Submitted 2019-06-24 · 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: "Simultaneous state estimation and control for nonlinear systems subject to bounded disturbances".

Dev: In this work, a moving horizon approach is used to address the output–feedback control problem for nonlinear systems subject to bounded disturbances.

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

Paper discussion segment 1 — Rosa and Dev discuss title and authors of the paper 'Simultaneous state estimation and control for nonlinear systems subject to bounded disturbances': Rosa: So, looking at the title, "Simultaneous state estimation and control for nonlinear systems subject to bounded disturbances," it tells us immediately that this work is focused on handling real-world complexity where things aren't perfectly predictable. It’s about managing uncertainty in physical systems that aren't just simple linear equations anymore.

Dev: I see why; "nonlinear systems" means the physics are complicated, and "bounded disturbances" means we have noise and errors that we can't eliminate entirely, so the AI has to be robust against those things.

Taro: It’s interesting how they framed it as an output-feedback control problem, which is super relevant because in many industrial settings, you don't get direct access to every internal variable; you only see what comes out of the system.

Rosa: Exactly, Taro; that output-feedback aspect makes it much more practical for real-world robotics where sensors are always noisy and sometimes intermittent. It’s not just about having perfect knowledge internally.

Dev: And when you read the authors' names, I’m always looking to see if they have a background in both estimation techniques and advanced control theory, because this paper seems to blend those two fields very tightly.

Taro: I think their combined expertise is what allows them to bridge that gap between pure theoretical estimation and practical control application, which is where most autonomy research struggles.

Rosa: That's a great point; it shows a deep understanding of the entire pipeline, not just optimizing one small piece of the puzzle in isolation. It’s about seeing the whole process as one interconnected optimization task.

Paper discussion segment 2 — Rosa and Dev discuss the paper's summary of the paper 'Simultaneous state estimation and control for nonlinear systems subject to bounded disturbances': Dev: The summary really hammers home that the main contribution is formulating this entire process as a single, infinite-horizon optimization problem that gets broken down into a manageable finite-horizon receding horizon problem.

Rosa: That’s the key takeaway, Dev; they aren't just proposing an estimator and then an MPC controller; they are solving for the optimal future state trajectory and the control inputs all at once within that single framework.

Taro: So, when things go sideways in a mission—say, a sudden gust of wind or unexpected friction—the system isn't just trying to correct the error after it happens; it’s planning around that potential issue from the very beginning.

Dev: That proactive planning is what I find compelling; if you are estimating your state and planning your controls simultaneously, you build in a level of foresight that separate modules simply can't achieve when conditions change rapidly.

Rosa: It means the system stays much more stable because the control action it calculates is already informed by its best current estimate of where it is going, not just a guess from a previous time step.

Taro: That integrated planning capability drastically improves resilience; it allows for smoother maneuvering when facing unexpected disruptions, which is crucial when you're operating far from the training environment.

Dev: It directly addresses the loop rate concerns we usually have; if the optimization is well-structured, it should yield a solution fast enough to maintain a stable control cycle even with complex dynamics involved.

Paper discussion segment 3 — Rosa and Dev discuss the improvements the paper suggests of the paper 'Simultaneous state estimation and control for nonlinear systems subject to bounded disturbances': Rosa: One of the most important things they suggest is linking the lengths of their forward and backward windows directly to closed-loop stability, which is a really strong theoretical guarantee they provide for this approach.

Dev: That’s significant; it means you have a clear mathematical condition—Theorem one—that tells you exactly what window sizes are needed to ensure the system stays bounded, provided those detectability conditions are met.

Taro: So, it moves the discussion from just "it might work" to "here's the math that proves *why* it works under certain conditions," which is essential for trusting this kind of AI in critical applications.

Rosa: It gives us a concrete design parameter to tune; we’re not just guessing window sizes anymore, we have a derived requirement based on the system's dynamics and noise characteristics.

Dev: I like that they derive Nc, the control horizon length, and it includes terms related to the system parameters like delta(L-one) and L/L-one which shows they’ve done some deep analysis into how much information those windows need.

Taro: And when I think about deployment outside the lab, this mathematical guarantee is what gives me confidence that we can trust the AI to handle bounded disturbances reliably in a real operational setting.

Rosa: It’s definitely a huge step forward; it moves this from being just a clever algorithm to having provable stability bounds, which is exactly what we need for serious deployment discussions.

Conclusion: Dev: To wrap up our discussion on "Simultaneous state estimation and control for nonlinear systems subject to bounded disturbances," the paper establishes that coupling estimation and control via a unified optimization framework offers superior performance compared to running them as separate modules.

Rosa: Exactly, Dev; it really shows that coupling those two tasks is the right way to manage the inherent uncertainty in these systems without falling into those common decoupling failures we see elsewhere. It’s a solid piece of engineering that moves us toward more robust robotic platforms.

Taro: I’m still thinking about how this resilience translates to truly autonomous missions where conditions are constantly shifting; it suggests a foundation for much tougher navigation when dealing with unpredictable environments.

Dev: And from my side, the main promise is that we can build systems that are more predictable in their failure modes because the controller isn't acting on outdated or inaccurate state estimates.

Rosa: I’m curious, Dev, when we look at these results, do you see any immediate hurdles for deploying this kind of system outside of a controlled lab setting?

Dev: The main hurdle remains the real-time execution; we still need to ensure that solving this complex optimization doesn't introduce unacceptable latency that would compromise the control loop rate.

Taro: And what about long-term operation? If we can prove boundedness, does that mean these systems can run reliably for extended periods in the field without constant recalibration?

Rosa: The paper’s focus on stability and boundedness suggests they are aiming for practical reliability, but it is important to remember that the real-world performance will depend heavily on those initial assumptions about detectability conditions.

Dev: So, while the theory is solid, we’ll need rigorous testing to confirm that those theoretical bounds hold up when we introduce genuine, unmodeled disturbances in the field.

Taro: I think the future research mentioned about adaptive laws is where things get interesting; that could be what allows this approach to handle even more unpredictable scenarios than just bounded noise.

Rosa: Definitely; it points toward a system that can truly learn and adjust its own estimation strategy as it encounters new types of uncertainty as it operates.

Dev: That's the kind of adaptability we need, Rosa, but we have to be careful that the adaptive mechanism doesn't introduce instability during the learning phase itself.

Taro: It seems like this work on "Simultaneous state estimation and control for nonlinear systems subject to bounded disturbances" is setting a very strong baseline for how AI can handle complex physical realities in real-time.

Rosa: I agree, Taro; it’s a solid piece of engineering that moves us toward more robust robotic platforms.

Dev: Yeah, and the focus on loop rate constraints is crucial because we don't want theoretical stability if the hardware can't keep up with the demands of a fast control cycle.

Taro: Next time, I’m looking forward to seeing how these principles apply to those other papers we looked at on arXiv, especially how this unified approach compares to those finite-horizon approximations in linear-quadratic games.

Rosa: We certainly will; it's going to be a fascinating comparison of robust nonlinear control versus traditional game theory solutions.

Instituto de Investigacion en Senales, Sistemas e Inteligencia Computacional

eess.SY, cs.SY

Submitted: 2019-06-24

Updated: 2026-09-24

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 76/100

The gist: In this work, a moving horizon approach is used to address the output–feedback control problem for nonlinear systems subject to bounded disturbances.

Terminology

Lyapunov function — 5× in this paper · explained in 2 episodes
American Control Conference — 3× in this paper
Robustness — 3× in this paper · explained in 5 episodes
Closed-Loop System — 2× in this paper · explained in 4 episodes
Adaptability — 1× in this paper · explained in 1 episode
Adaptation — 1× in this paper · explained in 1 episode
Adaptive Mechanism — 1× in this paper · explained in 1 episode
Decoupling — 1× in this paper · explained in 2 episodes
Failure Modes — 1× in this paper · explained in 3 episodes
Objective Function — 1× in this paper · explained in 2 episodes

Summary

In this work, a moving horizon approach is used to address the output–feedback control problem for nonlinear systems subject to bounded disturbances. The controller is formulated as an optimization-based problem that simultaneously estimates the state trajectory and computes future control inputs. This optimization minimizes a criterion involving finite forward and backward horizons with respect to the unknown initial state, measurement noises, and control input variables while being maximized with respect to the unknown future disturbances.

The novelty of this work lies in linking the lengths of the forward and backward windows with closed-loop stability, assuming detectability and decoding sufficient conditions to assure system stabilizability. Simulation examples are carried out to compare the performance of simultaneous and independent estimation and control approaches as well as to show the effects of simultaneously solving the control and estimation problems.

The problem is formulated in discrete-time nonlinear systems where the behavior is given by:

xk+1 = f (xk, uk) + wk

yk = h (xk) + vk,

The estimation and control problem attempts to simultaneously find the optimal state x̂kk and the optimal sequence of control inputs û which will steer the system to the desired operation zone. It is in an infinite-horizon optimization problem given by:

min ΨEC,k,∞:= x̂0k,ŵ,û s.t.

"j=0 to ∞ k X j=k



x̂j+1k = f x̂jk, ûjk + ŵjk,"

lc x̂jk, ûjk − lwc ŵjk

yj = h x̂jk + v̂jk,

x̂j+1 x X, uhat j+1 U, what j+1 W, vhat j+1 V

The infinite-horizon problem is reformulated into a receding finite-horizon problem:

min x̂k−Ne k,ŵ,û ΨEC,k−Ne + Nc:= Γk−Ne (χ) + s.t.

"le ŵjk, v̂jk + j=k−Ne to k+N Nc:= lc x̂jk, ûjk − 1/Nc X j=k to k+Nc



x̂j+1k = f x̂jk, ûjk + ŵjk,"

y=h xhat j+ vhat j

The criterion is decomposed as:

ΨEC,k−Ne + Nc:= ϕΨE,k−Ne + (1 − ϕ)ΨC,k−Nc,

where:

"ΨE,k−Ne:= Γk−Ne (χ) + k X j=k-Ne to k ΨC,k−Nc:= Υk+Nc (Ξ) + le ŵjk, v̂jk, k+N c −1 X j=k to k

lc x̂jk, ûjk − lwc ŵjk"

The goal is to estimate the initial state x̂0k and disturbances what j in Z[k−Ne, k−1] such that an estimate xhat kk is obtained to compute the control inputs uj in Z[k, k+Nc-1] that drive the system states to the desired region.

The stability analysis relies on two assumptions:

"The first assumption requires that the optimization criterion include an adaptive arrival cost (Sánchez et al. 2017). This assumption allows to ensure the boundedness of the state estimate and to obtain a bound for the estimation error set if the parameters of the estimation problem are properly chosen (Deniz et al. 2019)."

"The second assumption requires that the backward (estimation) and forward (control) horizons are sufficiently large so that enough information is obtained in order to find state estimates and control inputs compatible with dynamics, noises and constraints."

Theorem 1 states the necessary conditions to guarantee the feasibility and stability of the optimization problem, resulting in closed-loop trajectories along which the states remain bounded. The required control horizon Nc is given by:

Nc = [1 + ln(ln(δ(L−1)/ (L−1)) / L/L-1]

The proof involves analyzing the difference in costs at two consecutive sampling times, leading to the inequality:

∆Ψ ≤ − lc x̂kk, ûkk (1 − δω) + π E,

where ω:= Υk+Nc (Ξ) / lc x̂kk, ûjk quantifies improvements in the control cost and disturbance controllability, and πE:= -Γk−Ne (χ) + lwe ŵjk − le ŵjk, v̂jk quantifies the changes in the estimation cost by measuring the amount of information left behind the estimation window.

The paper presents two examples:

Example 1 applies to a continuous-time nonlinear scalar system, analyzing constraints and disturbances on closed-loop stability and performance.
Example 2 discusses simulations for a van der Pol oscillator, comparing the performance of simultaneous and independent MHE–MPC controllers under different parameters for Ne and Nc, showing the superior performance of the simultaneous MHE–MPC controller in regulating both states compared to the independent approach. The computational burden of the simultaneous MHE–MPC is lower than that of the independent one.

The conclusion is that this output-feedback approach for nonlinear systems subject to bounded disturbances using MHE–MPC combines state estimation and control into a single optimization, which is solved at each sampling time, and Theorem 1 provides the necessary conditions to guarantee the feasibility and stability of the optimization problem, resulting in the boundedness of system states as a function of the windows lengths Ne and Nc. Future work may involve designing forward window properties to improve estimation process and designing an adaptive law for ϕ such that estimation and control problems keep balanced.

The paper concludes with: The proposed approach combines the state estimation and control problems into a single optimization, which is solved at each sampling time. (Page 23)

Theorem 1 states the necessary conditions to guaranty the feasibility and stability of the optimization problem, and therefore the boundedness of system states, as a function of the windows lengths Ne and Nc. (Page 23)

These results require the compatibility between the robust estimated and controllable sets (Assumption 1) and the existence of a relaxed closed–loop Lyapunov function for the disturbed system (Assumption 18). (Page 23)

These conditions imply forward (Nc) and backward (Ne) horizons to find state estimates and control actions that are consistent with the system dynamics, constraints and disturbances. (Page 23)

The paper also highlights that the simultaneous MHE–MPC controller manages to regulate the system states for all noise realizations, whereas the independent MHE and MPC strategy fails to satisfy Assumption 1 because its design procedure applies the separation principle, which entails the automatic satisfaction of Assumption 1 and it does not include the constraints information in the selection of Ne and Nc." (Page 19)

The simultaneous MHE–MPC controller does [include constraints information in the selection of Ne and Nc]. (Page 19)

The computational burden comparison shows: The computational burden of the simultaneous MHE–MPC is lower than the independent one, as can be seen in Figure 6. (Page 21)

The execution times were averaged over 100 trials. The lower time, in the beginning, is due to the backward window corresponding to the estimation has not achieved yet its full length. (Page 21)

The paper also notes that the independent MHE and MPC approach is more sensitive to disturbances, requiring conservative values of Nc to guarantee the closed-loop stability. (Page 19)

This figure shows that the independent MHE and MPC approach is more sensitive to disturbances, requiring conservative values of Nc to guarantee the closed-loop stability. (Page 19)

The paper also notes that the simultaneous MHE–MPC controller manages to regulate the system states for all noise realizations. (Page 19)

On the other hand, the simultaneous MHE–MPC controller manages to regulate the system states for all noise realizations. (Page 19)

This problem is caused by the failure of the independent MHE and MPC to satisfy Assumption 1. (Page 19)

In fact, its design procedure applies the separation principle, which entails the automatic satisfaction of Assumption 1 and it does not include the constraints information in the selection of Ne and Nc. (Page 19)

On the other hand, the simultaneous MHE–MPC controller does [include constraints information in the selection of Ne and Nc]. (Page 19)

The paper also notes that the computational burden of the simultaneous MHE–MPC is lower than the independent one, as can be seen in Figure 6. (Page 21)

The execution times were averaged over 100 trials. The lower time, in the beginning, is due to the backward window corresponding to the estimation has not achieved yet its full length. (Page 21)

The independent MHE and MPC approach is more sensitive to disturbances, requiring conservative values of Nc to guarantee the closed-loop stability. (Page 19)

This figure shows that the independent MHE and MPC approach is more sensitive to disturbances, requiring conservative values of Nc to guarantee the closed-loop stability. (Page 19)

The simultaneous MHE–MPC controller manages to regulate the system states for all noise realizations. (Page 19)