Feasibility of Simultaneous Input-Output Constraints for Tracking in a Class of LTI Systems: Part I

arXiv:2610.02151 · eess.SY, cs.SY · Submitted 2026-10-01 · 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: "Feasibility of Simultaneous Input-Output Constraints for Tracking in a Class of LTI Systems".

Dev: Simultaneous satisfaction of input and output constraints for tracking in linear time-invariant (LTI) systems with multiple inputs and integral action is addressed by deriving necessary and sufficient conditions for a…

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

Title and authors: Rosa: So, to recap, this paper introduces a Control Barrier Function based governor designed specifically for LTI systems with multiple inputs and integral action to simultaneously handle output constraints and input constraints while tracking a desired command. Dev, how does that high-level summary translate into something concrete for our loop rate concerns?

Dev: Well, the core idea is using these CBFs—both high-order ones for the outputs like z and lower-order ones for inputs like u —to generate a modified command signal via a quadratic program. My main concern, Rosa, is that this whole process has to happen incredibly fast; we need to make sure that solving that program doesn't introduce unacceptable latency into our control loop.

Taro: From my side of things, I'm focused on the safety guarantees they offer; the paper establishes necessary and sufficient conditions for when this governor actually works, which is pretty significant because it moves beyond just "it might work." I’m really interested in that formal proof showing that if those specific conditions are met, we get forward invariance.

Rosa: That focus on forward invariance is huge for me, Taro; it means if our robotic system enters a safe zone defined by the constraints, the math guarantees it will stay there forever without needing constant external intervention or complex re-planning. Dev, can you elaborate on what those formal conditions look like in practical terms regarding failure modes?

Dev: The conditions are tied to that feasibility set F and Proposition one which states we need nu k(S) at least zero across all constraints defined by the plant matrices. If that minimum value drops below zero somewhere in our operating set S, the governor's proposed signal won't exist, and that signals a constraint violation or instability risk.

Taro: That makes sense; it’s essentially a mathematical check to see if the desired tracking performance clashes with the physical limits of the system. So, if we can find those right parameters alpha one we have a solid mathematical foundation for our control strategy, even when things get weird.

Rosa: Exactly! The authors demonstrate that they can always find these parameters through a systematic design procedure involving that algorithm, even if the initial setup is challenging. That systematic solvability is what makes this approach so appealing; it’s not just a theoretical existence proof, it’s a recipe for building the controller.

Dev: I agree about the design procedure being solvable, Rosa; that roadmap helps us move away from trial-and-error tuning when we're trying to deploy this on hardware with strict timing requirements. If we can use Algorithm one to find those thresholds for alpha two alpha u, and alpha g systematically, it reduces the guesswork significantly.

Taro: And that leads me to thinking about the real-world application; imagine a complex multi-joint robot needing to track a path while simultaneously managing motor saturation limits and sensor noise—this paper gives us the language to formalize exactly how those competing needs interact.

Rosa: That’s what I’m picturing; it moves us closer to systems that don't just follow commands but actively manage their own safety boundaries in real-time. This work opens up possibilities for building much more sophisticated autonomous agents that can operate reliably in environments where physical limits are constantly being tested.

The paper's summary: Taro: So, to summarize, the authors propose a systematic design procedure for tuning the parameters of their I-O Control Barrier Function governor, which is crucial for ensuring that safety and performance goals are met at once. Rosa, what’s the big takeaway from that design procedure?

Rosa: The main point is that they've shown we can systematically adjust those free parameters—like alpha one to guarantee feasibility under SIOCF on the set SE, which essentially gives us a predictable way to tune the controller rather than just guessing settings. Dev, how does this systematic tuning approach affect the practical deployment of these controllers?

Dev: It really helps by turning a complex, non-linear optimization problem into an iterative design process where we can control the trade-off between tracking accuracy and constraint satisfaction more deliberately. However, the paper flags that we still have to deal with the underlying complexity of integral action states, which might introduce its own kind of transient failure modes if our initial assumptions about those dynamics aren't perfectly aligned with reality.

Rosa: That’s a fair point about the integral action; I wonder how long these guarantees hold up in a truly dynamic, non-linear environment outside of the idealized LTI setup they started with? Taro, what do you think about pushing those constraints when the system is under duress?

Taro: The paper shows that when SIOCF isn't satisfied initially, we can increase the relaxation factor to enlarge our feasible set SE; this implies that we can intentionally accept a larger tracking error or a more aggressive input constraint relaxation to maintain safety. This is important for autonomy because it means the system has an explicit mechanism for prioritizing survival when things get messy.

Dev: That’s exactly what I was thinking; it gives us an explicit strategy for constraint management, which is better than just having a generic safety layer that kicks in blindly. We can quantify the cost of relaxing the input constraint and make an informed decision on how much performance we're willing to sacrifice.

Rosa: So, instead of just hoping a controller works, we have a structured way to design one that handles conflicting demands between tracking and safety. This moves us away from brittle controllers toward more resilient ones for field robotics.

Taro: I think the real implication here is that this framework provides a rigorous mathematical language for how autonomous systems should manage their operational envelopes under uncertainty, which is vital when dealing with unpredictable external factors in navigation or manipulation tasks.

Dev: It’s definitely a solid step forward because it provides the theoretical backing needed to trust these kinds of constraint-aware control laws for longer periods in demanding operational scenarios.

Rosa: This work really shows how we can build controllers that are not only precise but also inherently safe and robust against actuator limitations, which is something every field roboticist needs to see implemented.

The paper's improvements: Taro: So, to wrap up this discussion on "Feasibility of Simultaneous Input-Output Constraints for Tracking in a Class of LTI Systems: Part I," we’ve seen how this Control Barrier Function governor provides a formal method to balance tracking performance with hard safety limits. Dev, what do you see as the biggest practical impact of proving these necessary and sufficient conditions?

Dev: The biggest practical impact is moving us toward controllers that are designed with explicit constraint awareness from the start rather than having safety layers bolted on later, which is a huge win for loop rate stability and failure modes. It gives us a roadmap for designing systems that inherently manage those input and output limitations simultaneously.

Taro: I think this work provides the mathematical rigor we need to push autonomy into more constrained physical spaces where uncertainty is high; having these formal guarantees allows us to design agents that can operate in environments where the world misbehaves without instantly breaking safety protocols.

Rosa: That’s a powerful thought, Taro; it means we can build systems that are designed to survive unexpected behavior, not just those that perform perfectly in a vacuum. Dev, any final thoughts on the deployment timeline for this kind of robust control?

Dev: I think the immediate challenge remains translating these formal conditions into fast enough computations for real-time hardware, but the design procedure they laid out gives us a very clear path to optimizing those parameters efficiently.

Taro: We should definitely keep an eye on how they address the integral action states in future work; that’s where I think we can really test the limits of this governor's robustness against long-term drift.

Rosa: Agreed, Taro; seeing how this holds up when those dynamics evolve over extended periods is what I want to see next, especially concerning field deployments outside of a perfectly controlled lab setting.

Dev: Well, moving on from this paper, we’ve got some interesting stuff coming up in the papers about trajectory generation using Bernstein-Fourier approximants for optimal path planning. That sounds like a great topic for our next discussion.

Conclusion: Rosa: So we've looked at "Feasibility of Simultaneous Input-Output Constraints for Tracking in a Class of LTI Systems: Part I," and it really lays out how to use Control Barrier Functions to manage tracking goals alongside hard limits on inputs and outputs for systems with integral action.

Dev: That’s right, Rosa; the core concept is building a modified command signal through a quadratic program constrained by high-order CBFs that handle both output and input limits. I still have my head in the loop rate concerns, though; we need to make sure solving that program doesn't introduce latency into our real-time hardware execution.

Taro: For me, the formal proof of necessary and sufficient conditions for forward invariance is what really sells it; it gives us a solid mathematical foundation to push autonomy into more constrained physical spaces where uncertainty is high.

Rosa: It does, Taro; that guarantee means if our robotic system enters a safe zone defined by those constraints, the math ensures it stays there forever without needing constant external re-planning when things go wrong.

Dev: I agree about the robustness; it gives us an explicit strategy for constraint management instead of just relying on a generic safety layer that kicks in blindly. We can quantify the cost of relaxing input constraints and make an informed decision on how much performance we're willing to sacrifice.

Taro: That's exactly what makes it important for autonomy because it provides a rigorous mathematical language for how systems should manage their operational envelopes under uncertainty, which is vital when dealing with unpredictable external factors in navigation or manipulation tasks.

Rosa: This work shows how to build controllers that are not only precise but also inherently safe and robust against actuator limitations, which is something every field roboticist needs to see implemented.

Dev: It's definitely a solid step forward because it provides the theoretical backing needed to trust these kinds of constraint-aware control laws for longer periods in demanding operational scenarios. The challenge remains in implementing this with low latency and handling the integral action state correctly in a real-time control loop.

Taro: We should definitely keep an eye on how they address those integral action states in future work; that's where I think we can really test the limits of this governor's robustness against long-term drift.

Dev: Well, moving on from this paper, we've got some interesting stuff coming up in the papers about trajectory generation using Bernstein-Fourier approximants for optimal path planning. That sounds like a great topic for our next discussion.

Massachusetts Institute of Technology · The Boeing Company

eess.SY, cs.SY

Submitted: 2026-10-01

Updated: 2026-10-08

Comments: 14 pages, submitted to ACC 2027; v2: revised abstract, added the arXiv number of the companion paper (Part II, arXiv:2610.04742)

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

Importance score: 83/100

The gist: Simultaneous satisfaction of input and output constraints for tracking in linear time-invariant (LTI) systems with multiple inputs and integral action is addressed by deriving necessary and

Key concepts

Control Barrier Function (CBF): A mathematical tool used to ensure system states stay within a safe region by defining functions that must remain non-negative. These functions are constructed based on desired constraints, allowing the controller to actively prevent unsafe behavior in real-time.
I-O CBF Governor: A specific control strategy designed to generate the necessary command signal ($y_g$) for a system. It uses two pairs of CBFs—one set for output constraints and another for input constraints—to ensure both tracking goals and physical limits are met.
SIOCF Condition: This is the core mathematical requirement proving feasibility. It states that the I-O CBF Governor works correctly if, for all bounded desired signals, there exists a feasible set of inputs ($y_g$) that keeps the system within its safe set $S$ and respects all constraints.
Quadratic Program (QP): A mathematical optimization problem used to solve for the modified command signal ($y_g$). The governor minimizes an error term while subject to constraints derived from the CBFs, ensuring the resulting signal is both optimal for tracking and safe for physical limits.

Terminology

Control Barrier Function — 16× in this paper
Closed-Loop System — 7× in this paper · explained in 4 episodes
alpha — 5× in this paper · explained in 1 episode
Conference on Decision and Control — 4× in this paper
Core — 3× in this paper · explained in 2 episodes
Failure Modes — 3× in this paper · explained in 3 episodes
Robustness — 3× in this paper · explained in 5 episodes
Control Barrier Function (CBF) — 2× in this paper · explained in 5 episodes
Accuracy — 1× in this paper · explained in 1 episode
American Control Conference — 1× in this paper
Closed-Loop Control — 1× in this paper · explained in 2 episodes
Control Barrier Functions (CBF) — 1× in this paper · explained in 2 episodes
Curse of Dimensionality — 1× in this paper · explained in 6 episodes
Modular Architecture — 1× in this paper · explained in 4 episodes

Summary

Simultaneous satisfaction of input and output constraints for tracking in linear time-invariant (LTI) systems with multiple inputs and integral action is addressed by deriving necessary and sufficient conditions for a Control Barrier Function (CBF)-based governor to guarantee safety, boundedness, and forward invariance.

Problem Formulation

The paper considers a class of LTI plants with integral action described by the state-space equation:

x˙ p = Apxp + Bpu, y = Cregxp + Dregu, e˙I = y − yg. The objective is to find a modified command signal yg such that all states remain bounded, the output tracks a desired signal ycmd as closely as possible, and constraints on the input u and relevant outputs z are satisfied at all times. This leads to Problem 1: given the closed-loop system (5) and constraint requirement (2), find a modified command yg that ensures closed-loop stability, tracking, and constraint satisfaction.

I-O CBF Governor Design

The proposed solution is an I-O Control Barrier Function Governor designed to generate the modified command signal yg. This governor is constructed using two pairs of CBFs:

Output constraints are handled by high-order CBFs:

h+1(¯x):= zmax − Czx, h−1(¯x):= Czx¯ − zmin.

Input constraints are handled by higher-order CBFs:

h+u(¯x):= umax + Kfullx, h−u(¯x):= −Kfullx¯ − umin.

The governor generates the command signal yg via a quadratic program (6a):

yg(¯x, ycmd) = arg min ζ∈Rm ζ − ycmd2 s.t. ψ±2(¯x, ζ) ≥ 0, ψ±u(¯x, ζ) ≥ 0, ζ∞ ≤ αg.

The feasibility of this governor is determined by the set F(¯x), defined by the constraints in (7):

F(¯x):= ζ ∈ Rm: Aζ ζ ≤ b(¯x), where b(¯x):= Bζx¯ + b0 and Aζ, Bζ are matrices derived from plant parameters and CBF gains. Feasibility of (9) implies that all I-O constraints (2) are satisfied.

Necessary and Sufficient Conditions for Feasibility

The core result is the establishment of necessary and sufficient conditions for when input and output (I-O) constraints must be satisfied simultaneously with closed-loop boundedness using the CBF Governor. This is formalized by Definition 1, SIOCF: the I-O CBF Governor satisfies SIOCF on a non-empty closed set S ⊆ Ω if, for all bounded piecewise continuous ycmd and ∀x¯(t0) ∈ S, the solutions of (5) satisfy existence and uniqueness of yg (P1), x¯(t) ∈ S ∀t ≥ t0 (P2), z(t) ∈ Z and u(t) ∈ U ∀t ≥ t0 (P3), and yg(t)∞ ≤ αg ∀t ≥ t0 (P4).

The SIOCF condition on S is formalized by Proposition 1: F(¯x) ≠ ∅ ∀x¯ ∈ S if and only if min νk(S) ≥ 0, where νk(S):= inf x¯∈S λk⊤b(¯x), k = 1,..., N.

Solution Strategy and Design Procedure

The paper proposes a systematic design procedure for choosing the free parameters of the CBF governor, denoted as α1:= [α1 α2 αu αg]⊤. This design procedure is summarized in Algorithm 1 and involves three steps:

Determine initial parameters based on initial conditions X0 and construct an RPI set E and SE(ϱ).
Determine parameters (α2, αu, αg) that alter the quadratic program (6) but do not affect the set Ω or SE(ϱ).
If SIOCF is not satisfied in Steps 1 and 2, relax the input constraint by increasing ϱ to enlarge SE(ϱ), choosing thresholds α⋆g and ϱ⋆ defined in (24).

The paper proves that this design procedure is always solvable, demonstrating that the SIOCF condition can always be satisfied by expanding the design space where the input constraints are relaxed. The final result shows that for a given α1, there exist thresholds for (α2, αu, αg) such that the SIOCF condition holds on SE(ϱ).

Improvements for AI systems

Based on the provided scientific paper, here are specific improvements to AI systems that could be achieved by implementing the proposed I-O Control Barrier Function (CBF) Governor:

The core capability derived from this paper is a robust mechanism for tracking desired commands in complex Linear Time-Invariant (LTI) systems while strictly enforcing safety constraints on both system outputs and control inputs.

Here are the specific improvements and functionalities:

Simultaneous Input-Output Constraint Satisfaction for Tracking: The system can track a desired trajectory or set of reference signals, such as a robotic arm's end-effector position or an autonomous vehicle's path, while guaranteeing that the physical actuators never exceed their torque/force limits (input constraints) and that critical sensors/outputs never hit dangerous thresholds (output constraints) simultaneously.
Robustness to Model Uncertainty: The system can maintain constraint satisfaction even when the underlying LTI model contains unmodeled dynamics or parameter variations, provided the safety function and controller gains are designed within the derived feasibility conditions (especially when using the relaxed input scaling factor, ϱ).
Adaptive Constraint Handling via Governor Tuning: The system can dynamically adjust its control action to satisfy constraints in real-time. If a constraint becomes tighter or a reference signal demands high effort, the CBF Governor automatically modifies the command signal (the modified command signal) to maintain safety, effectively acting as an intelligent anti-windup mechanism tailored for integral action systems.
Optimal Constraint Relaxation Strategy: The system can perform smart constraint management by minimizing the necessary relaxation of input constraints (i.e., finding the smallest possible factor ϱ > 1) required to maintain safety while still achieving tracking performance, as systematically explored in Section V-B of the paper.
Guaranteed Forward Invariance: For critical subsystems, the control law ensures that if the system starts within a safe operating region (set S), it is mathematically guaranteed to remain within that set for all future time, providing strong long-term stability guarantees for safety-critical applications like medical devices or aerospace control.
High-Order System Management: The architecture supports systems with multiple inputs and integral action (e.g., chemical processes, complex multi-joint robots) by utilizing a sophisticated set of high-order CBFs (HOCBFs) and first-order input CBFs, allowing for precise management of coupled dynamics.

Sources

Feasibility and Explicit Safety Filters for Control Barrier Functions in Linear Systems