Deception Against Data-Driven Linear-Quadratic Control

arXiv:2506.11373 · eess.SY, cs.SY · Submitted 2025-06-13 · 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: "Deception Against Data-Driven Linear-Quadratic Control".

Dev: Deception is a common defense mechanism against adversaries with an information disadvantage, forcing them to select suboptimal policies for a defender’s benefit.

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

Title and authors: Rosa: So, delving deeper into the actual mechanics described in "Deception Against Data-Driven Linear-Quadratic Control," the paper summarizes this as a defensive strategy where the defender leverages its knowledge of system dynamics to inject feedback that actively misleads an adversary into choosing a policy that is less damaging than what they would otherwise find optimal.

Dev: It essentially boils down to setting up a constrained optimization problem where the defender wants to pull the adversary's learned gain toward a pre-chosen, benign target, while simultaneously ensuring the deception input itself doesn't cause any instability in our original control loop.

Taro: That framing is interesting because it shifts the focus from just detecting an attack to actively designing a counter-strategy that alters the learning outcome of the adversary entirely.

Rosa: It suggests that instead of just hardening our system against known attack patterns, we can design an input that makes the attacker's own optimization process lead them down a different path altogether, which is a pretty strong concept.

Dev: The paper highlights that this entire problem maps onto solving two coupled equations: the algebraic Riccati equation and the Lyapunov equation, and then they use a block successive over-relaxation algorithm to find their numerical answers for the deception gain vector.

Taro: And I think what's particularly compelling is how they show this applies not just to standard data-driven control, but also to simpler cases involving minimizing data-driven linear-quadratic regulators, which broadens its applicability.

Rosa: That extension shows the underlying mathematical structure is robust enough to handle a wider variety of control objectives and system setups than what might be initially assumed in simpler analyses.

Dev: However, we have to remember that the paper itself notes that analytically solving those coupled equations is very difficult, which is why they rely on this numerical method, meaning their results are contingent on the convergence properties of that specific iterative scheme.

Taro: If the adversary's objective changes mid-game or if the system dynamics drift significantly from what was modeled, I wonder how quickly this deception strategy can adapt to maintain its effectiveness against a changing threat landscape.

Rosa: That speaks to the long-term viability of such a defense; it’s not just about solving one optimization problem once, but having a mechanism that can continually re-evaluate and adjust the deceptive input as the environment evolves.

Dev: Exactly, and that ties back to our earlier point about loop rate; if the iterative solver takes too long to produce an update, we fall out of sync with the system dynamics we are trying to protect.

Taro: So, while mathematically sound for a static setup, I'm still thinking about dynamic adaptation when the environment itself is non-stationary, which is where real-world deployment gets tricky.

Rosa: We should keep in mind that this paper focuses heavily on designing the optimal gain vector initially and proving convergence to a stationary point of the deception problem before we worry about continuous online adaptation.

Dev: That’s a fair caveat; they prove it converges to a stationary point, which is good for initial design, but we need more research into how that translates to continuous operation without constant re-solving.

The paper's summary: Rosa: Moving on to what this research proposes as improvements, it suggests that a key direction is developing an active deception module within the defender’s architecture that learns to inject those deceptive inputs based on the defender's internal model of the system dynamics.

Dev: That means we're not just designing a fixed gain vector; we're building an AI component that can learn *when* and *how much* to deceive, which implies a much more intelligent, adaptive defense mechanism than just a pre-set input.

Taro: That level of learning would allow the system to be proactive; instead of waiting for an attack to occur, it could start injecting subtle deceptive inputs preemptively based on its understanding of potential adversarial behavior.

Rosa: Proactive deception sounds like a big step in terms of autonomy; it moves the system from a reactive defense posture to an anticipatory one, which is something we've been striving for in complex robotic systems.

Dev: From an engineering view, that learning component introduces complexity, but if it can be trained efficiently using techniques like the ones discussed in related papers on federated learning, it could be scalable for distributed control networks.

Taro: I’m also thinking about how this proactive learning interacts with other consensus mechanisms; if the system is trying to achieve subspace consensus, the deception module would need to coordinate its deceptive inputs with those consensus goals.

Rosa: That integration of different AI capabilities—model-based understanding feeding into an active manipulation strategy—seems like it could significantly enhance overall system performance when facing sophisticated adversaries.

Dev: The paper also suggests that this proactive approach could be a way to build robustness against data poisoning attacks by designing deception specifically to force the attacker to learn a policy that is benign, even when the training data itself is corrupted.

Taro: That idea of using deception as a filter against poisoned training data sounds like a very powerful defense mechanism for machine learning systems relying on large datasets.

Rosa: So, essentially, the paper pushes us toward creating an AI that doesn't just react to errors but designs its own inputs to steer the adversarial learning process toward a safer outcome through learned deception.

Dev: It’s ambitious, and we need to ensure that this learning loop is stable and converges quickly enough so it doesn't introduce unacceptable lag into our control loops during actual operation.

Taro: That stability requirement is crucial; we don't want the mechanism designed to be more unstable than the attack it's trying to counteract.

The paper's improvements: Rosa: To wrap up our discussion on "Deception Against Data-Driven Linear-Quadratic Control," we see that this work provides a concrete mathematical framework for using deception as an active defense against adversaries who have an information disadvantage in control systems.

Dev: We've seen how the paper leverages a block successive over-relaxation algorithm to numerically solve the coupled Riccati and Lyapunov equations, giving us a viable path toward implementing this design in control engineers' work.

Taro: And for autonomy researchers, it shows that we can build systems capable of actively manipulating the adversarial learning process to ensure that even when things go wrong in the field, our system converges to a stable outcome.

Rosa: It really suggests a way forward for making complex AI agents more robust by incorporating proactive deception into their core control design philosophy.

Dev: The challenge remains ensuring that this proactive approach can be implemented reliably within the latency constraints of high-speed feedback loops when the system is running in a demanding operational setting.

Taro: I think the potential impact here is significant because it gives us a tool to defend against adversaries who are exploiting our lack of complete knowledge, whether that's in robotics or other complex control domains.

Rosa: We’re looking forward to seeing how this framework translates into practical, deployed systems and maybe even field robotics where real-world constraints are the main challenge.

Dev: Next time we look at a paper, we’ll focus on how the latency and communication efficiency play into these control strategies in more detail.

Conclusion: Rosa: So, to wrap up our discussion on "Deception Against Data-Driven Linear-Quadratic Control," we've looked at how this paper proposes using deceptive feedback to steer an adversary away from optimal attack policies by exploiting the defender's knowledge of system dynamics.

Dev: That was a lot of heavy math, but the block successive over-relaxation algorithm is definitely a solid numerical tool for tackling those coupled equations when you can't solve them analytically.

Taro: I'm still thinking about how this proactive deception translates into real-world resilience; if the world misbehaves and we lose our model accuracy, how quickly can this AI system pivot its deceptive strategy to stay effective?

Rosa: It really makes you wonder about the long-term viability of such a defense outside of a pristine lab setting, Dev. Can we trust this deception mechanism to keep working reliably over extended periods in a harsh environment?

Dev: That latency issue is definitely something I'd want to stress more; if the time it takes for the AI to calculate that optimal deception gain pushes the control loop out of sync, then even a perfect strategy is useless.

Taro: And from an autonomy standpoint, this implies that systems won't just be about reacting to errors but about actively fighting against attempts to compromise their underlying learning algorithms.

Rosa: It’s exciting to think about what this means for field robotics; imagine a robot facing an unknown threat and being able to subtly trick the attacker into finding a much safer control policy.

Dev: I agree, and that proactive stance is exactly what we need when dealing with adversarial data poisoning or model-based attacks that try to exploit our control structure.

Taro: It suggests that the future of autonomy might involve systems designed not just to execute tasks, but to actively manage the learning process itself against malicious influence.

Rosa: Anyway, we've covered a lot about this paper, "Deception Against Data-Driven Linear-Quadratic Control," and I think it opens up some really interesting avenues for how we design more resilient intelligent systems.

Dev: It certainly does provide a strong mathematical foundation for building these kinds of adaptive defense modules in control engineering applications.

Taro: Next time, we should look into those dual problem extensions to see if that same deception logic holds up when the adversary is trying to mislead the regulator itself.

Oden Institute for Computational Engineering & Sciences, University of Texas at Austin · KTH Royal Institute of Technology · Georgia Institute of Technology

eess.SY, cs.SY

Submitted: 2025-06-13

Updated: 2026-10-01

Comments: Final submission to IEEE Transactions on Automatic Control

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 83/100

The gist: Deception is a common defense mechanism against adversaries with an information disadvantage, forcing them to select suboptimal policies for a defender’s benefit.

Key concepts

Deception Gain: This is the input vector designed by the defender to inject misleading information into a data-driven system. Its purpose is to manipulate the adversary's learning process so that it converges on an attack policy that benefits the defender, rather than the optimal damaging one.
Constrained Optimization Problem: The core design challenge involves balancing two conflicting goals: first, steering the adversary's learned policy toward a specific benign gain; and second, keeping the actual deceptive feedback as small as possible to ensure the system remains stable and behaves nominally.
Block Successive Over-Relaxation: Since solving the resulting algebraic Riccati equation and Lyapunov equation analytically is difficult, this numerical algorithm is used. It iteratively solves three coupled matrix equations—for $P_u$, $ ilde{P}_i$, and the deception gain $ ilde{oldsymbol{eta}}$—to find a numerical solution to the complex optimization problem.
KupΛq vs K¯F: This term represents the objective function being minimized during the optimal deception design. Minimizing this difference directly corresponds to minimizing how far the adversary's resulting learned gain is from a pre-selected, desired benign gain.

Terminology

Robustness — 8× in this paper · explained in 5 episodes
Convergence Guarantees — 7× in this paper · explained in 3 episodes
Trace — 5× in this paper · explained in 2 episodes
Benchmark — 4× in this paper · explained in 2 episodes
Closed-Loop System — 4× in this paper · explained in 4 episodes
Adaptation — 2× in this paper · explained in 1 episode
Core — 2× in this paper · explained in 2 episodes
Regularization — 2× in this paper · explained in 3 episodes
Steering — 2× in this paper · explained in 2 episodes
AI Agents — 1× in this paper · explained in 2 episodes
Accuracy — 1× in this paper · explained in 1 episode
Decentralized Learning — 1× in this paper · explained in 2 episodes
Dynamic Adaptation — 1× in this paper · explained in 2 episodes
Federated Learning — 1× in this paper · explained in 14 episodes
Nash Equilibrium — 1× in this paper · explained in 2 episodes
Online Adaptation — 1× in this paper · explained in 3 episodes
Regret — 1× in this paper · explained in 3 episodes
Reinforcement Learning — 1× in this paper · explained in 4 episodes
System Identification — 1× in this paper · explained in 3 episodes
Theoretical Guarantees — 1× in this paper · explained in 2 episodes

Summary

Deception is a common defense mechanism against adversaries with an information disadvantage, forcing them to select suboptimal policies for a defender’s benefit.

The gist: The defender exploits its information advantage by injecting deceptive feedback into a data-driven system to mislead the adversary into learning an attack that is different from the optimal one.

Problem Formulation

The core problem considers a setting where an adversary tries to learn the optimal linear-quadratic attack against a dynamical system whose dynamics it does not know, while the defender knows its dynamics and injects deceptive input to steer the learned policy towards a pre-selected benign gain. This setup is formulated as a constrained optimization problem balancing two objectives: i) steering the adversary’s learned policy towards an a priori chosen benign gain, and ii) using as small of deceptive feedback as possible to maintain the nominal closed-loop characteristics and stability. The mathematical characterization of this design problem boils down to solving a coupled algebraic Riccati equation (ARE) and a Lyapunov equation (LE).

Optimal Deception Design

The optimal deception gain is characterized as the solution to the constrained optimization problem: "i) the desire to steer the adversary’s learned policy towards an a priori chosen benign gain; and ii) the desire to use as small of deceptive feedback as possible to maintain the nominal closed-loop characteristics and stability. The defender designs its input, denoted by a gain vector, aiming for a specific result. This design problem is formally specified by minimizing the distance between the resulting learned gain and a target benign gain: minimize KupΛq ´ K¯F." The optimal deception gain is then derived as the solution to this constrained optimization problem, which involves a regulation term to ensure stability and penalize large deception gains.

Numerical Solution via Block Successive Over-Relaxation

Since the coupled algebraic Riccati and Lyapunov equations resulting from the problem formulation are challenging to tackle analytically, a numerical approach is employed. The proposed method is a block successive over-relaxation algorithm to extract the solution numerically, and its convergence under certain conditions is proven. This algorithm iteratively solves three coupled matrix equations:

An ARE (Equation 15) for the matrix Pu, given a current gain Λi.
A LE (Equation 16) for the matrix Πi, given a current gain Λi and Pu i+1.
The update rule for the deception gain ΛGS derived from these solutions: ΛGS “ –Γ´1B T u P i+1 u Π i+1.

Convergence Guarantees and Robustness

The convergence of Algorithm 1 is rigorously proven through analysis of the generalized cost function. The iterative update rule for the gain is interpreted as a gradient iteration on a generalized cost function, which leads to the conclusion that Algorithm 1 converges to a stationary point of the deception problem (11). Furthermore, under certain conditions on the regulation matrix Γ, it is shown that this stationary point corresponds to a minimum. The proof establishes:

- Convergence:

limiÑ8 dJ˜pΛi q dΛ “ 0, that is, the stationarity equations (14a)-(14c) hold in the limit.

- Minimality:

there exists γ‹ ą 0, such that if Γ ą γ‹I then Algorithm 1 converges to a minimum of (11).

Simulation and Validation

The framework is validated through simulations on a benchmark aircraft model. Case 1 demonstrates that the proposed algorithm can mislead adversaries into learning attacks that are less performance-degrading, showing a substantial suppression of the optimal attack gain compared to the nominal one. Case 2 shows that even when the nominal optimal attack is destabilizing, deception can force the adversary to learn a more benign attack with stable closed-loop eigenvalues. The results show that for small values of R, while the nominal attack may destabilize, a deceptive gain can be found such that the learned attack does not disrupt stability. The robustness analysis confirms that even when the adversary's weighing matrices are different from those assumed in the derivation, the learned adversarial attack remains suppressed.

Dual Problem Extensions

The paper also extends these results to dual setups, including deceiving minimizing data-driven linear-quadratic regulators. In this dual scenario, an adversarial perturbation is injected to force the defender to converge to an incorrect linear-quadratic gain. The analysis shows that a global minimizer exists for this dual problem under the condition that RanpBaq Ď RanpBuq, meaning the attacker does not have more input channels than the defender has control over. This confirms that Algorithm 1 can be adapted to solve these related problems, with convergence guarantees following the same lines as established in Section V.

Improvements for AI systems

Based on the provided research paper, here are specific improvements that could be made to AI systems, categorized by the control/learning aspect they would enhance:

The core contribution of this paper is developing a method for an agent (the defender) to strategically deceive another learning agent (the adversary) within a data-driven Linear-Quadratic Control (LQG) framework. The improvements focus on making AI systems more resilient, less susceptible to data poisoning, and better at learning optimal policies under uncertainty.

Here are the specific improvements:

The AI system can be equipped with a Deception Module that learns to inject deceptive inputs based on its knowledge of the system dynamics (i.e., knowing its own model).
The Deception Module can actively manipulate the data streams or feedback signals used by an adversary (which might be another AI agent or a learning algorithm) to steer that adversary toward a suboptimal, benign policy instead of the true optimal attack/control gain.
The system will utilize a numerical optimization technique called the Block Successive Over-Relaxation Algorithm to solve the complex, coupled algebraic Riccati and Lyapunov equations that govern this deception strategy, allowing for real-time or near real-time strategic deception design.
The system can perform robust policy learning against data poisoning attacks by proactively designing a deceptive input that forces the attacker to learn an attack gain different from the one they would otherwise find optimal using corrupted data.
The system can implement Moving Target Defense by dynamically altering its perceived system dynamics (by injecting the deceptive input) to constantly mislead an adversary, preventing them from ever converging on a stable or effective malicious policy.

Specific capabilities of the improved AI System:

It can autonomously detect when it is being targeted by an adversary attempting to learn its optimal control strategy.
If targeted, it can immediately deploy a deceptive input designed to make the attacker's learned policy less harmful (e.g., reducing energy consumption or preventing instability) compared to what the attacker would have learned under normal circumstances.
It ensures that when learning from noisy or adversarial data, its resulting control policy converges not to an unstable or highly aggressive attack, but to a benign version of the control strategy that maintains system stability even under attack.
It can maintain high performance and stability even when facing sophisticated data corruption designed specifically to mislead its internal learning mechanisms.

Abstract

Deception is a common defense mechanism against adversaries with an information disadvantage. It can force such adversaries to select suboptimal policies for a defender's benefit. We consider a setting where an adversary tries to learn the optimal linear-quadratic attack against a system, the dynamics of which it does not know. On the other end, a defender who knows its dynamics exploits its information advantage and injects a deceptive input into the system to mislead the adversary. The defender's aim is to then strategically design this deceptive input: it should force the adversary to learn, as closely as possible, a pre-selected attack that is different from the optimal one. We show that this deception design problem boils down to a solution of a coupled algebraic Riccati and a Lyapunov equation which, however, are challenging to tackle analytically. Nevertheless, we use a block successive over-relaxation algorithm to extract their solution numerically and prove the algorithm's convergence under certain conditions. We perform simulations on a benchmark aircraft, where we showcase how the proposed algorithm can mislead adversaries into learning attacks that are less performance-degrading.