arXiv subjects

Anders Rantzer

Publications and source records attributed to Anders Rantzer.

At least 19 recordsLinked to original sources

arXiv2026

Optimal input design via Frank-Wolfe

We study optimal input design over a finite horizon for linear dynamical systems. The goal is to minimize a weighted inverse-covariance (information) criterion subject to an energy budget. The set of covariances achievable by causal policies is convex but lacks a tractable explicit description, ruling out projection-based methods. We show that Frank--Wolfe applies naturally: each linear minimization subproblem is a budget-constrained finite-horizon linear quadratic (LQ) problem, solvable by a Riccati recursion and one-dimensional bisection over a Lagrange multiplier. Using smoothness of the objective over the feasible set, we establish an $\mathcal{O}(1/M)$ convergence rate for the objective value, while strong convexity yields an $\mathcal{O}(1/\sqrt{M})$ rate for the iterates. We further extend the framework to input design for system identification with unknown dynamics and adaptive online LQR, and illustrate the approach numerically.

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arXiv2026

Dual Control: On Exploration-Exploitation in Linear Systems

The term "dual control" refers to the dual objective of simultaneously balancing exploration and exploitation. Problems of this kind have been studied for nearly a century. This paper is devoted to theory and methodology relevant for optimal control of linear time-invariant systems whose parameters are initially unknown and must be learned by active probing. We review the main ideas underlying four major research directions: Multi-armed bandits, self-tuning regulators, regret rate minimizing controllers, and minimax optimal dual controllers. The first three have a long history and rich literature, whereas the fourth provides a promising framework for robust dual control.

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arXiv2026

Minimax optimal dual control of positive systems: an exact solution for scalar input-sign uncertainty

While recent advances in minimax dual control have led to exact solutions for uncertain general linear time-invariant systems as well as (sub)optimal dual controllers, corresponding results for linear positive systems are still lacking. This paper aims to fill this gap and thereby pave the way toward scalable dual control algorithms. We study the general minimax optimal dual control problem for positive linear systems with unknown dynamics and reformulate it as a standard zero-sum dynamic game. By allowing randomized control inputs, we solve the corresponding Bellman equation exactly for the scalar case with sign uncertainty in the input. This yields an implicit dual control policy that is optimal both in terms of cost and $\ell_1$-gain. The optimal dual policy uses exploration in a specific region of the hyperstate space to conduct optimal probing. Outside this exploration regime, the controller reduces to a deterministic certainty equivalence policy, indicating that sufficient information has been obtained to identify the correct input direction. In addition, these results allow us to analyze fundamental limitations of minimax dual control for positive systems and provide a foundation for more general dual control problems for positive systems for future work.

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arXiv2026

Minimax adaptive control for finite sets of positive linear systems

We present a minimax adaptive control framework for discrete-time positive linear systems with parametric uncertainty and adversarial disturbances. The uncertainty in the system dynamics is assumed to lie in a finite set of possible plants. We formulate the problem as a dynamic game between the controller, which minimizes the cost, and an adversary, which selects both the disturbances and the plant dynamics to maximize the cost. An equivalent reformulation of the original game transforms the problem into a standard minimax two-player zero-sum dynamic game. This enables the problem to be addressed via minimax dynamic programming. We provide an explicit solution to the Bellman inequality, yielding stabilizing, positivity preserving policies without requiring an initially stabilizing controller. The resulting controller enjoys robustness guarantees in the form of bounded $\ell_1$-gain from disturbances to errors. Once the uncertain parameters have been sufficiently estimated, the controller behaves like a standard $\mathcal H_\infty$-type optimal controller for positive linear systems. The theoretical findings are supported by numerical experiments illustrating the resulting adaptive controller in action.

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arXiv2026

L1 Optimal Control of Continuous-Time Stochastic Positive Systems

We present an L1-optimal control problem class with linear nonnegative costs subject to multiplicative It\^o diffusion processes with elementwise linear input constraints. Forward invariance of the positive orthant is established for the considered stochastic dynamics, and a simulation method consistent with this invariance property is proposed. Both finite-horizon and discounted infinite-horizon stochastic L1-optimal control problems are considered. These problems admit explicit solutions characterized by a vector-valued ordinary differential equation in the finite-horizon case and by an algebraic equation in the infinite-horizon case. Notably, the optimal value function and feedback policy coincide with those of the corresponding deterministic problem, demonstrating robustness to multiplicative stochastic uncertainty. A portfolio example illustrates our results.

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arXiv2026

Sparse State Feedback Control for Industrial Applications

We present an optimization-based methodology for designing sparse state-feedback controllers for industrial applications that are suited for linear control, and demonstrate the framework by designing a level controller for an industrial rougher flotation bank at the Aitik mine. In contrast to the dense linear-quadratic (LQ) controller gains currently operating at the concentrator, our approach enforces a sparsity pattern that is consistent with the interaction structure of the flotation bank and accounts for the worst-case expected inflow disturbances during tuning, while optimizing controller performance through the Integral Absolute Error (IAE) index. The non-zero elements of the sparse gain matrices are optimized using a coordinate search algorithm that handles bound constraints and preserves closed-loop stability. The resulting sparse controller achieves improved load disturbance rejection in the flotation cells compared to the LQ controller. These improvements are consistently observed in both linear and nonlinear simulations. In addition, the imposed structure, results in gain matrices that are easier to adjust and interpret. Importantly, the sparse controllers generated for the Aitik mine are directly suitable for industrial deployment and offer an effective alternative to the existing dense LQ design.

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arXiv2026

Model Predictive Control for Constrained Linear Positive Systems on Graphs

Positive systems describing networks with inherently non-negative states and inputs arise naturally in routing, logistics, and compartmental modelling. We consider problems modelled as positive linear systems in incidence form with linear cost. The addition of capacity constraints on states (storage) and inputs (flows between nodes) significantly increases the problem complexity. Leveraging the analytic structure of the unconstrained problem, an explicit suboptimal admissible controller is constructed. This yields graph-computable performance bounds and a minimum stabilising horizon length for a model predictive controller without terminal conditions. A convex program enables efficient computation of the optimal bound and horizon. These results highlight how system structure enables explicit MPC guarantees that are typically not available.

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arXiv2026

Minimax optimal dual control -- The single input case

An explicit solution is derived for the Bellman inequality corresponding to minimax optimal dual control. The minimizing player determines control action as a function of past state measurements and inputs. The maximizing player selects disturbances and model parameters for the underlying linear time-invariant dynamics. The optimal minimizing policy is a dual controller that optimizes the tradeoff between exploration and exploitation. Once sufficient data has been collected, the policy becomes a deterministic certainty equivalence controller. However, when data is insufficient, the policy introduces a randomized term to improve excitation.

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arXiv2026

On Integral Linear Constraints on Convex Cones

In this paper, we consider integral linear constraints and the dissipation inequality with linear supply rates for certain sets of trajectories confined pointwise in time to a convex cone which belongs to a finite-dimensional normed vector space. Such constraints are then shown to be satisfied if and only if a bounded linear functional exists which satisfies a conic inequality. This is analogous to the typical situation in which a quadratic supply rate over the entire space is related to a linear matrix inequality. A connection is subsequently drawn precisely to linear-quadratic control: by proper choice of cone, the main results can be applied to produce a known L1-gain analogue to the bounded real lemma in positive systems theory, as well as a non-strict version of the Kalman-Yakubovich-Popov Lemma in linear-quadratic control.

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arXiv2026

Positive Observers Revisited

The paper shows that positive linear systems can be stabilized using positive Luenberger-type observers. This is achieved by structuring the observer as monotonically converging upper and lower bounds on the state. Analysis of the closed-loop properties under linear observer feedback gives conditions that cover a larger class than previous observer designs. The results are applied to nonpositive systems by enforcing positivity of the dynamics using feedback from the upper bound observer. The setting is expanded to include stochastic noise, giving conditions for convergence in expectation using feedback from positive observers.

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arXiv2025

Minimax Optimal Adaptive Control for Systems on Cones

The theory of optimal control on positive cones has recently identified several new problem classes where the Bellman equation can be solved explicitly, in analogy with classical linear quadratic control. In this paper, the idea is extended to minimax adaptive control, yielding exact solutions to instances of the Bellman equation for dual control. In particular, this allows for optimization of the fundamental tradeoff between exploration and exploitation.

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arXiv2025

On Minimax Optimal Dual Control for Fully Actuated Systems

A multi-variable adaptive controller is derived as the explicit solution to a minimax dynamic game. The minimizing player selects the control action as a function of past state measurements and inputs. The maximizing player selects disturbances and model parameters for the underlying linear time-invariant dynamics. This leads to a Bellman equation that can be solved explicitly for the case with unitary B-matrix known up to a sign and no input penalty. The minimizing policy is a dual controller that optimizes the tradeoff between exploration and exploitation.

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arXiv2025

Linear Regulator-Based Synchronization of Positive Multi-Agent Systems

This paper addresses the positive synchronization of interconnected systems on undirected graphs. For homogeneous positive systems, a static feedback protocol design is proposed, based on the Linear Regulator problem. The solution to the algebraic equation associated to the stabilizing policy can be found using a linear program. Necessary and sufficient conditions on the positivity of each agent's trajectory for all nonnegative initial conditions are also provided. Simulations on large regular graphs with different nodal degree illustrate the proposed results.

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arXiv2025

Nonlinear Dynamical Unbalanced Optimal Transport: Relaxation and Duality

In this paper, we introduce a generalized dynamical unbalanced optimal transport framework by incorporating limited control input and mass dissipation, addressing limitations in conventional optimal transport for control applications. We derive a convex dual of the problem using dual optimal control techniques developed before and during the 1990s, transforming the non-convex optimization into a more tractable form.At the core of this formulation is the smooth sub-solutions to an HJB equation. A first-order algorithm based on the dual formulation is proposed to solve the problem numerically.

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arXiv2025

A Minimax Optimal Controller for Positive Systems

We present an explicit solution to the discrete-time Bellman equation for minimax optimal control of positive systems under unconstrained disturbances. The primary contribution of our result relies on deducing a bound for the disturbance penalty, which characterizes the existence of a finite solution to the problem class. Moreover, this constraint on the disturbance penalty reveals that, in scenarios where a solution is feasible, the problem converges to its equivalent minimization problem in the absence of disturbances.

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arXiv2024

Adaptive Control of Positive Systems with Application to Learning SSP

An adaptive controller is proposed and analyzed for the class of infinite-horizon optimal control problems in positive linear systems presented in (Ohlin et al., 2024b). This controller is derived from the solution of a "data-driven algebraic equation" constructed using the model-free Bellman equation from Q-learning. The equation is driven by data correlation matrices that do not scale with the number of data points, enabling efficient online implementation. Consequently, a sufficient condition guaranteeing stability and robustness to unmodeled dynamics is established. The derived results also provide a quantitative characterization of the interplay between excitation level and robustness to unmodeled dynamics. The class of optimal control problems considered here is equivalent to Stochastic Shortest Path (SSP) problems, allowing for a performance comparison between the proposed adaptive policy and model-free algorithms for learning the stochastic shortest path, as demonstrated in the numerical experiment.

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arXiv2024

On PI-control in Capacity-Limited Networks

This paper concerns control of a class of systems where multiple dynamically stable agents share a nonlinear and bounded control-interconnection. The agents are subject to a disturbance which is too large to reject with the available control action, making it impossible to stabilize all agents in their desired states. In this nonlinear setting, we consider two different anti-windup equipped proportional-integral control strategies and analyze their properties. We show that a fully decentralized strategy will globally, asymptotically stabilize a unique equilibrium. This equilibrium also minimizes a weighted sum of the tracking errors. We also consider a light addition to the fully decentralized strategy, where rank-1 coordination between the agents is introduced via the anti-windup action. We show that any equilibrium to this closed-loop system minimizes the maximum tracking error for any agent. A remarkable property of these results is that they rely on extremely few assumptions on the interconnection between the agents. Finally we illustrate how the considered model can be applied in a district heating setting, and demonstrate the two considered controllers in a simulation.

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arXiv2024

A Cone-preserving Solution to a Nonsymmetric Riccati Equation

In this paper, we provide the following simple equivalent condition for a nonsymmetric Algebraic Riccati Equation to admit a stabilizing cone-preserving solution: an associated coefficient matrix must be stable. The result holds under the assumption that said matrix be cross-positive on a proper cone, and it both extends and completes a corresponding sufficient condition for nonnegative matrices in the literature. Further, key to showing the above is the following result which we also provide: in order for a monotonically increasing sequence of cone-preserving matrices to converge, it is sufficient to be bounded above in a single vectorial direction.

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