arXiv subjects

Johan Löfberg

Publications and source records attributed to Johan Löfberg.

14 recordsLinked to original sources

arXiv2026

Static output-feedback stabilization is NP-hard

We prove that unrestricted static output-feedback stabilization is NP-hard, already in the single-input multiple-output case. More precisely, we give a polynomial-time many-one reduction from the NP-complete Betweenness problem. Given an instance with $n$ elements and $m$ betweenness constraints, the reduction constructs integer matrices $A\in\mathbb Z^{D\times D}$, $B\in\mathbb Z^{D\times1}$, and $C\in\mathbb Z^{n\times D}$ such that the instance is satisfiable if and only if there exists an unrestricted real row vector $K\in\mathbb{R}^{1\times n}$ for which $A+BKC$ is Hurwitz. The state dimension is $D=6(m+n)$, and every matrix entry has logarithmic bit length in the input size. The reduction uses one universal cubic source polynomial whose Routh determinant enforces the sign condition encoding betweenness, while fixed anchor values provide a mechanism for excluding spurious stabilizing gains of arbitrarily large magnitude. Frequency separation then compresses the simultaneous source conditions into one common affine polynomial, which is realized directly as the characteristic polynomial of $A+BKC$. As a by-product, deciding whether an affinely parameterized rational polynomial family contains a Hurwitz member on a rational box is NP-hard.

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arXiv2026

Second Order Zarankiewicz Number

We introduce the \emph{second order Zarankiewicz number} $z_2(m,n)$ for irreducible doubly simple biquadratic forms with $|E_1|=z(m,n)$, together with the intermediate recursive-line and signed parameters $z_{RL}(m,n)$ and $z_{SL}(m,n)$. They satisfy the unconditional hierarchy \[ \operatorname{BSR}(m,n) \ge z_2(m,n) \ge z_{SL}(m,n) \ge z_{RL}(m,n) \ge z_{wL}(m,n) \ge z(m,n), \] where $z_{RL}$ is defined by the strengthened recursive rectangle criterion $(RW3^+)$ together with the conditions $(S)$ and $C_4$-freeness of $G_1$, and $z_{SL}$ is defined by the signed criterion $(RW3^\pm)$. We show that $(RW3^+)$ is sound and strictly weaker than the literal weak cross-cell test $(W3)$ on the weak-admissible class, and that $(RW3^\pm)$ is likewise sound. At the smallest four-column cases we obtain \[ z_2(5,4)=z_{SL}(5,4)=z_{RL}(5,4)=13>12=z_{wL}(5,4), \] and \[ z_2(6,4)=z_{SL}(6,4)=z_{RL}(6,4)=16>14=z_{wL}(6,4). \] In three columns this yields \[ z_2(m,3)=z_{RL}(m,3)=2m \qquad \text{for all } m\ge 3, \] with strict separation from $z_{wL}(m,3)$ for every $m\ge10$, and exact gap $\lfloor(m-3)/3\rfloor$ for $m\ge16$. Further finite computations give $z_{RL}(5,5)=17$, $z_{RL}(7,4)=19$, and $z_{RL}(7,7)\ge32>28=z_{wL}(7,7)$. Along $N=2p$ with $p$ an odd prime, we obtain the cubic asymptotic separation \[ z_2\!\left(\binom{N}{2},N\right)-z_{wL}\!\left(\binom{N}{2},N\right)\ge \left(\frac{1}{16}-o(1)\right)N^3. \] For the exceptional incidence case $p=3$ we prove $z_{SL}(15,6)=z_2(15,6)=60$. Concurrent work of Chen and Chen, using the recursive-line definition introduced here, incorporates the finite recursive-line values of this manuscript and establishes further exact four-column values together with an eventual formula for all $m\ge15$. The conjectural equality $z_2=z_{RL}$ is supported by the exact two-column, three-column, four-column, and odd-prime incidence families.

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arXiv2026

The Global Asymptotic Stability Problem for Linear MPC Is Undecidable

We prove that deciding global asymptotic stability for constrained finite-horizon linear model predictive control is undecidable. This holds at horizon one with identity state, input, and terminal weights, unique optimizers, and global feasibility. Separate reductions cover predicted-state boxes, hard input boxes, and quadratically softened input boxes. A fourth reduction fixes the state and input dimensions to three and six. Hence undecidability is not caused by long horizons, growing dimensions, failures of recursive feasibility, or nonuniqueness.

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arXiv2026

Exact and Asymptotic Values for Weak Limited Augmented Zarankiewicz Numbers in the $m\times 3$ Case

We determine the exact weak limited augmented Zarankiewicz numbers $z_{wL}(m,3)$ for all $m\ge 3$: \[ z_{wL}(m,3)= \begin{cases} m+3+\left\lceil \dfrac{m}{2}\right\rceil+1, & 9\le m\le 15,\\[2mm] m+3+\left\lfloor \dfrac{2m-4}{3}\right\rfloor, & m\ge 16, \end{cases} \] with $z_{wL}(3,3)=6$, $z_{wL}(4,3)=8$, and $z_{wL}(m,3)=2m$ for $5\le m\le 9$. In particular, \[ \lim_{m\to\infty} \frac{z_{wL}(m,3)}{m} = \frac{5}{3}. \] The proof is fully analytic, relying on a uniform base classification, two constructive lower-bound families (staircase and $5m/3$), and a sharp upper-bound argument based on a peeling lemma and the analysis of two W2-sensitive boundary cases. Numerical MILP computations were used only as proof-mining tools to identify the structural lemmas; the final theorem is unconditional. We also extend the known range of the original limited numbers $z_L(m,3)$ through $m=13$, where the gap to $z_{wL}(m,3)$ is only 2 or 3.

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arXiv2026

A kernel proof of the De Cock-De Moor Lyapunov identity

We prove the rank-one Lyapunov spectral identity recorded as Problem 9.1 in the 2004 collection of unsolved problems in mathematical systems and control theory. Let $P,Q,R$ solve the coupled discrete Lyapunov and Sylvester equations associated with $A$ and its rank-one update $A_2=A+vw^\top$. When the displayed inverses exist, we show that $P^{-1}RQ^{-1}R^\top$ and $(I+PQ)^{-1}$ have the same characteristic polynomial. A rank-one determinant factorization of the equation for $Q$ produces a scalar bilinear kernel. Evaluating it at the eigenvalues of $A$ and at their reciprocals gives $RQ^{-1}R^\top=BQ^{-1}B=P-BPB$, after which the two target matrices are the same two factors in opposite order. Polynomial continuation extends the identity from a nonempty open set of admissible systems to the full admissible domain and yields a determinant corollary without stability assumptions; when the spectra of $A$ and $A_2$ are disjoint, $Z=b(A)^{-1}P$ gives an explicit similarity. In the Schur-stable realization setting, the result recovers the associated principal-angle and past/future canonical-correlation spectra.

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arXiv2026

Sparse Feedback Implementation for Sender-Receiver Transportation Linear-Quadratic Control

We study a sparse linear-quadratic problem for transportation dynamics. The sparsity pattern has a natural directed-graph representation in which vertices are storage locations and edges are transportation links. The goal is to compute the optimal control signal without applying the usually dense optimal feedback gain directly. We show that the optimal feedback gain can be factorized as the product of a sparse matrix and the inverse of another sparse matrix from the right. In contrast to an existing factorization that uses an inverse from the left, the proposed factors use graph-adapted state coordinates. When the underlying graph is a tree, we give, for every edge orientation, a closed formula for the Riccati matrix that determines these factors and a graph-based bound on their possible nonzero entries. This factorization can reduce the online cost relative to dense feedback multiplication and permits distributed evaluation. The main message is that linear quadratic control need not appear dense when expressed in graph-adapted coordinates, and that these coordinates can reveal intrinsic sparsity.

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arXiv2026

A characteristic function framework for chance constraint programming in stochastic model predictive control

The computation of chance constraints in stochastic model predictive control is often numerically challenging due to the non-Gaussian nature of the disturbances. To overcome this problem, we propose an optimization computational framework applicable to non-Gaussian disturbances. This framework employs a numerical inversion method, utilizing the characteristic function of the disturbance distribution to compute the probability in the chance constraint as well as its gradient. To improve efficiency, it vectorizes integral points and reuses intermediate computations in Gauss-Kronrod quadrature. The framework is implemented within the YALMIP toolbox to perform chance constraint calculations for arbitrary non-Gaussian disturbances, applicable to both single-component distributions and mixture models. It allows the user to simply specify a distribution type and its parameters for the disturbance and directly compute the probability and its gradient to solve the optimization problem. The method is validated through a numerical example of a stochastic model predictive control application.

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arXiv2026

Stochastic Model Predictive Control with Online Risk Allocation and Feedback Gain Selection

Stochastic Model Predictive Control addresses uncertainties by incorporating chance constraints that provide probabilistic guarantees of constraint satisfaction. However, simultaneously optimizing over the risk allocation and the feedback policies leads to intractable nonconvex problems. This is due to (i) products of functions involving the feedback law and risk allocation in the deterministic counterpart of the chance constraints, and (ii) the presence of the nonconvex Gaussian quantile (probit) function. Existing methods rely on two-stage optimization, which is nonconvex. To address this, we derive disjunctive convex chance constraints and select the feedback law from a set of precomputed candidates. The inherited compositions of the probit function are replaced with power- and exponential-cone representable approximations. The main advantage is that the problem can be formulated as a mixed-integer conic optimization problem and efficiently solved with off-the-shelf software. Moreover, the proposed formulations apply to general chance constraints with products of exclusive disjunctive and Gaussian variables. The proposed approaches are validated with a path-planning application.

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arXiv2023

Fast or Cheap: Time and Energy Optimal Control of Ship-to-Shore Cranes

This paper addresses the trade-off between time- and energy-efficiency for the problem of loading and unloading a ship. Container height constraints and energy consumption and regeneration are dealt with. We build upon a previous work that introduced a coordinate system suitable to deal with container avoidance constraints and incorporate the energy related modeling. In addition to changing the coordinate system, standard epigraph reformulations result in an optimal control problem with improved numerical properties. The trade-off is dealt with through the use of weighting of the total time and energy consumption in the cost function. An illustrative example is provided, demonstrating that the energy consumption can be substantially reduced while retaining approximately the same loading time.

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arXiv2022

Time-optimal control of cranes subject to container height constraints

The productivity and efficiency of port operations strongly depend on how fast a ship can be unloaded and loaded again. With this in mind, ship-to-shore cranes perform the critical task of transporting containers into and onto a ship and must do so as fast as possible. Though the problem of minimizing the time spent in moving the payload has been addressed in previous studies, the different heights of the container stacks have not been the focus. In this paper, we perform a change of variable and reformulate the optimization problem to deal with the constraints on the stack heights. As consequence, these constraints become trivial and easy to represent since they turn into bound constraints when the problem is discretized for the numerical solver. To validate the idea, we simulate a small-scale scenario where different stack heights are used. The results confirm our idea and the representation of the stack constraints become indeed trivial. This approach is promising to be applied in real crane operations and has the potential to enhance their automation.

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arXiv2022

Robust Lattice-based Motion Planning

This paper proposes a robust lattice-based motion-planning algorithm for nonlinear systems affected by a bounded disturbance. The proposed motion planner utilizes the nominal disturbance-free system model to generate motion primitives, which are associated with fixed-size tubes. These tubes are characterized through designing a feedback controller, that guarantees boundedness of the errors occurring due to mismatch between the disturbed nonlinear system and the nominal system. The motion planner then sequentially implements the tube-based motion primitives while solving an online graph-search problem. The objective of the graph-search problem is to connect the initial state to the final state, through sampled states in a suitably discretized state space, such that the tubes do not pass through any unsafe states (representing obstacles) appearing during runtime. The proposed strategy is implemented on an Euler-Lagrange based ship model which is affected by significant wind disturbance. It is shown that the uncertain system trajectories always stay within a suitably constructed tube around the nominal trajectory and terminate within a region around the final state, whose size is dictated by the size of the tube.

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arXiv2020

Estimation-aware model predictive path-following control for a general 2-trailer with a car-like tractor

The design of the path-following controller is crucial for reliable autonomous vehicle operation. This design problem is especially challenging for a general 2-trailer with a car-like tractor due to the vehicle's unstable joint-angle kinematics in backward motion. Additionally, advanced sensors placed in the rear of the tractor have been proposed to solve the joint-angle estimation problem. Since these sensors typically have a limited field of view, the estimation solution introduces restrictions on the joint-angle configurations that can be estimated with high accuracy. To explicitly consider these constraints in the controller, a model predictive path-following control approach is proposed. Two approaches with different computation complexity and performance are presented. In the first approach, the joint-angle constraints are modeled as a union of convex polytopes, making it necessary to incorporate binary decision variables. The second approach avoids binary variables at the expense of a more conservative controller. In simulations and field experiments, the performance of the proposed path-following control approach is compared with a previously proposed control strategy.

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arXiv2016

Stability analysis of Model Predictive Controllers using Mixed Integer Linear Programming

It is a well known fact that finite time optimal controllers, such as MPC does not necessarily result in closed loop stable systems. Within the MPC community it is common practice to add a final state constraint and/or a final state penalty in order to obtain guaranteed stability. However, for more advanced controller structures it can be difficult to show stability using these techniques. Additionally in some cases the final state constraint set consists of so many inequalities that the complexity of the MPC problem is too big for use in certain fast and time critical applications. In this paper we instead focus on deriving a tool for a-postiori analysis of the closed loop stability for linear systems controlled with MPC controllers. We formulate an optimisation problem that gives a sufficient condition for stability of the closed loop system and we show that the problem can be written as a Mixed Integer Linear Programming Problem (MILP)

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arXiv2012

Model Reduction using a Frequency-Limited H2-Cost

We propose a method for model reduction on a given frequency range, without the use of input and output filter weights. The method uses a nonlinear optimization approach to minimize a frequency limited H2 like cost function. An important contribution in the paper is the derivation of the gradient of the proposed cost function. The fact that we have a closed form expression for the gradient and that considerations have been taken to make the gradient computationally efficient to compute enables us to efficiently use off-the-shelf optimization software to solve the optimization problem.

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