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An optimal control problem on finite-dimensional positive cones is stated. Under a critical assumption on the cone, the corresponding Bellman equation is satisfied by a linear function, which can be computed by convex optimization. A separate theorem relates the assumption on the cone to the existence of minimal elements in certain subsets of the dual cone. Three special cases are derived as examples. The first one, where the positive cone is the set of positive semi-definite matrices, reduces to standard linear quadratic control. The second one, where the positive cone is a polyhedron, reduces to a recent result on optimal control of positive systems. The third special case corresponds to linear quadratic control with additional structure, such as spatial invariance., Comment: 16 pages, to be published in the proceedings for the 2024 Conference on Decision and Control (CDC)

In this paper we consider calibration of hydraulic models for district heating systems based on operational data. We extend previous theoretical work on the topic to handle real-world complications, namely unknown valve characteristics and hysteresis. We generate two datasets in the Smart Water Infrastructure laboratory in Aalborg, Denmark, on which we evaluate the proposed procedure. In the first data set the system is controlled in such a way to excite all operational modes in terms of combinations of valve set-points. Here the best performing model predicted volume flow rates within roughly 5 and 10 \% deviation from the mean volume flow rate for the consumer with the highest and lowest mean volume flow rates respectively. This performance was met in the majority of the operational region. In the second data set, the system was controlled in order to mimic real load curves. The model trained on this data set performed similarly well when evaluated on data in the operational range represented in the training data. However, the model performance deteriorated when evaluated on data which was not represented in the training data.

This work concerns reduction of the peak flow rate of a district heating grid, a key system property which is bounded by pipe dimensions and pumping capacity. The peak flow rate constrains the number of additional consumers that can be connected, and may be a limiting factor in reducing supply temperatures when transitioning to the 4th generation of district heating. We evaluate a full year of operational data from a subset of customer meters in a district heating system in Germany. We consider the peak flow rate reduction that could be achieved with full a posteriori knowledge of this data. Three strategies for reducing the peak flow rate are investigated: A load shifting demand response strategy, an upper limitation in substation return temperatures, and an upper limitation on each substation's volume flow rate. We show that imposing up to to 18 % load flexibility for the customers provides an equal reduction in the peak system flow rate under the load shifting strategy. The limited return temperature strategy is less efficient at curtailing the peak flow rate, but provides an overall reduction of volume flow rates. Finally, the flow rate limitation method can introduce new, higher flow rate peaks, reducing performance.

We investigate optimal mass transport problem of affine-nonlinear dynamical systems with input and density constraints. Three algorithms are proposed to tackle this problem, including two Uzawa-type methods and a splitting algorithm based on the Douglas-Rachford algorithm. Some preliminary simulation results are presented to demonstrate the effectiveness of our approaches.

The strategy of pre-training a large model on a diverse dataset, then fine-tuning for a particular application has yielded impressive results in computer vision, natural language processing, and robotic control. This strategy has vast potential in adaptive control, where it is necessary to rapidly adapt to changing conditions with limited data. Toward concretely understanding the benefit of pre-training for adaptive control, we study the adaptive linear quadratic control problem in the setting where the learner has prior knowledge of a collection of basis matrices for the dynamics. This basis is misspecified in the sense that it cannot perfectly represent the dynamics of the underlying data generating process. We propose an algorithm that uses this prior knowledge, and prove upper bounds on the expected regret after $T$ interactions with the system. In the regime where $T$ is small, the upper bounds are dominated by a term that scales with either $\texttt{poly}(\log T)$ or $\sqrt{T}$, depending on the prior knowledge available to the learner. When $T$ is large, the regret is dominated by a term that grows with $\delta T$, where $\delta$ quantifies the level of misspecification. This linear term arises due to the inability to perfectly estimate the underlying dynamics using the misspecified basis, and is therefore unavoidable unless the basis matrices are also adapted online. However, it only dominates for large $T$, after the sublinear terms arising due to the error in estimating the weights for the basis matrices become negligible. We provide simulations that validate our analysis. Our simulations also show that offline data from a collection of related systems can be used as part of a pre-training stage to estimate a misspecified dynamics basis, which is in turn used by our adaptive controller.

This is a draft paper originally posted on Arxiv as a documentation of a plenary lecture at CDC2023. The core material has been accepted for publication at L4DC 2024. Certainty equivalence adaptive controllers are analysed using a ``data-driven Riccati equation'', corresponding to the model-free Bellman equation used in Q-learning. The equation depends quadratically on data correlation matrices. This makes it possible to derive simple sufficient conditions for stability and robustness to unmodeled dynamics in adaptive systems. The paper is concluded by short remarks on how the bounds can be used to quantify the interplay between excitation levels and robustness., Comment: 1 figure

We present a novel class of minimax optimal control problems with positive dynamics, linear objective function and homogeneous constraints. The proposed problem class can be analyzed with dynamic programming and an explicit solution to the Bellman equation can be obtained, revealing that the optimal control policy (among all possible policies) is linear. This policy can in turn be computed through standard value iterations. Moreover, the feedback matrix of the optimal controller inherits the sparsity structure from the constraint matrix of the problem statement. This permits structural controller constraints in the problem design and simplifies the application to large-scale systems. We use a simple example of voltage control in an electric network to illustrate the problem setup., Comment: 6 pages, 1 figure, accepted as a joint submission in ACC and L-CSS

We present a method for optimal control with respect to a linear cost function for positive linear systems with coupled input constraints. We show that the optimal cost function and resulting sparse state feedback for these systems can be computed by linear programming. Our framework admits a range of network routing problems with underlying linear dynamics. These dynamics can be used to model traditional graph-theoretical problems like shortest path as a special case, but can also capture more complex behaviors. We provide an asynchronous and distributed value iteration algorithm for obtaining the optimal cost function and control law., Comment: Submitted to ECC2024

We consider control of multiple stable first-order \review{agents} which have a control coupling described by an M-matrix. These agents are subject to incremental sector-bounded \review{input} nonlinearities. We show that such plants can be globally asymptotically stabilized to a unique equilibrium using fully decentralized proportional-integral controllers equipped with anti-windup and subject to local tuning rules. In addition, we show that when the nonlinearities correspond to the saturation function, the closed loop asymptotically minimizes a weighted 1-norm of the agents state mismatch. The control strategy is finally compared to other state-of-the-art controllers on a numerical district heating example.

Control of network systems with uncertain local dynamics has remained an open problem for a long time. In this paper, a distributed minimax adaptive control algorithm is proposed for such networks whose local dynamics has an uncertain parameter possibly taking finite number of values. To hedge against this uncertainty, each node in the network collects the historical data of its neighboring nodes to decide its control action along its edges by finding the parameter that best describes the observed disturbance trajectory. Our proposed distributed adaptive controller is scalable and we give both lower and upper bounds for its $\ell_{2}$ gain. Numerical simulations demonstrate that once each node has sufficiently estimated its local uncertainty, the distributed minimax adaptive controller behaves like the optimal distributed $\mathcal{H}_{\infty}$ controller in hindsight., Comment: arXiv admin note: text overlap with arXiv:2210.00081

We present an online learning analysis of minimax adaptive control for the case where the uncertainty includes a finite set of linear dynamical systems. Precisely, for each system inside the uncertainty set, we define the model-based regret by comparing the state and input trajectories from the minimax adaptive controller against that of an optimal controller in hindsight that knows the true dynamics. We then define the total regret as the worst case model-based regret with respect to all models in the considered uncertainty set. We study how the total regret accumulates over time and its effect on the adaptation mechanism employed by the controller. Moreover, we investigate the effect of the disturbance on the growth of the regret over time and draw connections between robustness of the controller and the associated regret rate.

This paper examines the local exponential stability (LES) of trajectories for nonlinear systems on Riemannian manifolds. We present necessary and sufficient conditions for LES of a trajectory on a Riemannian manifold by analyzing the complete lift of the system along the given trajectory. These conditions are coordinate-free which reveal fundamental relationships between exponential stability and incremental stability in a local sense. We then apply these results to design tracking controllers and observers for Euler-Lagrangian systems on manifolds; a notable advantage of our design is that it visibly reveals the effect of curvature on system dynamics and hence suggests compensation terms in the controller and observer. Additionally, we revisit some well-known intrinsic observer problems using our proposed method, which largely simplifies the analysis compared to existing results.

Recent work [Ran22] formulated a class of optimal control problems involving positive linear systems, linear stage costs, and elementwise constraints on control. It was shown that the problem admits linear optimal cost and the associated Bellman's equation can be characterized by a finite-dimensional nonlinear equation, which is solved by linear programming. In this work, we report exact dynamic programming (DP) theories for the same class of problems. Moreover, we extend the results to a related class of problems where the norms of control are bounded while the optimal costs remain linear. In both cases, we provide conditions under which the solutions are unique, investigate properties of the optimal policies, study the convergence of value iteration, policy iteration, and optimistic policy iteration applied to such problems, and analyze the boundedness of the solution to the associated optimization programs. Apart from a form of the Frobenius-Perron theorem, the majority of our results are built upon generic DP theory applicable to problems involving nonnegative stage costs.

This paper aims at coordinating interconnected agents where the control input of each agent is limited by the control input of others. In that sense, the systems have to share a limited resource over a network. Such problems can arise in different areas and it is here motivated by a simplified district heating example. When the shared resource is insufficient for the combined need of all systems, the resource will have to be shared in an optimal fashion. In this scenario, we want the systems to automatically converge to an optimal equilibrium. The contribution of this paper is the proposal of a control architecture where each separate system is controlled by a local PI controller. The controllers are then coordinated through a global rank-one anti-windup signal. It is shown that the equilibrium of the proposed closed-loop system minimizes the infinity-norm of stationary state deviations. Convergence is investigated via a randomized numerical study, and a district heating-inspired example illustrates the utility of the proposed method.

Using hydraulic models in control design in district heating networks can increase pumping efficiency and reduce sensitivity to hydraulic bottlenecks. These models are usually white-box, as they are obtained based on full knowledge of the district heating network and its parameters. This type of model is time-consuming to obtain, and might differ from the actual behavior of the system. In this paper, a method is proposed to obtain a grey-box hydraulic model for tree-shaped district heating systems: hydraulic parameters are estimated based on pressure measurements in only two locations. While previous works only estimate parameters related to pressure losses in pipes, this work also includes customers valves in the grey-box model structure, an important inclusion for control-oriented applications. Finally, a numerical example illustrates the proposed method on a small district heating network, showing its ability to obtain an accurate model on the basis of noisy measurements.

This paper deals with a distributed implementation of minimax adaptive control algorithm for networked dynamical systems modeled by a finite set of linear models. To hedge against the uncertainty arising out of finite number of possible dynamics in each node in the network, it collects only the historical data of its neighboring nodes to decide its control action along its edges. This makes our proposed distributed approach scalable. Numerical simulations demonstrate that once each node has sufficiently estimated the uncertain parameters, the distributed minimax adaptive controller behaves like the optimal distributed H-infinity controller in hindsight.

In this paper, we apply an optimal LQ controller, which has an inherent structure that allows for a distributed implementation, to an irrigation network. The network consists of a water reservoir and connected water canals. The goal is to keep the levels close to the set-points when farmers take out water. The LQ controller is designed using a first-order approximation of the canal dynamics, while the simulation model used for evaluation uses third-order canal dynamics. The performance is compared to a P controller and an LQ controller designed using the third-order canal dynamics. The structured controller outperforms the P controller and is close to the theoretical optimum given by the third-order LQ controller for disturbance rejection.

We present a constructive approach to bounded $\ell_2$-gain adaptive control with noisy measurements for linear time-invariant scalar systems with uncertain parameters belonging to a finite set. The gain bound refers to the closed-loop system, including the learning procedure. The approach is based on forward dynamic programming to construct a finite-dimensional information state consisting of $\mathcal H_\infty$-observers paired with a recursively computed performance metric. We do not assume prior knowledge of a stabilizing controller., Comment: Submitted to L4DC 2022

We study an optimal control problem for a simple transportation model on a path graph. We give a closed form solution for the optimal controller, which can also account for planned disturbances using feed-forward. The optimal controller is highly structured, which allows the controller to be implemented using only local communication, conducted through two sweeps through the graph.

H-infinity optimal control and estimation are addressed for a class of systems governed by partial differential equations with bounded input and output operators. Diffusion equations are an important example in this class. Explicit formulas for the optimal state feedback controller as well as the optimal state estimator are given. Unlike traditional methods for H-infinity synthesis, no iteration is needed to obtain the optimal solution. Moreover, the optimal performance for both the state feedback and state estimation problems are explicitly calculated. This is shown to be useful for problems of H-infinity optimal actuator and sensor location. Furthermore, the results can be used in testing and bench-marking of general purpose algorithms for H-infinity synthesis. The results also apply to finite-dimensional systems.

In this paper we consider distributed adaptive stabilization for uncertain multivariable linear systems with a time-varying diagonal matrix gain. We show that uncertain multivariable linear systems are stabilizable by diagonal matrix high gains if the system matrix is an H-matrix with positive diagonal entries. Based on matrix measure and stability theory for diagonally dominant systems, we consider two classes of uncertain linear systems, and derive a threshold condition to ensure their exponential stability by a monotonically increasing diagonal gain matrix. When each individual gain function in the matrix gain is updated by state-dependent functions using only local state information, the boundedness and convergence of both system states and adaptive matrix gains are guaranteed. We apply the adaptive distributed stabilization approach to adaptive synchronization control for large-scale complex networks consisting of nonlinear node dynamics and time-varying coupling weights. A unified framework for adaptive synchronization is proposed that includes several general design approaches for adaptive coupling weights to guarantee network synchronization., Comment: 16 Pages and 7 figures

A new approach to feedback control design based on optimal control is proposed. Instead of expensive computations of the value function for different penalties on the states and inputs, we use a control Lyapunov function that amounts to be a value function of an optimal control problem with suitable cost design and then study combinations of input and state penalty that are compatible with this value function. This drastically simplifies the role of the Hamilton-Jacobi-Bellman equation, since it is no longer a partial differential equation to be solved, but an algebraic relationship between different terms of the cost. The paper illustrates this idea in different examples, including $\mathcal{H}_{\infty}$ control and optimal control of coupled oscillators., Comment: 6 pages, 4 figures

We address the issue of establishing standard forms for nonnegative and Metzler matrices by considering their similarity to nonnegative and Metzler Hessenberg matrices. It is shown that for dimensions $n \geq 3$, there always exists a subset of nonnegative matrices that are not similar to a nonnegative Hessenberg form, which in case of $n=3$ also provides a complete characterization of all such matrices. For Metzler matrices, we further establish that they are similar to Metzler Hessenberg matrices if $n \leq 4$. In particular, this provides the first standard form for controllable third order continuous-time positive systems via a positive controller-Hessenberg form. Finally, we present an example which illustrates why this result is not easily transferred to discrete-time positive systems. While many of our supplementary results are proven in general, it remains an open question if Metzler matrices of dimensions $n \geq 5$ remain similar to Metzler Hessenberg matrices., Comment: Accepted for publication in Special Matrices

For linear time-invariant systems with uncertain parameters belonging to a finite set, we present a purely deterministic approach to multiple-model estimation and propose an algorithm based on the minimax criterion using constrained quadratic programming. The estimator tends to learn the dynamics of the system, and once the uncertain parameters have been sufficiently estimated, the estimator behaves like a standard Kalman filter., Comment: Accompanying code: https://github.com/kjellqvist/MinimaxEstimation.jl

The 4th generation of district heating systems face a potential problem where lowered water temperatures lead to higher flow rates, which requires higher hydraulic capacity in terms of pipe and pump sizes. This increases the effect of the already existing issue of hydraulic bottlenecks, causing peripheral units (customers) to experience reduced flow rates. A coordinating control strategy is presented in this work aimed at reducing the effect of such bottlenecks on the comfort of customers. This is done by distributing the flow deficit over many units rather than a few. Previous works mainly focus on MPC-structured controllers that depend on complex system models and online optimization techniques. This work proposes a method that requires little information about models for individual units and minimal IT communication between control systems. The proposed method is compared with a traditional control strategy and an optimal baseline in a simulation study. This shows that the proposed method can decrease the worst case indoor temperature deviations.

In inverse optimal control, the optimality of a given feedback stabilizing controller is a byproduct of the choice of a meaningful, a posteriori defined, cost functional. This allows for a simple tuning comparable to linear quadratic control, also for nonlinear controllers. Our work illustrates the usefulness of this approach in the control of converter-based power systems and networked systems in general, and thereby in finding controllers with topological structure and known optimality properties. In particular, we design an inverse optimal feedback controller that stabilizes the phase angles of voltage-source controlled DC/AC converters at an induced steady state with zero frequency error. The distributed angular droop controller yields active power to angle droop behavior at steady state. Moreover, we suggest a practical implementation of the controller and corroborate our results through simulations on a three-converter system and a numerical comparison with standard frequency droop control., Comment: 8 pages, 5 figures

An adaptive controller with bounded l2-gain from disturbances to errors is derived for linear time-invariant systems with uncertain parameters restricted to a finite set. The gain bound refers to the closed loop system, including the non-linear learning procedure. As a result, robustness to unmodelled dynamics (possibly nonlinear and infinite-dimensional) follows from the small gain theorem. The approach is based on a new zero-sum dynamic game formulation, which optimizes the trade-off between exploration and exploitation. An explicit upper bound on the optimal value function is stated in terms of semi-definite programming and a corresponding simple formula for an adaptive controller achieving the upper bound is given. Once the uncertain parameters have been sufficiently estimated, the controller behaves like standard H-infinity optimal control., Comment: arXiv admin note: text overlap with arXiv:1912.03550

For linear time-invariant systems having a state matrix with uncertain sign, we formulate a minimax adaptive control problem as a zero sum dynamic game. Explicit expressions for the optimal value function and the optimal control law are given in terms of a Riccati equation. The optimal control law is adaptive in the sense the past data is used to estimate the uncertain sign for prediction of future dynamics. Once the sign has been estimated, the controller behaves like standard H-infinity optimal state feedback.

The relaxation systems are an important subclass of the passive systems that arise naturally in applications. We exploit the fact that they have highly structured state-space realisations to derive analytical solutions to some simple H-infinity type optimal control problems. The resulting controllers are also relaxation systems, and often sparse. This makes them ideal candidates for applications in large-scale problems, which we demonstrate by designing simple, sparse, electrical circuits to optimally control large inductive networks and to solve linear regression problems.

This paper proposes a convex approach to the Frisch-Kalman problem that identifies the linear relations among variables from noisy observations. The problem was proposed by Ragnar Frisch in 1930s, and was promoted and further developed by Rudolf Kalman later in 1980s. It is essentially a rank minimization problem with convex constraints. Regarding this problem, analytical results and heuristic methods have been pursued over a half century. The proposed convex method in this paper is shown to be accurate and demonstrated to outperform several commonly adopted heuristics when the noise components are relatively small compared with the underlying data.

Machine and reinforcement learning (RL) are increasingly being applied to plan and control the behavior of autonomous systems interacting with the physical world. Examples include self-driving vehicles, distributed sensor networks, and agile robots. However, when machine learning is to be applied in these new settings, the algorithms had better come with the same type of reliability, robustness, and safety bounds that are hallmarks of control theory, or failures could be catastrophic. Thus, as learning algorithms are increasingly and more aggressively deployed in safety critical settings, it is imperative that control theorists join the conversation. The goal of this tutorial paper is to provide a starting point for control theorists wishing to work on learning related problems, by covering recent advances bridging learning and control theory, and by placing these results within an appropriate historical context of system identification and adaptive control., Comment: Tutorial paper, 2019 IEEE Conference on Decision and Control, to appear

Internal positivity offers a computationally cheap certificate for external (input-output) positivity of a linear time-invariant system. However, the drawback with this certificate lies in its realization dependency. Firstly, computing such a realization requires to find a polyhedral cone with a potentially high number of extremal generators that lifts the dimension of the state-space representation, significantly. Secondly, not all externally positive systems posses an internally positive realization. Thirdly, in many typical applications such as controller design, system identification and model order reduction, internal positivity is not preserved. To overcome these drawbacks, we present a tractable sufficient certificate of external positivity based on second-order cones. This certificate does not require any special state-space realization: if it succeeds with a possibly non-minimal realization, then it will do so with any minimal realization. While there exist systems where this certificate is also necessary, we also demonstrate how to construct systems, where both second-order and polyhedral cones as well as other certificates fail. Nonetheless, in contrast to other realization independent certificates, the present one appears to be favourable in terms of applicability and conservatism. Three applications are representatively discussed to underline its potential. We show how the certificate can be used to find externally positive approximations of nearly externally positive systems and demonstrated that this may help to reduce system identification errors. The same algorithm is used then to design state-feedback controllers that provide closed-loop external positivity, a common approach to avoid over- and undershooting of the step response. Lastly, we present modifications to generalized balanced truncation such that external positivity is preserved where our certificate applies.

We address a class of systems for which the solution to an H-infinity optimal control problem can be given on a very simple closed form. In fact, both the control law and optimal performance value are explicitly given. The class of systems include models for large-scale systems such as temperature dynamics in buildings, buffer networks and transportations systems. Furthermore, the structure of the control law is suitable for distributed control of such large-scale systems, which is illustrated through examples., Comment: 24 pages, 1 figure

It is proved that network realizability of controllers can be enforced without conservatism using convex constraints on the closed loop transfer function. Once a network realizable closed loop transfer matrix has been found, a corresponding controller can be implemented using a network structured version of Internal Model Control., Comment: 3 pages

Sufficient and necessary conditions for the stability of positive feedback interconnections of negative imaginary systems are derived via an integral quadratic constraint (IQC) approach. The IQC framework accommodates distributed-parameter systems with irrational transfer function representations, while generalising existing results in the literature and allowing exploitation of flexibility at zero and infinite frequencies to reduce conservatism in the analysis. The main results manifest the important property that the negative imaginariness of systems gives rise to a certain form of IQCs on positive frequencies that are bounded away from zero and infinity. Two additional sets of IQCs on the DC and instantaneous gains of the systems are shown to be sufficient and necessary for closed-loop stability along a homotopy of systems., Comment: Submitted to Automatica, A preliminary version of this paper appeared in the Proceedings of the 2015 European Control Conference

We develop a simple routine unifying the analysis of several important recently-developed stochastic optimization methods including SAGA, Finito, and stochastic dual coordinate ascent (SDCA). First, we show an intrinsic connection between stochastic optimization methods and dynamic jump systems, and propose a general jump system model for stochastic optimization methods. Our proposed model recovers SAGA, SDCA, Finito, and SAG as special cases. Then we combine jump system theory with several simple quadratic inequalities to derive sufficient conditions for convergence rate certifications of the proposed jump system model under various assumptions (with or without individual convexity, etc). The derived conditions are linear matrix inequalities (LMIs) whose sizes roughly scale with the size of the training set. We make use of the symmetry in the stochastic optimization methods and reduce these LMIs to some equivalent small LMIs whose sizes are at most 3 by 3. We solve these small LMIs to provide analytical proofs of new convergence rates for SAGA, Finito and SDCA (with or without individual convexity). We also explain why our proposed LMI fails in analyzing SAG. We reveal a key difference between SAG and other methods, and briefly discuss how to extend our LMI analysis for SAG. An advantage of our approach is that the proposed analysis can be automated for a large class of stochastic methods under various assumptions (with or without individual convexity, etc)., Comment: To Appear in Proceedings of the Annual Conference on Learning Theory (COLT) 2017

A multi-variable PI (proportional integrating) controller is proved to be optimal for an important class of control problems where performance is specified in terms of frequency weighted H-infinity norms. The problem class includes networked systems with a subsystem in each node and control action along each edge. For such systems, the optimal PI controller is decentralized in the sense that control action along a given network edge is entirely determined by states at nodes connected by that edge.

We consider a class of monotone systems in which the control signal multiplies the state. Among other applications, such bilinear systems can be used to model the evolutionary dynamics of HIV in the presence of combination drug therapy. For this class of systems, we formulate an infinite horizon optimal control problem, prove that the optimal control signal is constant over time, and show that it can be computed by solving a finite-dimensional non-smooth convex optimization problem. We provide an explicit expression for the subdifferential set of the objective function and use a subgradient algorithm to design the optimal controller. We further extend our results to characterize the optimal robust controller for systems with uncertain dynamics and show that computing the robust controller is no harder than computing the nominal controller. We illustrate our results with an example motivated by combination drug therapy., Comment: 15 pages, 2 figures

The problem of low-rank approximation with convex constraints, which appears in data analysis, system identification, model order reduction, low-order controller design and low-complexity modelling is considered. Given a matrix, the objective is to find a low-rank approximation that meets rank and convex constraints, while minimizing the distance to the matrix in the squared Frobenius norm. In many situations, this non-convex problem is convexified by nuclear norm regularization. However, we will see that the approximations obtained by this method may be far from optimal. In this paper, we propose an alternative convex relaxation that uses the convex envelope of the squared Frobenius norm and the rank constraint. With this approach, easily verifiable conditions are obtained under which the solutions to the convex relaxation and the original non-convex problem coincide. An SDP representation of the convex envelope is derived, which allows us to apply this approach to several known problems. Our example on optimal low-rank Hankel approximation/model reduction illustrates that the proposed convex relaxation performs consistently better than nuclear norm regularization and may outperform balanced truncation., Comment: Accepted for publication in IEEE Transactions on Automatic Control

Controllers with a diagonal-plus-low-rank structure constitute a scalable class of controllers for multi-agent systems. Previous research has shown that diagonal-plus-low-rank control laws appear as the optimal solution to a class of multi-agent H2 coordination problems, which arise in the control of wind farms. In this paper we show that this result extends to the case where the information exchange between agents is subject to limitations. We also show that the computational effort required to obtain the optimal controller is independent of the number of agents and provide analytical expressions that quantify the usefulness of information exchange.

We address H-infinity structured static state feedback and give a simple form for an optimal control law applicable to linear time invariant systems with symmetric and Hurwitz state matrix. More specifically, the control law as well as the minimal value of the norm can be expressed in the matrices of the system's state space representation, given separate cost on state and control input. Thus, the control law is transparent, easy to synthesize and scalable. Furthermore, if the plant possess a compatible sparsity pattern it is also distributed. Examples of such sparsity patterns are included. Furthermore, we give an extension of the optimal control law that incorporate coordination among subsystems. We demonstrate by a numerical example that the derived optimal controller is equal in performance to an optimal controller derived by the riccati equation approach., Comment: 6 pages, 5 figures. Submitted to 2016 American Control Conference

Closed-loop positivity of feedback interconnections of positive monotone nonlinear systems is investigated. It is shown that an instantaneous gain condition on the open-loop systems which implies feedback well-posedness also guarantees feedback positivity. Furthermore, the notion of integral linear constraints (ILC) is utilised as a tool to characterise uncertainty in positive feedback systems. Robustness analysis of positive linear time-varying and nonlinear feedback systems is studied using ILC, paralleling the well-known results based on integral quadratic constraints., Comment: 5 pages, 1 figure

Distributed algorithms for solving coupled semidefinite programs (SDPs) commonly require many iterations to converge. They also put high computational demand on the computational agents. In this paper we show that in case the coupled problem has an inherent tree structure, it is possible to devise an efficient distributed algorithm for solving such problems. This algorithm can potentially enjoy the same efficiency as centralized solvers that exploit sparsity. The proposed algorithm relies on predictor-corrector primal-dual interior-point methods, where we use a message-passing algorithm to compute the search directions distributedly. Message-passing here is closely related to dynamic programming over trees. This allows us to compute the exact search directions in a finite number of steps. Furthermore this number can be computed a priori and only depends on the coupling structure of the problem. We use the proposed algorithm for analyzing robustness of large-scale uncertain systems distributedly. We test the performance of this algorithm using numerical examples., Comment: 14 pages and 6 figurs. Submitted to IEEE Transactions on Automatic Control