Keywords:System Identification Abstract: We present a technique to design controllers from data for systems whose model is imprecisely known. The technique is based on collecting measurements of low complexity from the systems and using them for the synthesis of controllers, which is reduced to the solution of data-dependent semidefinite programs. The method provides stability certificates in the presence of perturbations on the dataset.

Keywords:Optimization : Theory and Algorithms Abstract: Mini Course on Measure differential equations: modeling and numerical solution organised by Benoît Bonnet, Didier Henrion, Swann Marx, and Francesco Rossi This mini-course focuses on numerical methods for solving differential equations on measures, with applications in optimization and control, as well as modeling multi-scale multi-agent systems. Covered material include the moment sums of squares hierarchy and Eulerian schemes.

Part one - modeling: In the first part of the mini-course we describe the optimization and control problems that can be modelled in terms of partial differential equations (PDEs) on probability measures. We present the main existing wellposedness theories which are based on optimal mass transportation, establish the correspondence between non-linear ordinary differential equations (ODEs) and linear transport PDEs, and discuss some of the recent progresses made in this area of the literature. We also present a large number of domains in which measure differential dynamics have been successfully applied, e.g. multi-anticipative road traffic, crowd dynamics, collective behavior in animal groups, supply chains, age-structured populations.

Part two - solving: In the second part of the mini-course we introduce some numerical methods for solving measure PDEs. On the one hand, we describe briefly the moment sums of squares a.k.a. Lasserre hierarchy, originally introduced for polynomial optimization, and later on extended to optimal control of ODEs and more recently to numerical computation of non-linear PDEs. On the other hand, we describe Euler schemes for measure differential equations with non-local terms. The idea is to discretize the measure into a grid, and to let it evolve following some adapted ODE. Convergence is based on optimal transportation strategies that provide a metric tool (the Wasserstein distance) for the study of measure dynamics.

Confirmed speakers:

Nastassia Pouradier-Duteil - continuum limits of collective dynamics with time-varying weights Benoît Bonnet - differential inclusions in measure spaces Claudia Totzeck - optimization methods for multi-agent systems Didier Henrion - tutorial on the moment-SOS aka Lasserre hierarchy Swann Marx - conservation laws solved with the Lasserre hierarchy and the Christoffel-Darboux polynomial

Keywords:Dissipativity, Nonlinear Systems and Control, Mathematical Theory of Networks and Circuits Abstract: Following the seminal work of Zames, the input-output theory of the 70s acknowledged that incremental properties (e.g. incremental gain) are the relevant quantities to study in nonlinear feedback system analysis. Yet, non-incremental analysis has dominated the use of dissipativity theory in nonlinear control from the 80s. Results connecting dissipativity theory and incremental analysis are scattered and progress has been limited. This abstract investigates whether this theoretical gap is of fundamental nature and considers new avenues to circumvent it.

Keywords:Dissipativity, Transportation Systems, Systems on Graphs Abstract: We present a nonlinear ODE-based thermo-hydraulic model of a district heating system with multiple heat producers, consumers and storage devices. We analyze the conditions under which the hydraulic and thermal subsystems of the model exhibit shifted passivity properties. For the hydraulic subsystem, our claims on passivity draw on the monotonicity of the vector field associated with the district heating system's flow dynamics, which mainly codifies viscous friction effects on the system's pressures. For the temperature dynamics, we propose a storage function based on the ectropy function of a thermodynamic system, recently used in the passivity analysis of heat exchanger networks.

Keywords:Port-Hamiltonian Systems, Control of Distributed Parameter Systems, Stability Abstract: In this extended abstract we show, on the one dimensional (1D) heat equation example, how Boundary Controlled Irreversible Port Hamiltonian Systems (BC-IPHS) formulations and systems thermodynamic fundamental properties can be used for control design purposes.

Keywords:Dissipativity, Port-Hamiltonian Systems, Physical Systems Theory Abstract: The Second Law of thermodynamics implies that no thermodynamic system with a single heat source at constant temperature can convert heat into mechanical work in a repeatable manner. First, we note that this is equivalent to cyclo-passivity at the mechanical port of the thermodynamic system which is constrained by constant temperature at the thermal port. Second, we address the general system-theoretic question which physical systems with two power ports share this property, called one-port cyclo-passivity. Recently, sufficient conditions for one-port cyclo-passivity have been obtained, based on the structure of the interconnection matrix in the port-Hamiltonian formulation. We elaborate on these conditions and provide some extensions. Next we focus on control strategies which go beyond the classical Carnot cycle in order to convert energy in case of one-port cyclo-passivity, and apply this to a number of multiphysics systems.

Keywords:Coding Theory Abstract: We investigate Singleton-like bounds in the Lee metric and characterize extremal codes. We then focus on Plotkin-like bounds in the Lee metric and present a new bound that extends and refines a previously known bound, which it out-performs in the case of non-free codes. We then compute the density of codes that meet this bound. Finally, we fill a gap in the characterization of Lee-equidistant codes.

Keywords:Coding Theory, Information Theory Abstract: Cameron-Liebler sets of lines in a finite 3-dimensional space PG(3,q) originate from the study by Cameron and Liebler in 1982 of groups of collineations with equally many orbits on the points and the lines of PG(3,q). These objects have some interesting equivalent characterizations, and are examples of Boolean functions of degree one and completely regular codes. In this talk, we focus on these objects from a geometric perspective, and report on several existence and non-existence results, including a recent so-called modular equality for the parameter of Cameron-Liebler sets of k-spaces in finite n-dimensional projective spaces.

Keywords:Coding Theory, Information Theory Abstract: Cameron-Liebler sets of lines in a finite 3-dimensional space mathrm{PG}(3,q) originate from the study by Cameron and Liebler in 1982 of groups of collineations with equally many orbits on the points and the lines of mathrm{PG}(3,q). These objects have some interesting equivalent characterizations, and are examples of Boolean functions of degree one. In this talk, we focus on these objects and their generalisation from a geometric perspective, and report on several existence and non-existence results, including a lower bound on the existence of the parameter x (besides trivial examples).

Keywords:Coding Theory, Information Theory, System Identification Abstract: Based on a recently presented approach to non-adaptive group testing in terms of residuated pairs on Boolean lattices, we extend our investigation into group testing to the noisy case.

In this context, several concepts and techniques from traditional Coding Theory over finite fields need to be adapted and refined. We will touch a variety of relevant notions and concepts and provide some first sporadic examples of error-correcting group testing schemes.

The possible applications are diverse and reach from simple search of defective items in production chains to mass testing under pandemic conditions.

Keywords:Coding Theory Abstract: The aim of this paper is to explore a possible route to the construction, or to a non-existence proof, of a 2-analog of the Fano Plane by viewing its codewords as binary linear [7,3] codes.

Keywords:Coding Theory Abstract: The fact that further constructions often don't bring breakthrough results motivates us to combine designs to be particles of other combinatorial structures. One of them are mosaics of designs, where instead of having a matrix presenting incidences of a design, one might fill the matrix with incidences of more than one design. Another way of combining designs are design cubes. They can be thought of as 3-dimensional incidence 0-1 matrices, such that each 2-dimensional incidence submatrix satisfies the properties of a design. In this paper we shall be concentrated on designing and combining t-designs although we are aware of the fact that the ideas presented here might work and be interesting for other sorts of combinatorial designs.

Keywords:Nonlinear Systems and Control, Infinite Dimensional Systems Theory, Operator Theoretic Methods in Systems Theory Abstract: In his 1932 paper, Carleman proposes a linearization method to transform a given finite-dimensional nonlinear system that is defined by an analytic function into an equivalent infinite-dimensional linear system with (usually) unbounded operators. Finite truncation of the transformed system has been used to study dynamical properties, learning, and control of such nonlinear systems. One of the important problems in this context is to quantify the effectiveness of such finitely truncated models. In this paper, we provide explicit error bounds and prove that the trajectory of the truncated system stays close to that of the original nonlinear system over a quantifiable time interval. This is particularly important in several applications, including Model Predictive Control, to choose proper truncation lengths for a given sampling period and employ the resulting truncated system as a good approximation of the nonlinear system.

Keywords:System Identification, Operator Theoretic Methods in Systems Theory, Nonlinear Systems and Control Abstract: Conventionally, data driven identification and control problems for higher order dynamical systems are solved by augmenting the system state by the derivatives of the output to formulate first order dynamical systems in higher dimensions. However, solution of the augmented problem typically requires knowledge of the full augmented state, which requires numerical differentiation of the original output, frequently resulting in noisy signals. This manuscript develops the theory necessary for a direct analysis of higher order dynamical systems using higher order Liouville operators. Fundamental to this theoretical development is the introduction of signal valued RKHSs and new operators posed over these spaces. Ultimately, it is observed that despite the added abstractions, the necessary computations are remarkably similar to that of first order DMD methods using occupation kernels.

Keywords:Operator Theoretic Methods in Systems Theory, System Identification, Machine Learning and Control Abstract: Given a function over a potentially high dimensional domain, the active subspace method seeks an affine subspace inside which the functions changes the most on average. This is done by finding the eigenvectors of a covariance matrix incorporating gradient information. In a similar vein, the active manifold method finds a manifold M and if information on f is recovered along M then f can be recovered on the connected component of a level set touching M. An inherent limitation of the Active subspace technique is that it only considers affine subspaces (which may still be high dimensional). Inspired by methods in occupation kernel dynamic mode decomposition, we develop a notion of active subspace taking place in a Hilbert space which contains sufficient complexity to describe highly nonlinear level sets. In this learning problem, only function values along trajectories following the gradient direction of the function are required to determine this decomposition.

Keywords:Optimal Control, Operator Theoretic Methods in Systems Theory, Nonlinear Systems and Control Abstract: In this effort, a novel operator theoretic framework is developed for data-driven solution of optimal control problems. The developed methods focus on the use of trajectories (i.e., time-series) as the fundamental unit of data for the resolution of optimal control problems in dynamical systems. Trajectory information in the dynamical systems is embedded in a reproducing kernel Hilbert space (RKHS) through what are called occupation kernels. The occupation kernels are tied to the dynamics of the system through the densely defined Liouville operator. The pairing of Liouville operators and occupation kernels allows for lifting of nonlinear finite-dimensional optimal control problems into the space of infinite-dimensional linear programs over RKHSs.

Keywords:Optimal Control, Optimization : Theory and Algorithms, Process Control Abstract: The operation of gas pipeline networks leads to problems of optimal boundary control for systems governed by the isothermal Euler equations. In this contribution we consider a problem of optimal Dirichlet control with an objective function of integral type that is given as the sum of a tracking term for a desired stationary state and the corresponding control cost that is also given by a time-integral. We study the well-posedness and exact controllability properties of the system. We study regular solutions that generate a field of non-intersecting characteristic curves without rarefaction fan. We present an integral turnpike result and a result about the turnpike phenomenon with interior decay for such an optimal control problem where the control cost is given by the H2-norm.

Keywords:Optimal Control, Control of Distributed Parameter Systems Abstract: We analyse the turnpike properties for a general, linear-quadratic (LQ) optimal control problem. We assume that the system under consideration is governed by an infinite-dimensional differential equation with a generator A of a strongly continuous semi-group. The objective function is the sum of a control cost and a tracking term for an observation of the state.

The novelty of the results is twofold. Firstly, it obtains positive turnpike results for systems that are (partially) uncontrollable. Secondly, it provides turnpike results for optimal averaged control associated to a family of problems that depend on a random parameter, which is the first turnpike type result that extends the averaged controllability approach to optimal control problems. In both cases, the results do not require assumptions on stabilizability and detectability, which are most commonly used in the study of turnpike phenomena.

Examples supporting the theoretical findings will be presented as well.

Keywords:Dissipativity, Optimal Control, Nonlinear Systems and Control Abstract: Recent results in the literature have provided connections between the turnpike property, near optimality of closed- loop solutions, and strict dissipativity. In this talk, based on the recent paper Grüne and Krügel (2021) (to which we refer for all proofs), we consider optimal control problems with discounted stage cost. In contrast to non-discounted optimal control problems, it is more likely that several asymptotically stable optimal equilibria coexist. Due to the discounting and transition cost from a local to the global equilibrium, it may be more favourable staying in a local equilibrium than moving to the global “cheaper” equilibrium. In this talk, we propose a local notion of dis- counted strict dissipativity and a local turnpike property, both depending on the discount factor. Using these con- cepts, we investigate the local behaviour of (near-)optimal trajectories and develop conditions on the discount factor to ensure convergence to a local asymptotically stable optimal equilibrium.

Keywords:Optimal Control, Control of Distributed Parameter Systems, Hybrid Systems Abstract: This paper studies the integral turnpike and turnpike in the average for a class of random ordinary differential equations. We prove that, under suitable assumptions on the matrices that define the system, the optimal solutions for an optimal distributed control tracking problem remain, in an average sense, sufficiently close to the associated random stationary optimal solution for most of the time horizon.

Keywords:Control of Distributed Parameter Systems, Optimal Control, Large Scale Systems Abstract: In this paper, we consider the problem of optimally guiding a large-scale swarm of underwater vehicles that is tasked with the indirect control of an advection-diffusion environmental field. The microscopic vehicle dynamics are governed by a stochastic differential equation with drift. The drift terms model the self-propelled velocity of the vehicle and the velocity field of the currents. In the mean-field setting, the macroscopic vehicle dynamics are governed by a Kolmogorov forward equation in the form of a linear parabolic advection-diffusion equation. The environmental field is governed by an advection-diffusion equation in which the advection term is defined by the fluid velocity field. The vehicles are equipped with on-board actuators that enable the swarm to act as a distributed source in the environmental field, modulated by a scalar control parameter that determines the local source intensity. In this setting, we formulate an optimal control problem to compute the vehicle velocity and actuator intensity fields that drive the environmental field to a desired distribution within a specified amount of time. After proving an existence result for the solution of the optimal control problem, we discretize and solve the problem using the Finite Element Method (FEM). Finally, we show through numerical simulations the effectiveness of our control strategy in regulating the environmental field to zero or to a desired distribution in the presence of a double-gyre flow field.

Keywords:Control of Distributed Parameter Systems, Nonlinear Systems and Control, Hybrid Systems Abstract: In this article, we derive physics-based operating modes based on degradative governing equations, which are used to ensure safe use and minimal degradation during long-term cycling. The fast-charging protocols are efficiently and deterministically simulated using a mixed continuous-discrete (aka hybrid) approach to fast charging. This simultaneously solves the battery system of equations and the constraint-based control problem. The approach is evaluated using a Porous Electrode Theory-based model that includes solid-electrolyte interface (SEI) capacity fade. Three physics-based charging protocols are compared to a conventional constant current-constant voltage (CC-CV) protocol. Given identical levels of capacity fade after 500 cycles, the physics-based protocols uniformly reach a greater charge capacity compared to CC-CV after charging for 10 and 15 minutes. The computational cost of simulating physics-based charging protocols is only about 30% greater than the CC-CV method. The fast charging framework is easily extendable to other battery models, irrespective of model complexity.

Keywords:Control of Distributed Parameter Systems, Optimization : Theory and Algorithms, System Identification Abstract: Owing to the wide availability of efficient convex optimization algorithms, convex relaxation of optimum sensor selection problems has gained in popularity. Generally, however, there is a performance gap between the optimal solution of the original combinatorial problem and the heuristic solution of the respective relaxed continuous problem. This gap can be small in many cases, but there is no guarantee that this is always the case. That is why the D-optimality criterion is often extended by addition of some kind of sparsity-enforcing penalty term. Unfortunately, the problem convexity is then lost and the question of how to control the influence of this penalty so as not to excessively deteriorate the optimal relaxed solution remains open. This work proposes an alternative problem formulation, in which the sparsity-promoting term is directly minimized subject to the constraint that the D-efficiency of the sensor selection is no less than a given threshold. This offers direct control of the degree of optimality of the produced solution. An efficient computational scheme based on the majorization-minimization algorithm is proposed, which reduces to solving a sequence of low-dimensional convex optimization problems via generalized simplicial decomposition. A numerical example illustrating the effectiveness of the proposed approach is also reported.

Keywords:Optimal Control, Dissipativity, Infinite Dimensional Systems Theory Abstract: We analyze strict dissipativity of generalized linear quadratic optimal control problems on Hilbert spaces. Here, the term ``generalized'' refers to cost functions containing both quadratic and linear terms. We characterize strict pre-dissipativity with a quadratic storage function via coercivity of a particular Lyapunov-like quadratic form. Further, we show that under an additional algebraic assumption, strict pre-dissipativity can be strengthened to strict dissipativity. Last, we relate the obtained characterizations of dissipativity with exponential detectability.

Keywords:Control of Distributed Parameter Systems, Infinite Dimensional Systems Theory, Stability Abstract: This work concerns the internal stabilization of underactuated linear systems of m heat equations in cascade, where the control is placed internally in the first equation only and the diffusion coefficients are distinct. Combining the modal decomposition method with a recently introduced state transformation approach for observation problems, a proportional-type stabilizing control is given explicitly. It is based on a transformation for the ODE system corresponding to the comparatively unstable modes into a target one, where calculation of the stabilization law is independent of the arbitrarily large number of them and it is achieved by solving generalized Sylvester equations recursively. This provides a finite-dimensional counterpart of a recently introduced infinite-dimensional one, which led to Lyapunov stabilization. The present approach answers to the problem of stabilization with actuators not appearing in all the states and when boundary control results do not apply.

Keywords:Nonlinear Systems and Control, Optimal Control, Optimization : Theory and Algorithms Abstract: Recently, volumetric displays based on acoustic levitation have demonstrated the capability to produce mid-air content using the Persistence of Vision (PoV) effect. In these displays, acoustic traps are used to rapidly move a small levitated particle along a prescribed path. This note is based on our recent work OptiTrap (Paneva et al., 2022), the first structured numerical approach for computing trap positions and timings via optimal control to produce feasible and (nearly) time-optimal trajectories that reveal generic levitated graphics. While previously, feasible trap trajectories needed to be tuned manually for each shape and levitator, relying on trial and error, OptiTrap automates this process by allowing for a systematic exploration of the range of contents that a given levitation display can render. This represents a crucial milestone for future content authoring tools for acoustic levitation displays and advances volumetric displays closer toward real-world applications.

Keywords:Feedback Control Systems, Hybrid Systems, Optimal Control Abstract: Event-driven Intermittent control (IC) has been used as a framework to explain relevant aspects of human movement. The events, which are generated by states crossing predefined thresholds, give rise to state trajectories that mimic those of a continuous controller, by only using feedback information at event times. Here we present the results of using an optimisation approach to identify the parameters of an intermittent controller from experimental data, where users performed one dimensional mouse movements in a reciprocal pointing task. The results show that IC is able to reproduce both, the dynamical features and the variability of the pointing task across participants. We then introduce probabilistic elements in the IC framework in the form of Gaussian processes as an additional method to represent human movement variability.

Keywords:Model Predictive Control Abstract: We investigate the ability of Model Predictive Control (MPC) to generate human-like movements during interaction with mid-air user interfaces, i.e., pointing in virtual or augmented reality, using a state-of-the-art biomechanical model. The model is partly a black box implemented in the MuJoCo physics engine, requiring either gradient-free optimization algorithms or gradient approximation. This makes it even more important to choose the objective function or the MPC horizon length wisely. We introduce three objective functions suggested in the literature and identify optimal cost weights such that the simulated trajectories best match real ones obtained from motion capturing, i.e., we tackle an inverse optimal control problem. For the best performing objective function, we then analyze the effects of the horizon length and of the cost weights. This model-based approach enables the analysis of interaction techniques, e.g., in terms of ergonomics and effort, without the need for extensive user studies.

Keywords:Systems Biology, Optimal Control, Adaptive Control Abstract: Recent experiments have suggested that the nervous system adapted feedback control strategies during an ongoing, perturbed movement. These findings raised the possibility that a function of motor adaptation could be to complement feedback control online, but this idea had not been tested with biologically realistic properties of the human motor system, considering in particular the non-linear limb dynamics and the presence of transmission delays in the neural feedback loop. This study addresses this question by showing that online adaptive control is indeed feasible in a simplified nonlinear model of the human arm, featuring a delay of 60ms as observed in experiments. It is shown that online adaptation can reduce the impact of non-linear effects arising due to limb dynamics within a single movement. Strikingly, the directions that most benefited from online adaptation correlated with known directional biases characterising the distribution of reaching representations in the primate's brain. Further, it is demonstrated that, for some movement directions, it is possible to learn to produce relatively straight hand paths with end-point errors comparable with human performance within tens of trials. These simulation results provide support to the hypothesis that a function of adaptation in the human sensorimotor system is to compensate online for unmodelled disturbances arising in novel or non-linear environments.

Keywords:Feedback Control Systems Abstract: We compare the effectiveness of different auditory cues for attracting attention to spatial targets around a mobile user, using a commercial 3D audio headset instrumented with GPS and inertial sensors. We compare two approaches to spatial audio feedback with a baseline case that only provides `on target' feedback: 1. hints as single sounds played from a 3D location and 2. frequency modulation of inter-pulse gaps based on proximity. We illustrate the difference in user control behaviour created by the different forms of feedback with phase plots. Single 3D sound hints provided the best improvement over the baseline case of no hint. Frequency modulation of pulses performed more poorly for larger targets. The choice of sound has a significant effect on targeting performance and there is a significant trade-off between efficient targeting and aesthetically-pleasing audio.