Keywords:Infinite Dimensional Systems Theory Abstract: We briefly recall the basics about Lax-Phillips semigroups for well-posed linear systems, and the definition of well-posed nonlinear systems via nonlinear Lax Phillips semigroups. Then we concentrate on two results concerning well-posed nonlinear systems: We investigate a special class of nonlinear systems that are obtained by modifying the second order differential equation that is part of the description of conservative linear systems "out of thin air" introduced by M. Tucsnak and G. Weiss in 2003. The differential equation contains a nonlinear damping term that is maximal monotone and possibly set- valued. We show that this new class of nonlinear systems is incrementally scattering passive (hence well-posed). Our approach uses the theory of maximal monotone operators and the Crandall-Pazy theorem about nonlinear contraction semigroups, which we apply to the Lax-Phillips semigroup of the system. We investigate the class of incrementally scattering passive nonlinear systems, as defined in some earlier papers of ours. We show that these can be defined by a differential inclusion and a formula defining the current output in term of the current state and the current input. Our approach uses the theory of maximal monotone operators. The talk is based on joint work with Shantanu Singh.

Keywords:Mathematical Theory of Networks and Circuits, Linear Systems, Stability Abstract: We consider the differential-algebraic systems obtained by modified nodal analysis of linear RLC circuits from a systems theoretic viewpoint. We derive expressions for the set of consistent initial values and show that the properties of controllability at infinity and impulse controllability do not depend on parameter values but rather on the interconnection structure of the circuit. We further present circuit topological criteria for behavioral stabilizability.

Keywords:Networked Control Systems, Systems on Graphs, Mechanical Systems Abstract: Traditional linear passive vibration-absorber networks, such as the tuned mass damper (TMD), often contain springs, dampers, and masses. Recently there has been a growing trend to supplement or replace the masses with inerters. When considering the absorbers without a mass, a structure-immittance approach was proposed to identify possible configurations consisting of springs, dampers, and inerters. This approach can characterise the full class of network layouts with pre-determined numbers of each element type, and also prescribe the allowed value range for each element. More recently, a mass-included passive absorber, the tuned-mass-damper-inerter, was introduced, showing significant performance benefits on vibration suppression. With the aim to further explore the potential of numerous mass-included passive absorber layouts, a more generalised methodology was developed. Using this methodology, a full class of absorber layouts with a mass and a pre-determined number of inerters, dampers, and springs connected in series and parallel can be systematically investigated. A 3-storey building model is used to demonstrate the advantages of the proposed approaches for the cases without and with a mass, where the performance improvements can be up to 21.6% and 65.6%, respectively, compared to the TMD.

Keywords:Large Scale Systems, Linear Systems, Mathematical Theory of Networks and Circuits Abstract: The disturbance suppression problem for a chain of masses is discussed. The particular focus is placed on synthesising mechanical networks between masses that effectively suppress the disturbance propagation along the chain of any length. This study is motivated by the problem of controlling multi-agent systems where agents may leave or join the network. That is, the size of the network may change over time. In this work, we give the explicit expressions of scalar transfer functions from disturbance to an intermass displacement as a function of the number of masses, N, and discuss the methodology of synthesising a controller such that the H^infty norm is upper bounded by a prescribed value for any N.

Keywords:Mechanical Systems, Mathematical Theory of Networks and Circuits, Dissipativity Abstract: This paper continues the work of Georgiou, Jabbari and Smith on lossless adjustable mechanical devices. Defining equations and mechanical constructions of lossless adjustable springs and inerters for translational and rotational devices will be recalled. The role played by the lossless adjustable two-port transformer will be highlighted. A mechanical design will be described for a lossless adjustable rotational two-port transformer involving a double-cone arrangement, movable carriage and a pair of counter-rotating balls.

Keywords:Nonlinear Systems and Control, Mechanical Systems, System Identification Abstract: In recent years, network synthesis theory has been successfully applied to vibration absorber design, to identify optimum mechanical networks providing performance improvements. These identified mechanical networks consist of ideal linear modelling elements, such as springs, dampers and inerters. For real-life applications, the essential next step is to transfer these linear mechanical networks into physical absorber designs. There are two major challenges for this step: firstly, in order to achieve practical physical realisations, multidomain physical components (mechanical, hydraulic, pneumatic and electrical) need to be considered; and secondly, nonlinearities and other parasitic properties of physical components must be taken into consideration or potentially be made full use of. To this end, this paper, using a nonlinear mechanical network-based model for a bespoke mechanical-hydraulic device, demonstrates the feasibility of resolving both challenges.

Keywords:Mathematical Theory of Networks and Circuits, Linear Systems, Algebraic Systems Theory Abstract: The synthesis of bandpass microwave filters is based on the use of equivalent circuit models made of coupled resonators. These couplings are usually supposed to be independent of the frequency. We present in this paper a circuit model including possibly frequency varying couplings. After presenting some of its properties we consider the associated synthesis problem and show how techniques such as Groebner basis computation and Schur analysis based extraction techniques can be used to solve the latter exhaustively.

Keywords:Port-Hamiltonian Systems, Dissipativity, Stability Abstract: Gas transport in one-dimensional pipe networks can be described as an abstract dissipative Hamiltonian system, for which quantitative stability bounds can be derived by means of relative energy estimates for subsonic flow. This allows to establish convergence to the parabolic limit problem in the practically relevant high friction regime. The stability estimates carry over almost verbatim to a mixed finite element approximations with an implicit Euler time discretization, leading to order optimal convergence rates that are uniform the high friction limit. All results are proven in detail for the flow on a single pipe, but by the port-Hamiltonian formalism, they naturally extend to pipe networks.

Keywords:Port-Hamiltonian Systems, Infinite Dimensional Systems Theory, Numerical and Symbolic Computations Abstract: Mixed finite element (FE) approaches have proven very useful for the structure-preserving discretization of port-Hamiltonian (PH) distributed parameter systems, but non-uniform boundary conditions (BCs) were treated in an implicit manner up to now. We apply our recent approach from structure mechanics, which relies on the weak imposition of both Neumann and Dirichlet BCs based on a suitable variational principle, to the class of PH systems of two conservation laws. We illustrate (a) starting with the integral conservation laws the transition to an exterior calculus representation suitable for FE approximation according to Farle et al. (2013). Based thereon, we show (b) the variational formulation with weakly imposed BCs of both types. We discuss (c) on a simple example on a quadrilateral mesh the structure and the variables of the resulting FE models compared to the equations derived from a direct discrete approach on dual cell complexes. We (d) provide the corresponding FEniCS code for download.

Keywords:Port-Hamiltonian Systems, Numerical and Symbolic Computations, Infinite Dimensional Systems Theory Abstract: In this contribution, port-Hamiltonian systems with non-homogeneous mixed boundary conditions are discretized in a structure-preserving fashion by means of the Partitioned FEM. At the discrete level, an explicit pHs is obtained. The general construction relies on a weak imposition of the boundary conditions by means of the Hellinger-Reissner variational principle, as recently proposed in [Thoma et al., 2021]. The case of linear hyperbolic wave-like systems, including the elastodynamic problem and the Maxwell equations in 3D, is then illustrated in detail. A numerical example is worked out on the case of the wave equation.

Keywords:Numerical and Symbolic Computations, Infinite Dimensional Systems Theory, Mechanical Systems Abstract: We construct a structure-preserving finite element method and time-stepping scheme for inhomogeneous, incompressible magnetohydrodynamics (MHD). The method preserves energy, cross-helicity (when the fluid density is constant), magnetic helicity, mass, total squared density, pointwise incompressibility, and the constraint div B = 0. to machine precision, both at the spatially and temporally discrete levels.

Keywords:Port-Hamiltonian Systems Abstract: This work demonstrates the discretization of the boundary-controlled Maxwell equations, recast as a port-Hamiltonian system (pHs). After a reminder on the Stokes-Dirac structure associated with the Maxwell system, we introduce different partitioned weak formulations that preserve the pHs structure, and its associated power balance, at the semi- discrete level. These weak formulations are compared through numerical applications to closed non-perfectly conducting cavities and open waveguides under transverse approximation.

Keywords:Quantum Control, Nonlinear Systems and Control, Optimal Control Abstract: Which quantum states can be reached by controlling open Markovian n-level quantum systems? Here, we address reachable sets of coherently controllable quantum systems with switchable coupling to a thermal bath of temperature T. --- The core problem reduces to a toy model of studying points in the standard simplex allowing for two types of controls: (i) permutations within the simplex, (ii) contractions by a dissipative semigroup [Dirr et al. (2019)]. By illustration, we put the problem into context and show how toy-model solutions pertain to the reachable set of the original controlled Markovian quantum system. Beyond the case T = 0 (amplitude damping) we present new results for 0 < T < 1 using methods of d-majorisation.

Keywords:Quantum Control Abstract: Motivated by reachability questions in coherently controlled open quantum systems coupled to a thermal bath, as well as recent progress in the field of thermo-/vector-majorization we generalize classical majorization from unital quantum channels to channels with an arbitrary fixed point D of full rank. Such channels preserve some Gibbs-state and thus play an important role in the resource theory of quantum thermodynamics, in particular in thermo-majorization.

Based on this we investigate D-majorization on matrices in terms of its topological and order properties, such as existence of unique maximal and minimal elements, etc. Moreover we characterize D-majorization in the qubit case via the trace norm and elaborate on why this is a challenging task when going beyond two dimensions.

Keywords:Nonlinear Systems and Control, Optimal Control, Quantum Control Abstract: We study time minimal control problem for quantum systems whose dynamics are governed by the Bloch equation with interaction. The dynamics of the quantum systems are analyzed as affine control systems on the Bloch ball using parametrizations of the density matrix. The influence of Coulomb energies during a process of population transfer for a quantum system with several energy levels is shown and time minimal trajectories are given.

Keywords:Quantum Control, Applications of Algebraic and Differential Geometry in Systems Theory, Operator Theoretic Methods in Systems Theory Abstract: The last two decades produced a substantial noncommutative (in the free algebra) real and complex algebraic geometry. The aim of the subject is to develop a systematic theory of equations and inequalities for noncommutative polynomials in operator variables. A problem leading very directly to such equations and inequalities is finding good quantum strategies for games. The talk will focus on quantum games and present recent results done jointly with Adam Bene Watts, Igor Klep, Vern Paulsen, Mousavi, Nezhadi, Russel, and Zehong Zhao.

Keywords:Large Scale Systems, Computations in Systems Theory, Nonlinear Systems and Control Abstract: We describe here a non-intrusive data-driven time-domain formulation of balanced truncation (BT) for bilinear control systems. We build on the recent method of Gosea et al. (2021) that recasts the classic BT method for linear time invariant systems as a data-driven method requiring only evaluations of either transfer function values or impulse responses. We extend the domain of applicability of this non-intrusive data driven method to bilinear systems, arguably the simplest nontrivial class of weakly nonlinear systems.

Keywords:Large Scale Systems, Linear Systems Abstract: This extended abstract proposes a data-driven model reduction approach on the basis of noisy data. Firstly, the concept of data reduction is introduced. In particular, we show that the set of reduced-order models obtained by applying a Petrov-Galerkin projection to all systems explaining the data characterized in a large-dimensional quadratic matrix inequality (QMI) can again be characterized in a lower-dimensional QMI. Next, we develop a data-driven generalized balanced truncation method that relies on two steps. First, we provide necessary and sufficient conditions such that systems explaining the data have common generalized Gramians. Second, these common generalized Gramians are used to construct projection matrices that allow to characterize a class of reduced-order models via generalized balanced truncation in terms of a lower-dimensional QMI by applying the data reduction concept. Additionally, we present an alternative procedure to compute an a priori error bound.

Max Planck Institute for Dynamics of Complex Technical Systems

Keywords:Large Scale Systems, Machine Learning and Control, System Identification Abstract: The design of controllers for general nonlinear PDE models is a difficult task because of the high dimensionality of the partially discretized equations. It has been observed that the embedding of nonlinear systems into the class of linear parameter varying systems (LPV) gives way to apply linear theory and methods from numerical linear algebra for controller design. The feasibility of the LPV approach hinges on the dimension of the inherent parametrization. In this work we propose and evaluate combinations of convolutional neural networks and clustering algorithms for very low-dimensional parametrizations of incompressible Navier-Stokes equations.

Keywords:Port-Hamiltonian Systems, Systems on Graphs, Large Scale Systems Abstract: This contribution is on the construction of structure-preserving, online-efficient reduced models for a class of nonlinear partial differential equations on networks, which inherit a port-Hamiltonian structure. The flow problem finds broad application, e.g., in the context of gas distribution networks. We propose a snapshot-based reduction approach that consists of a mixed variational Galerkin approximation combined with quadrature-type complexity reduction. Its main feature is that certain compatibility conditions are assured during the training phase, which make our approach structure-preserving. The resulting reduced models are locally mass conservative and inherit an energy-bound and port-Hamiltonian structure. We demonstrate the applicability and good stability properties of our approach using the example of the Euler equations on networks.

Keywords:Optimization : Theory and Algorithms Abstract: The relationship between nonnegative polynomials and sums of squares on semialgebraic set S is one of the central questions in real algebraic geometry. The (convex) dual side of this story is important in analysis, where it is known as the truncated S-moment problem, and it considers the truncated cones of moments which are dual to nonnegative polynomials, and ``pseudo-moments'' which are dual to sums of squares. We bring a new tool for understanding of these classical problems: tropicalization. While extensively studied in complex algebraic geometry, tropicalization is rarely applied to semialgebraic sets. We provide explicit combinatorial descriptions of tropicalizations of the moment and pseudo-moment cones, and demonstrate their usefulness in distinguishing between nonnegative polynomials and sums of squares, proving results limiting the power of sums of squares approximations of nonnegative polynomials. We believe that this just scratches the surface of applications of tropicalization in semi-algebraic geometry.

Keywords:Optimization : Theory and Algorithms Abstract: We study the signed valuations of convex semialgebraic sets defined over non-Archimedean fields. This is motivated by the efforts to understand the structure of semialgebraic sets that arise in convex optimization, such as the spectrahedra and the hyperbolicity cones. We give a full characterization of regular sets that are obtained as signed tropicalizations of convex semialgebraic sets, and we prove that the signed tropicalizations of hyperbolicity cones have a more restrictive structure. To obtain our results, we combine two recent advances in the area of tropical geometry: the study of signed valuations of general semialgebraic sets and the separation theorems for signed tropical convexities.

Keywords:Optimization : Theory and Algorithms Abstract: The moment-sum-of-squares (moment-sos) hierarchy is one of the most celebrated and widely applied methods for approximating the minimum of an n-variate polynomial over a feasible region defined by polynomial (in)equalities. A key feature of the hierarchy is that it can be formulated as a semidefinite program of size polynomial in the number of variables n. Although this suggests that it may therefore be computed in polynomial time, this is not necessarily the case. Indeed, as O’Donnell and later Raghavendra & Weitz show, there exist examples where the sos-representations used in the hierarchy have exponential bit-complexity. We study the computational complexity of the sos-hierarchy, complementing and expanding upon earlier work of Raghavendra & Weitz. In particular, we establish algebraic and geometric conditions under which polynomial-time computation is possible. As this work is still ongoing, our results should be treated as preliminary.

Keywords:Optimization : Theory and Algorithms Abstract: Studying convex cones inside the cone of positive semidefinite (PSD) polynomials is an important field of research in real algebraic geometry and polynomial optimization. In this work, we combine two such well established cones, which are sums of squares (SOS) and sums of nonnegative circuit polynomials (SONC) and consider PSD polynomials, that decompose into an SOS and a SONC part. We call the resulting set the SOS+SONC cone. For this newly established cone, we prove two separation results. The first one is an analogue to Hilbert’s 1888 Theorem for the SOS+SONC cone. The second one shows that whenever the SOS and SONC cones are proper subsets of the PSD cone, they are also proper subsets of the SOS+SONC cone.

Keywords:Quantum Control, Optimal Control Abstract: Quantum optimal control has numerous important applications, ranging from magnetic resonance imagining to laser control of chemical reactions and quantum computing. There are two major challenges: non-commutativity inherent in quantum systems and non-convexity of quantum optimal control problems involving more than three quantum levels. Here, under mild assumptions, we present the first globally convergent methods for quantum optimal control. We address the non-commutativity of the control Hamiltonian at different times by the use of Magnus expansion. To tackle the non-convexity, we employ non-commutative polynomial optimisation and non-commutative geometry. Our results also demonstrate that the use of Magnus expansion expands the reachable set for Hamiltonians that are not operator controllable. Further, we show that for any fixed precision, there exists an efficiently-solvable convexification of the Magnus-expanded polynomial functionals in the control signal. Quantum optimal control is hence approximable to any fixed precision in a model of computing, wherein one arithmetic operation with two real numbers can be performed within one unit of time that has been introduced by Blum, Shub, and Smale.

Keywords:Stability, Systems on Graphs, Networked Control Systems Abstract: We study internal stability in the context of diffusively-coupled control architecture, common in multi-agent systems (e.g. the celebrated consensus protocol). We derive a condition under which the system can be stabilized by no controller from that class. The condition says effectively that diffusively-coupled controllers cannot stabilize agents that share common unstable dynamics, directions included. This class always contains a group of homogeneous unstable agents, like integrators. We argue that the underlying reason is intrinsic cancellations of unstable agent dynamics by such controllers, even static ones, where directional properties play a key role. The intrinsic lack of internal stability explains the notorious behavior of some distributed control protocols when affected by measurement noise or exogenous disturbances.

Keywords:Large Scale Systems, Networked Control Systems, Systems on Graphs Abstract: Models of social influence may present discontinuous dynamical rules, which are unavoidable with topological interactions, i.e. when the dynamics is the outcome of interactions with a limited number of nearest neighbors.

Here, we show that classical solutions are not sufficient to describe the resulting dynamics. We first describe the time evolution of the interaction graph associated to Caratheodory solutions, whose properties depend on the dimension of the state space and on the number of considered neighbors. We then prove the existence of Caratheodory solutions for 2-nearest neighbors, via a constructive algorithm.

Keywords:Systems on Graphs, Mathematical Theory of Networks and Circuits Abstract: Consider discrete-time linear distributed averaging dynamics, whereby a finite number of agents in a network start with uncorrelated and unbiased noisy measurements of a common state of the world modeled as a scalar parameter, and iteratively update their estimates following a non-Bayesian learning rule. Specifically, let every agent update her estimate to a convex combination of her own current estimate and those of her neighbors in the network (this procedure is also known as the French-DeGroot model, or the consensus algorithm). As a result of this iterative averaging process, each agent obtains an asymptotic estimate of the state of the world, and the variance of this individual estimate depends on the matrix of weights the agents assign to themselves and to the others. We study a game-theoretic multi-objective optimization problem whereby every agent seeks to choose her self-confidence value in the convex combination in such a way to minimize the variance of her asymptotic estimate of the state of the world. Assuming that the relative influence weights assigned by the agents to their neighbors in the network remain fixed and form an irreducible relative influence matrix, we characterize the Pareto frontier of the problem, as well as the set of Nash equilibria in the resulting game.

Keywords:Transportation Systems, Mathematical Theory of Networks and Circuits Abstract: We study a network design problem (NDP) where the planner aims at selecting the optimal single-link intervention in a transportation network to minimize the total congestion. Our first result is to show that the NDP may be formulated in terms of electrical quantities on a related resistor network, in particular in terms of the effective resistance between adjacent nodes. We then suggest an approach to approximate such an effective resistance by performing only local computations, and exploit this approach to design an efficient algorithm to solve the NDP, without recomputing the equilibrium flow after the intervention. We then study the optimality of the proposed procedure for recurrent networks, and provide simulations over relevant networks.

Keywords:System Identification, Networked Control Systems, Linear Systems Abstract: This work focuses on the generic identifiability of dynamical networks with partial excitation and measurement: a set of nodes are interconnected by transfer functions according to a known topology, some nodes are excited, some are measured, and only a part of the transfer functions are known. Our goal is to determine whether the unknown transfer functions can be generically recovered based on the input-output data collected from the excited and measured nodes. Introducing the notion of generic local identifiability, we derive a necessary and sufficient algebraic condition, which can be checked efficiently by rank computation. Another notion, generic decoupled identifiability, allows to reflect on a larger network which decouples excitations and measurements. This yields a necessary path-based condition, and a sufficient one.

Keywords:Optimization : Theory and Algorithms Abstract: The paper proposes an algorithm that uses distributed online mirror descent algorithm for solving constrained online optimization problem with event triggered communication. The optimization is over a time horizon and the future objective functions are not apriori known to each agent. In the proposed algorithm, the communication between the agents, that happens in a distributed optimization framework, occurs only when the difference between the current state and the state when the last event has been triggered exceeds a threshold. The performance of the algorithm is analysed using a regret function. We establish a bound on the regret and provide sufficient conditions on the step-size and thresholding error such that the regret is sublinear. We demonstrate the reduction in the number of inter-agent communications using our proposed algorithm for an estimation problem in a dynamic environment.

Keywords:Model Predictive Control, Nonlinear Systems and Control, Adaptive Control Abstract: Funnel MPC, a novel Model Predictive Control (MPC) scheme, allows guaranteed output tracking of smooth reference signals with prescribed error bounds for nonlinear multi-input multi-output systems. To this end, the stage cost resembles the high-gain idea of funnel control. Without imposing additional output constraints or terminal conditions, the Funnel MPC scheme is initially and recursively feasible for systems with relative degree one and stable internal dynamics. Using an additional funnel for the derivative as a penalty term in the stage cost, these results can be also extended to single-input single-output systems with relative degree two.

Keywords:Model Predictive Control, Stochastic Control and Estimation Abstract: We propose a stochastic MPC scheme using an optimization over the initial state for the predicted trajectory. Considering linear discrete-time systems under unbounded additive stochastic disturbances subject to chance constraints, we use constraint tightening based on probabilistic reachable sets to design the MPC. The scheme avoids the infeasibility issues arising from unbounded disturbances by including the initial state as a decision variable. We show that the stabilizing control scheme can guarantee constraint satisfaction in closed loop, assuming unimodal disturbances. In addition to illustrating these guarantees, the numerical example indicates further advantages of optimizing over the initial state for the transient behavior.

Keywords:System Identification, Machine Learning and Control, Robust and H-Infinity Control Abstract: We present a novel targeted exploration strategy for the application of robust dual control. Unlike common greedy random exploration strategies considered in the related dual control literature, we introduce a targeted strategy in which the exploration inputs are a linear combination of sinusoids whose amplitudes are optimized based on an exploration criterion. Specifically, we leverage recent results on persistence of excitation using spectral lines to show how a (high probability) lower bound on the resultant persistence of excitation of the exploration data can be established. These results can be used to provide a priori lower bounds on the remaining model uncertainty after exploration. Given this exploration strategy and the corresponding uncertainty bounds, tools from robust control and gain-scheduling can be used to design a robust dual controller.

Keywords:Networked Control Systems, Computations in Systems Theory, Optimization : Theory and Algorithms Abstract: In this paper, we discuss hands-off feedback control of discrete-time linear time-invariant systems based on receding horizon control. Hands-off control, also known as sparse control, is a control that has a long time duration over which the control action is exactly zero whilst satisfying control objectives. To obtain the maximum hands-off control, the L1-norm optimization is adopted. For a model predictive control formulation, we need to numerically solve the L1 optimization with equality/inequality constraints. Although fast iterative algorithms are known to solve the optimization problem, they will often not be fast enough for control systems that need real-time computation. To obtain the control values in real time, we propose to stop the iteration for the L1 optimization after just one step. We prove that this strategy leads to practical stability of the closed-loop, provided the systems are open-loop stable. Simulation results show the effectiveness of the proposed method.

Keywords:Infinite Dimensional Systems Theory, Model Predictive Control, Linear Systems Abstract: Explicit model predictive control design is carefully developed for discrete-time linear plants on Hilbert spaces, and we highlight the role of the so-called Slater condition in the reliable explicit solution of the MPC optimization. We then proceed to present an explicit MPC algorithm that accounts for the stabilization and input constraints satisfaction. We do structure preserving temporal discretization of the infinite-dimensional parabolic PDE system by application of the Cayley transformation. The salient feature of explicit MPC design is the realization of the region-free approach in explicit MPC design with identification of active constraint sets to realize optimal stabilization and constraints satisfaction. Finally, the resulting design is illustrated by the application to the PDE model given by an unstable heat equation with boundary actuation and Neumann boundary conditions. The example demonstrates simultaneous stabilization and input constraints satisfaction on the one hand, and on the ability to deal with a relatively high plant dimension and a long optimization horizon on the other hand.

Keywords:Stochastic Control and Estimation, Optimal Control, Numerical and Symbolic Computations Abstract: We consider Dynamic programming equations associated to discrete time stochastic control problems with continuous state space, which arise in particular from monotone time discretizations of Hamilton-Jacobi-Bellman equations. We develop and study several numerical algorithms for solving such equations, combining tropical numerical methods and stochastic dual dynamic programming methods. We also compare these algorithms with the point based methods for solving Partially Observable Markov Decision Processes (POMDP).

Keywords:Optimal Control, Numerical and Symbolic Computations, Stochastic Control and Estimation Abstract: We investigate in this work a fully-discrete semi-Lagrangian approximation of second order possibly degenerate Hamilton-Jacobi-Bellman (HJB) equations on a bounded domain with oblique derivatives boundary conditions. These equations appear naturally in the study of optimal control of diffusion processes with oblique reflection at the boundary of the domain.

The proposed scheme is shown to satisfy a consistency type property, it is monotone and stable. Our main result is the convergence of the numerical solution towards the unique viscosity solution of the HJB equation. The convergence result holds under the same asymptotic relation between the time and space discretization steps as in the classical setting for semi-Lagrangian schemes on unbounded domains. We present some numerical results, in dimensions one and two, on unstructured meshes, that confirm the numerical convergence of the scheme.

Keywords:Hybrid Systems, Optimal Control, Stochastic Control and Estimation Abstract: We investigate the modelling of sailing races as hybrid stochastic games, either with zero or nonzero sum, where the first case is typical of match races and the second of fleet races. In particular, we provide models of growing complexity and dimension, study the optimal strategies in various racing situations and devise some fast and/or reduced memory implementation.

Keywords:Stochastic Control and Estimation, Optimal Control, Nonlinear Systems and Control Abstract: A class of nonlinear, stochastic staticization control problems (including minimization problems with smooth, convex, coercive payoffs) driven by diffusion dynamics with constant diffusion coefficient is considered. The nonlinearities are addressed through staticization-based duality. The second-order Hamilton-Jacobi partial differential equations (HJ PDEs) are converted into associated control problems with higher-dimensional states. In these problems, one component of the state propagates by deterministic, nonlinear dynamics, while the other component is a scaled Brownian motion. These components interact only through a bilinear terminal cost. This structure will be exploited to generate an efficient solution approach.

Keywords:Optimal Control, Optimization : Theory and Algorithms, Linear Systems Abstract: In this paper, we review the basics of compressed sensing and introduce its application to optimal control, called the maximum hands-off control. First, we present the mathematical formulation of compressed sensing and show a heuristic approach to the problem using the l1 norm with efficient numerical algorithms. Then, we introduce the maximum hands-off control, the sparsest control, or the L0 optimal control. We show mathematical properties of the maximum hands-off control, such as the equivalence between the L0 and L1 optimal controls, necessary conditions, and the existence. We also show the time discretization method to numerically compute the maximum hands-off control. Finally, we showcase some extensions of the maximum hands-off control.

Keywords:Optimal Control Abstract: Optimal control theory, algorithms, and software for analyzing and computing local solutions of linear and nonlinear optimal control problems have reached a high level of maturity, finding their way into industry. In the context of many applications, locally optimal control inputs can be computed within the milli- and microsecond range. This is in sharp contrast to the development of algorithms for locating global minimizers of non- convex optimal control problems, which is hindered by several key issues, including the overall complexity of generic optimal control problems and their curse of dimensionality. This talk reviews and discusses recent solutions that address these rather fundamental challenges including novel types of Branch & Lift methods as well as modern Koopman- Pontryagin operator based lifting methods for global optimal control. Various numerical experiments will be used to illustrate the effectiveness of these approaches. The talk concludes with an assessment of the state of the art and highlights important avenues for future research.

Keywords:Optimal Control Abstract: Optimal feedback synthesis for nonlinear dynamics -a fundamental problem in optimal control- is enabled by solving fully nonlinear Hamilton-Jacobi-Bellman type PDEs arising in dynamic programming. While our theoretical understanding of dynamic programming and HJB PDEs has seen a remarkable development over the last decades, the numerical approximation of HJB-based feedback laws has remained largely an open problem due to the curse of dimensionality. More precisely, the associated HJB PDE must be solved over the state space of the dynamics, which is extremely high-dimensional in applications such as distributed parameter systems or agent-based models. In this talk we will review recent approaches regarding the effective numerical approximation of very high-dimensional HJB PDEs. We will explore modern scienti fic computing methods based on tensor decompositions of the value function of the control problem, and the construction of data-driven schemes in supervised and semi-supervised learning environments. We will highlight some novel research directions at the intersection of control theory, scientific computing, and statistical machine learning.

Keywords:Control issues in Finance, Stochastic Modeling and Stochastic Systems Theory, Optimization : Theory and Algorithms Abstract: In this study, we develop a quantitative strategy for controlling cash-flow fluctuations of power utilities in electricity trading market using adequate financial instruments. In particular, we focus on hedging of thermal power generations and provide mixed positions of derivatives and forwards in a flexible manner, where we apply nonparametric regression techniques to find optimal payoff structure of derivatives and/or optimal units of forward contracts with fine granularity. An empirical backtest is conducted to illustrate our proposed hedging strategy.

Keywords: Forward contracts, derivatives, thermal power generators, hedging, nonparametric regressions.

Keywords:Control issues in Finance, Stochastic Modeling and Stochastic Systems Theory, Optimization : Theory and Algorithms Abstract: In this paper, we propose an online data-driven sliding window approach to solve a log-optimal portfolio problem. In contrast to many of the existing papers, this approach leads to a trading strategy with time-varying portfolio weights rather than fixed constant weights. We show, by conducting various empirical studies, that the approach possesses a superior trading performance to the classical log-optimal portfolio in the sense of having a higher cumulative rate of returns.

Keywords:Stochastic Modeling and Stochastic Systems Theory, Machine Learning and Control, Infinite Dimensional Systems Theory Abstract: Signature methods represent a non-parametric way for extracting characteristic features from time series data which is essential in machine learning tasks. This explains why these techniques become more and more popular in econometrics and mathematical finance. Indeed, signature based approaches allow for data-driven and thus more robust model selection mechanisms, while first principles like no arbitrage can still be easily guaranteed.

Here we focus on financial models whose dynamics are described by linear functions of the (time-extended) signature of a primary underlying process, which can range from a (market-inferred) Brownian motion to a general multidimensional tractable stochastic process. The framework is universal in the sense that any classical model can be approximated arbitrarily well and that the model characteristics can be learned from all sources of available data by simple methods. In view of option pricing and calibration, key quantities that need to be computed in these models are the expected value or Fourier Laplace transform of the signature of the primary underlying process. Surprisingly this can be achieved via techniques from affine and polynomial processes. These formulas can then be used in the calibration procedure to option prices, while calibration to time series data just reduces to a simple regression.

Keywords:Control issues in Finance, Stochastic Modeling and Stochastic Systems Theory, Stochastic Control and Estimation Abstract: It has been shown in the literature that for certain trading strategies based on control techniques, namely for the so-called simultaneously long short strategies under relatively weak market assumptions in continuous time, the so-called robust positive expectation property holds. This means that for such strategies, if the assumptions are fulfilled, in expectation positive profits can be proven. Of course, arguments such as trading costs or trading constraints can be used when discussing these unexpected results. But there are also risks inherent in the strategies themselves, such as short-selling risks, discretization risks, or momenta. In this talk, we will present these risks and show how they can possibly be controlled.

Keywords:Control issues in Finance, Feedback Control Systems Abstract: The takeoff point for this work is the emerging body of literature which addresses algorithmic trading in the framework of feedback control systems. In this setting, the buying and selling of equities period is governed by the action of a controller, using past history, to determine the time-varying investment level. Almost all of the papers to date begin with a underlying mathematical model structure for the stock-price dynamics and "theoretical" performance is studied. In many cases, the parameters of the price model are not assumed to be known in advance; they are estimated over time from the realized price path. In the literature, we also see many variations on this theme. For example, the investment-level controller may have no explicit reliance on an assumed price model and instead are updated by performance variables such as account market value or gains and losses over time. Subsequently, the authors of papers along these lines demonstrate the performance of their trading algorithms using various criteria. Given this context, we draw attention to use of the word "should" in the title. This is intentional because this short paper is an opinion piece; it does not contain new results. Instead, arguments are given that the control community will be well served if future papers devote greater attention to backtesting and standardization of benchmark data sets. It is argued that this will enable the results of one researcher to more easily be compared against those of another and increase the impact of control-theoretic papers on researchers and practitioners outside the control field. While it is true that a number of the control-inspired papers to date already include some backtest results, the use of widely varying data sets makes evaluation of worthiness of their "controller recipes" difficult or impossible.

Keywords:Mechanical Systems, Neural Networks, System Identification Abstract: During the last years, Hamiltonian neural networks (HNN) have been introduced to incorporate prior physical knowledge when learning dynamical systems. Hereby, the symplectic system structure is preserved despite the data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we enhance the HNN with a Lie algebra framework to detect and embed symmetries in the neural network. This approach allows to simultaneously learn the symmetry group action and the total energy of the system.

Keywords:System Identification, Mechanical Systems, Machine Learning and Control Abstract: Hamilton's principle is one of the most fundamental principle in physics. Incorporating the principle into data-driven models of dynamical systems guarantees that motions share important qualitative properties with the real system, such as energy or momentum conservation. To learn Lagrangian dynamics, we propose to learn inverse modified Lagrangians related to variational integrators instead of attempting to learn an exact Lagrangian, as is typically done in the literature. The key advantage is that inverse modified Lagrangians can be learned from snapshots of position data of observed trajectories directly without approximating velocities or acceleration data. This is beneficial when snapshot times are large. Moreover, when inverse modified Lagrangians are integrated using a variational method, discretisation errors are compensated for. Therefore, large step-sizes can be used while maintaining high accuracy and tiny energy errors.

Keywords:Dissipativity, Neural Networks, Stability Abstract: Building on the strong connection between dissipativity theory and Integral Quadratic Constraints, we show how feedback loops involving neural networks can be analysed computationally with respect to both stability and robustness. A basic building block is the ReLU (Rectified Linear Unit) nonlinearity and we present both old and new dissipation inequalities that are useful for its analysis.

Keywords:Large Scale Systems, Dissipativity, Computations in Systems Theory Abstract: In this paper we present a simple technique which can be systematically used to obtain non-conservative decomposition for a class of linear matrix inequalities (LMIs) with an additive structure. By non-conservative decomposition we mean a suitable replacement of an additive LMI with a set of equivalent inequalities which are coupled by common variables. The results are applied on several stability/dissipativity analysis problems to produce analysis LMIs suitable for distributed computation.

Keywords:Stability, Computations in Systems Theory, Stochastic Modeling and Stochastic Systems Theory Abstract: We consider orthogonal transformations of arbitrary square matrices to a form where all diagonal entries are equal. In our main results we treat the simultaneous transformation of two matrices and the symplectic orthogonal transformation of one matrix. A relation to the joint real numerical range is worked out, efficient numerical algorithms are developed and applications to stabilization by rotation and by noise are presented.

Keywords:Computations in Systems Theory Abstract: In this paper we develop the use of monads, a concept popularised in the functional language Haskell, to provide a general framework for studying control systems with uncertainty. We first develop the theory of monads, notably dependent products and conditioning. We then provide three main applications, namely computing the input-output behaviour, designing observers, and studying compositionality for uncertain systems.

Keywords:Large Scale Systems, Optimization : Theory and Algorithms Abstract: We study a dual decomposition algorithm for a distributed optimization problem with a communication structure corresponding to a hypergraph. We prove that in the case of quadratic objective functions the respective discrete-time dynamical system of a modified dual decomposition algorithm that makes use of the Hessians of the objective functions converges in only one iteration.

Keywords:Networked Control Systems, Linear Systems, Optimization : Theory and Algorithms Abstract: Controllability maximization problem under sparsity constraints is a node selection problem that selects inputs that are effective for control in order to minimize the energy to control for desired state. In this paper we discuss the equivalence between the sparsity constrained controllability metrics maximization problems and their convex relaxation. The proof is based on the matrix-valued Pontryagin maximum principle applied to the controllability Lyapunov differential equation.

Keywords:Applications of Algebraic and Differential Geometry in Systems Theory, Networked Control Systems, Nonlinear Systems and Control Abstract: Multi-agent systems are known to exhibit stable emergent behaviors, including polarization, over R^n or highly symmetric nonlinear spaces. In this article, we eschew linearity and symmetry of the underlying spaces, and study the stability of polarized equilibria of multi-agent gradient flows evolving on general hypermanifolds. The agents attract or repel each other according to the partition of the communication graph that is connected but otherwise arbitrary. The hypersurfaces are outfitted with geometric features styled "dimples" and "pimples" that characterize the absence of flatness. The signs of inter-agent couplings together with these geometric features give rise to stable polarization under various sufficient conditions.

Keywords:Nonlinear Systems and Control, Quantum Control, Applications of Algebraic and Differential Geometry in Systems Theory Abstract: For a symmetric Lie algebra g=k⊕p we consider a class of bilinear or more general control-affine systems on p defined by a drift vector field X and control vector fields ad_{ki} which gain fast and full control on the adjoint orbits of the corresponding compact group K. We show that under quite general assumptions on X such a control system is essentially equivalent to a natural reduced system on a maximal Abelian subspace a⊆p, and likewise to related differential inclusions defined on a.

Keywords:System Identification, Systems on Graphs Abstract: Spectral network identification aims at inferring the eigenvalues of the Laplacian matrix of a network from measurement data. This allows to capture global information on the network structure from local measurements at a few number of nodes. In this paper, we consider the spectral network identification problem in the generalized setting of a vector-valued diffusive coupling. The feasibility of this problem is investigated and theoretical results on the properties of the associated generalized eigenvalue problem are obtained. Finally, we propose a numerical method to solve the generalized network identification problem, which relies on dynamic mode decomposition and leverages the above theoretical results.

Max Planck Institute for Dynamics of Complex Technical Systems

Keywords:Nonlinear Systems and Control, Stochastic Control and Estimation, Machine Learning and Control Abstract: In the context of Koopman operator based analysis of dynamical systems, the generator of the Koopman semigroup is of central importance. Models for the Koopman generator can be used, among others, for system identification, coarse graining, and control of the system at hand.

A critical modeling choice is the subspace or dictionary used for Koopman estimation. In this talk, I will present recent advances allowing for the approximation of the generator on reproducing kernel Hilbert spaces (RKHS), and on tensor-structured subspaces by means of low-rank representations. Both approaches allow modelers to employ high-dimensional, or even infinite-dimensional approximation spaces, while controlling the computational effort at the same time. In both cases, I will discuss the algorithmic realization and computational complexity in detail. I will also discuss recent results on estimating the finite-data estimation error for Koopman generator models.

Max Planck Institute for Dynamics of Complex Technical Systems

Keywords:Machine Learning and Control, Optimal Control, Nonlinear Systems and Control Abstract: Extended Dynamic Mode Decomposition, embedded in the Koopman framework, is a widely-applied technique to predict the evolution of an observable along the flow of a dynamical (control) system. However, despite its popularity, the error analysis for control systems is still fragmentary. Here, we provide a complete and rigorous analysis of the approximation error for control systems. To this end, the approximation error is split up according to its two sources of error: the finite dictionary size (projection) and the finite amount of i.i.d. data used to generate the surrogate model (estimation). Then, invoking—among others—finite-elements techniques and the Chebyshev inequality, probabilistic error bounds are derived. Finally, we demonstrate the applicability of the novel error bounds in optimal control with state and control constraints.

Keywords:Operator Theoretic Methods in Systems Theory, Computations in Systems Theory, Optimal Control Abstract: We provide a data-driven framework for optimal control of a continuous-time stochastic dynamical system. The proposed framework relies on the linear operator theory involving linear Perron-Frobenius (P-F) and Koopman operators. Our first results involving the P-F operator provide a convex formulation to the optimal control problem in the dual space of densities. This convex formulation of the stochastic optimal control problem leads to an infinite-dimensional convex program. The finite-dimensional approximation of the convex program is obtained using a data-driven approximation of the P-F operator. Our second results demonstrate the use of the Koopman operator, which is dual to the P-F operator, for the stochastic optimal control design. We show that the Hamilton Jacobi Bellman (HJB) equation can be expressed using the Koopman operator. We provide an iterative procedure along the lines of a popular policy iteration algorithm based on the data-driven approximation of the Koopman operator for solving the HJB equation. The two formulations, namely the convex formulation involving P-F operator and Koopman based formulation using HJB equation, can be viewed as dual to each other where the duality follows due to the dual nature of P-F and Koopman operators.

Keywords:Machine Learning and Control, Nonlinear Systems and Control, Information Theory Abstract: This paper presents a fuzzy systems approach to the prediction of nonlinear time series and dynamical systems, based on a fuzzy model. The underlying mechanism governing a time series is perceived by a modified structure of the fuzzy system in order to capture the time series as well as the information about its successive time derivatives. The prediction task is carried out by a fuzzy predictor based on the extracted rules and by the Taylor ODE solver method. The approach has been applied to a benchmark problem: the Mackey- Glass chaotic time series. Furthermore, comparative studies with other fuzzy and neural network predictors were made and these suggest equal or even better performance of the herein presented approach.

Keywords:Linear Systems Abstract: Concepts of controllability and observability have been de fined for a class of decentralized systems known as coordinated linear systems. The classical duality result does not extend to these systems. In the present paper, we generalize these notions of controllability and observability to poset-causal systems. We introduce the dual system associated with a poset-causal system and extend the classical duality result using this notion of a dual system.

Keywords:Operator Theoretic Methods in Systems Theory, Multidimensional Systems, Linear Systems Abstract: We consider overdetermined multidimensional discrete-time systems where the evolution of the whole state vector is given by several update equations in several linearly independent directions. Such systems are overdetermined and we assume that they come equipped with compatibility difference equations for the input and output signals. As a consequence of these compatibility equations frequency domain analysis leads to function theory on a certain algebraic curve rather than to function theory in several complex variables. More precisely, the transfer function of the system is (under certain assumptions) a meromorphic bundle map on a compact Riemann surface.

In this talk we will discuss the corresponding realization problem and provide a solution which is the higher genus analogue of the classical Hankel realization.

Keywords:Linear Systems Abstract: In a seminal paper Foster showed that the impedances of lumped electrical circuits generated by inductances and capacitors are positive real odd functions (PRO for short). For multi-port electrical systems built from inductances and capacitors one obtains matrix-valued PRO functions, denoted PRO_m in the case of m by m matrix functions. Like PRO, the class of matrix functions PRO_m is also a convex invertible cone, i.e., a convex cone closed under inversion (in the form of involution). Given a minimal, Weierstrass descriptor realization for a function in PRO_m, we explicitly compute a minimal, Weierstrass descriptor realization for its involution, and through these formulas one can analyse the zero-pole structure of the function.

Keywords:Operator Theoretic Methods in Systems Theory, Mathematical Theory of Networks and Circuits, Linear Systems Abstract: There are four families of passive linear systems, described by: Positive Real, Bounded real, Discrete-time Positive real and Discrete-time Bounded real functions. We show that by using Quadratic Matrix Inequalities, starting with the state-space realization of one of the four families, the other three may be obtained through prescribed Linear Fractional Transformations.

Keywords:Operator Theoretic Methods in Systems Theory, Linear Systems Abstract: In recent years various papers appeared that concentrate on inverse problems associated with work of Ellis and Gohberg on orthogonal matrix Wiener functions. Here we study such an inverse problem restricting to rational matrix functions on the real line. The functions used in stating the problem are assumed to be given by minimal state space realizations, and the necessary and sufficient solution criterion as well as the formulas for the solution presented here are described in terms of the matrices of the state space realizations along with solutions to certain Lyapunov equations associated with the data.

Keywords:Optimization : Theory and Algorithms, Networked Control Systems Abstract: We propose a novel fully decentralized energy management scheme for aggregating distributed energy resources for grid flexibility services in wholesale electricity market. We model this problem as a multi-leader-multi-follower noncooperative game. Then a fully distributed algorithm in discrete-time is proposed to solve the problem and find the Nash Equilibrium(NE). In this algorithm, each aggregator only needs to exchange its estimate of the aggregate and an auxiliary variable with its neighbours. This scheme shows the scalability and efficiency in aggregating flexibility services from a large number of prosumers.

Keywords:Large Scale Systems, Systems on Graphs, Nonlinear Systems and Control Abstract: We deal with evolutionary game-theoretic learning processes for population games on networks with dynamically evolving communities. Specifically, we propose a novel framework in which a deterministic, continuous-time replicator equation on a community network is coupled with a closed migration process between the communities, in turn governed by an environmental feedback mechanism resulting in co-evolutionary dynamics. Through a rigorous analysis of the system of differential equations obtained, we characterize the equilibria of the coupled dynamical system. Moreover, for a class of population games ---matrix games--- a Lyapunov argument is used to establish an evolutionary folk theorem that guarantees convergence to the evolutionary stable states of the game. Numerical simulations are provided to illustrate and corroborate our theoretical findings.

Keywords:Optimization : Theory and Algorithms, Multidimensional Systems, Stability Abstract: We model a system made of n asymmetric firms participating in a market in which each firm chooses as its strategy a supply function relating its quantity to its price. Such strategy (Supply function equilibrium) is a generalization of models where firms can either set a fixed quantity (Cournot model) or set a fixed price (Bertrand model). Our goal is to study the payas-bid auction in this setting. Under the assumption of K-Lipschitz supply functions, we were capable of determining existence and characterization of Nash equilibria of the game.

Keywords:Networked Control Systems, Systems on Graphs, Stochastic Modeling and Stochastic Systems Theory Abstract: This extended abstract presents two network games applied to problems in the context of security -- robotic surveillance and agents' mutual influences on communication networks. The analysis depends on a method we termed as ``control of Markov chains.'' In these two games, agents' strategy can be regarded as designing a Markov chain. And we study the existence and properties of pure strategy Nash equilibria in different scenarios. The main advantage of the games is that the payoff functions for the agents will be properties derived from a Markov chain itself, such as the stationary distribution. Lastly, we discuss the possibilities of extending the analysis.

Keywords:Mathematical Theory of Networks and Circuits Abstract: This paper studies the power flow feasibility of DC power grids with constant-power loads. We introduce and motivate the concept of Braess' paradox for power flow feasibility, and show that this phenomenon can occur in most practical power grids with at least two source nodes.

Keywords:Nonlinear Systems and Control, Physical Systems Theory Abstract: The bifurcation behaviors of coupled Andronov-Hopf oscillators is analyzed. For this, an analytic solution of the bifurcation point of two linear, static coupled Andronov-Hopf oscillators is provided. This solution is rearranged so that the bifurcation can be interpreted geometrically by maximizing the area of an triangle defined by the system parameters. This geometric interpretation serves as the basis to address a network of linear, static coupled Andronov-Hopf oscillators, where a rather simple solution for the bifurcation point can be determined depending on a simple tree topology. With this, implications about the synchronization of this class of networks can be deduced.

Keywords:System Identification Abstract: This article is a resubmission of the full paper that was accepted for the presentation at the MTNS 2020. This article reports error analysis and asymptotic variance of a closed-loop subspace model identification method for a system described with the output-error state-space representation. For details, since the procedure of the identification method includes the QR factorization of stacked data Hankel matrices, this study investigates asymptotic properties of block entries of the triangular matrix obtained from the QR factorization. The set of the block entries is separated into two components, namely, the signal-based component and the noise-based component. The contributions are to derive asymptotic properties of both components and to obtain the asymptotic covariance matrix of the vectorization of the noise-based component.

Keywords:System Identification, Optimization : Theory and Algorithms, Signal Processing Abstract: In this article, we propose a novel system identification method for stable and sparse linear time-invariant systems. We adopt kernel-based regularization to take a priori information, such as the decay rate, of the target system into account. For promoting sparsity, we introduce the minimax concave penalty function, which is known to promote sparser results than the standard L1 penalty. The estimation problem is shown to be reduced to a convex optimization problem, which can be efficiently solved by the forward-backward algorithm. We show a numerical example of delayed FIR (finite impulse response) system identification to illustrate the effectiveness of the proposed method.

Keywords:Nonlinear Filtering and Estimation, Nonlinear Systems and Control Abstract: This paper is concerned with the development of a general scheme for box particle filtering, in which the likelihood computation is shown to be the most crucial step for the estimation strategy. An overview on Box Particle Filters and discussions about from assumptions used in the literature to the filters performance evaluation approach are in the scope of the paper. From this, we aim to produce a filter taking advantages from strong aspects of various existing box particle filters. A class of nonlinear L^2 functions is concerned. Also, a comparative study via an illustration example to highlight the efficiency of the proposed method is investigated.

Keywords:Linear Systems, Optimal Control, Optimization : Theory and Algorithms Abstract: This paper considers a deterministic estimation problem to find the input and state of a linear dynamical system which minimise a weighted integral squared error between the resulting output and the measured output. A completion of squares approach is used to find the unique optimum in terms of the solution of a Riccati differential equation. The optimal estimate is obtained from a two-stage procedure that is reminiscent of the Kalman filter. The first stage is an end-of-interval estimator for the finite horizon which may be solved in real time as the horizon length increases. The second stage computes the unique optimum over a fixed horizon by a backwards integration over the horizon. A related tracking problem is solved in an analogous manner. Making use of the solution to both the estimation and tracking problems a constrained estimation problem is solved which shows that the Riccati equation solution has a least squares interpretation that is analogous to the meaning of the covariance matrix in stochastic filtering. The paper shows that the estimation and tracking problems considered here include the Kalman filter and the linear quadratic regulator as special cases.

Keywords:Stochastic Control and Estimation, Networked Control Systems, Hybrid Systems Abstract: In this extended abstract, we consider the problem of analyzing the performance of distributed filters for continuous-time linear stochastic systems under certain information constraints. We associate an undirected and connected graph with the measurements of the system, where the nodes have access to partial measurements in continuous time. Each node executes a locally optimally filter based on the available measurements. In addition, a node communicates its estimate to a neighbor at some randomly drawn discrete time instants, and these activation times of the graph edges are governed by independent Poisson counters. When a node gets some information from its neighbor, it resets its state using a convex combination of the available information. Consequently, each node implements a filtering algorithm in the form of a stochastic hybrid system. We derive bounds on expected value of error covariance for each node, and show that they converge to a common value for each node if the mean sampling rates for communication between nodes are large enough.

Keywords:Stochastic Control and Estimation, Stochastic Modeling and Stochastic Systems Theory, Robust and H-Infinity Control Abstract: Linear, continuous-time systems with state-multiplicative noise are considered. The problem of H_infty Luenberger filtering for either deterministic norm-bounded or polytopic-type uncertain systems are solved via a simple LMI(s) condition. An illustrative example is given that demonstrates the tractability of our solution method in the robust uncertain case.

Keywords:Optimal Control, Neural Networks, Nonlinear Systems and Control Abstract: We propose new neural networks algorithms for the approximation of deterministic optimal control problems with maximum running cost. This problem is motivated by the approximation of general optimal control problems in the presence of state constraints. This problem is also related to Hamilton-Jacobi-Bellman equations with an obstacle term. Difficulties arise in particular because of the non-smoothness of the value to be approximated, and appropriate solutions are studied to deal with this specific issue. Numerical examples are given on front propagation problems in the presence of an obstacle, for average dimensions 2leq dleq 8.

Keywords:Optimal Control, Computations in Systems Theory, Optimization : Theory and Algorithms Abstract: Solving high-dimensional optimal control problems and their corresponding Hamilton-Jacobi partial differential equations is an important but challenging problem. In particular, handling optimal control problems with state-dependent running costs or constraints on the control presents an additional challenge. We present two representation formulas: one is a Hopf-type representation formula for solving a class of optimal control problems with certain non-smooth state-dependent running costs, and the other is a Lax-Oleinik-type representation formula for solving a class of optimal control problems with certain control constraints. Based on these formulas, we propose efficient algorithms that overcome the curse of dimensionality. As such, our proposed methods have the potential to serve as a building block for solving more complicated high-dimensional optimal control problems in real-time.

Keywords:Numerical and Symbolic Computations, Optimization : Theory and Algorithms, Stochastic Modeling and Stochastic Systems Theory Abstract: We develop a new algorithm named TTRISK to solve high-dimensional risk-averse optimization problems governed by differential equations (ODEs and/or PDEs) under uncertainty. As an example, we focus on the so-called Conditional Value at Risk (CVaR), but the approach is equally applicable to other coherent risk measures. Both the full and reduced space formulations are considered. The algorithm is based on low rank tensor approximations of random fields discretized using stochastic collocation. To avoid non-smoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and tensor approximation errors. Moreover, unbiased Monte Carlo CVaR estimate can be computed by using the smoothed CVaR as a control variate. To accelerate the computations, we introduce an efficient preconditioner for the KKT system in the full space formulation.The numerical experiments demonstrate that the proposed method enables accurate CVaR optimization constrained by large-scale discretized systems. In particular, the first example consists of an elliptic PDE with random coefficients as constraints. The second example is motivated by a realistic application to devise a lockdown plan for United Kingdom under COVID-19. The results indicate that the risk-averse framework is feasible with the tensor approximations under tens of random variables.

This is an extended abstract for a talk based on https://arxiv.org/abs/2111.05180

Keywords:Feedback Control Systems, Nonlinear Systems and Control Abstract: An optimized variant of the State Dependent Riccati Equations (SDREs) approach for nonlinear optimal feedback stabilization is presented. The proposed method is based on the construction of equivalent semilinear representations associated to the dynamics and their affine combination. The optimal combination is chosen to minimize the discrepancy between the SDRE control and the optimal feedback law stemming from the solution of the corresponding Hamilton Jacobi Bellman (HJB) equation. Numerical experiments assess effectiveness of the method in terms of stability of the closed-loop with near-to-optimal performance.