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We present a probabilistic formulation of risk aware optimal control problems for stochastic differential equations. Risk awareness is in our framework captured by objective functions in which the risk neutral expectation is replaced by a risk function, a nonlinear functional of random variables that accounts for the controller's risk preferences. We state and prove a risk aware minimum principle that gives necessary and sufficient conditions for optimality of generalized control processes taking values on probability measures defined on a given action space. We show that going from the risk neutral to the risk aware case, the minimum principle is modified by the introduction of one additional real-valued stochastic process that acts as a risk adjustment factor. This adjustment process is explicitly given as the expectation, conditional on the filtration at the given time, of an appropriately defined functional derivative of the risk function evaluated at the random total cost. The control model we employ differs from standard relaxed controls, and we elaborate on the differences, and benefits and drawbacks, of the control types; we further give conditions under which the generalized control can be realized using a strict control process. We present an application of the results for a portfolio allocation problem and show that the risk awareness of the objective function gives rise to a risk premium term that is characterized by the risk adjustment process described above. This suggests uses of our results in e.g. pricing of risk modeled by generic risk functions in financial applications. [ABSTRACT FROM AUTHOR]

This article addresses the approximate and exact controllability issues of switched infinite-dimensional linear systems. First, a controllability map that is related to switching sequence and a controllability Gramian operator are proposed, based on which two necessary and sufficient conditions are established for approximate and exact controllability, respectively. Second, it shows that the obtained conditions can degenerate into the existing controllability criteria of infinite-dimensional linear systems, switched finite-dimensional linear systems, and a single finite-dimensional linear system. Finally, the new results are applied to analyze the approximate controllability of switched linear parabolic systems and the exact controllability of switched linear wave systems. Two examples are given to illustrate the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]

This article studies the feedback stabilization problem of the motion of a tank that contains an incompressible, Newtonian, viscous liquid. The control input is the force applied on the tank and the overall system consists of two nonlinear partial differential equations and two ordinary differential equations. Moreover, a spill-free condition is required to hold. By applying the control Lyapunov functional methodology, a set of initial conditions (state space) is determined for which spill-free motion of the liquid is possible by applying an appropriate control input. Semi-global stabilization of the liquid and the tank by means of a simple feedback law is achieved, in the sense that for every closed subset of the state space, it is possible to find appropriate controller gains, so that every solution of the closed-loop system initiated from the given closed subset satisfies specific stability estimates. The closed-loop system exhibits an exponential convergence rate to the desired equilibrium point. The proposed stabilizing feedback law does not require measurement of the liquid level and velocity profiles inside the tank and simply requires measurements of 1) the tank position error and tank velocity, 2) the total momentum of the liquid, and 3) the liquid levels at the tank walls. The obtained results allow an algorithmic solution of the problem of the spill-free movement and slosh-free settlement of a liquid in a vessel of limited height (such as water in a glass) by a robot to a prespecified position, no matter how full the vessel is. [ABSTRACT FROM AUTHOR]

Accurately following a geometric desired path in a two-dimensional (2-D) space is a fundamental task for many engineering systems, in particular mobile robots. When the desired path is occluded by obstacles, it is necessary and crucial to temporarily deviate from the path for obstacle/collision avoidance. In this article, we develop a composite guiding vector field via the use of smooth bump functions and provide theoretical guarantees that the integral curves of the vector field can follow an arbitrary sufficiently smooth desired path and avoid collision with obstacles of arbitrary shapes. These two behaviors are reactive since path (re)planning and global map construction are not involved. To deal with the common deadlock problem, we introduce a switching vector field, and the Zeno behavior is excluded. Simulations are conducted to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we revisit the concepts and tools of harmonic analysis and control and provide a rigorous mathematical answer to the following question: When does an harmonic control have a representative in the time domain? By representative we mean a control in the time domain that leads by sliding Fourier decomposition to exactly the same harmonic control. Harmonic controls that do not have such representatives lead to erroneous results in practice. The main results of this article are: A one-to-one correspondence between ad hoc functional spaces guaranteeing the existence of a representative, a strict equivalence between the Caratheorody solutions of a differential system and the solutions of the associated harmonic differential model, and as a consequence, a general harmonic framework for linear time periodic systems and bilinear affine systems. The proposed framework allows to design globally stabilizing harmonic control laws. We illustrate the proposed approach on a single-phase rectifier bridge. Through this example, we show how one can design stabilizing control laws that guarantee periodic disturbance rejection and low harmonic content. [ABSTRACT FROM AUTHOR]

This article considers a discrete-time robust optimal control problem on matrix Lie groups. The underlying system is assumed to be perturbed by exogenous unmeasured bounded disturbances, and the control problem is posed as a min–max optimal control wherein the disturbance is the adversary and tries to maximize a cost that the control tries to minimize. Assuming the existence of a saddle point in the problem, we present a version of the Pontryagin maximum principle (PMP) that encapsulates first-order necessary conditions that the optimal control and disturbance trajectories must satisfy. This PMP features a saddle point condition on the Hamiltonian and a set of backward difference equations for the adjoint dynamics. We also present a special case of our result on Euclidean spaces. We conclude with applying the PMP to robust version of single axis rotation of a rigid body. [ABSTRACT FROM AUTHOR]

The Lyapunov equation for polytopic linear inclusions over the space of Minkowski functions of nonempty compact and convex sets that contain the origin as an interior point is studied. In particular, necessary and sufficient conditions for the characterization, existence, and uniqueness of its fundamental solution are derived. [ABSTRACT FROM AUTHOR]

The article introduces and solves a structural controllability problem for continuum ensembles of linear time-invariant systems. All the individual linear systems of an ensemble are sparse, governed by the same sparsity pattern. Controllability of an ensemble system is, by convention, the capability of using a common control input to simultaneously steer every individual systems in it. A sparsity pattern is structurally controllable if it admits a controllable linear ensemble system. A main contribution of the article is to provide a graphical condition that is necessary and sufficient for a sparsity pattern to be structurally controllable. Like other structural problems, the property of being structural controllable is monotone. We provide a complete characterization of minimal sparsity patterns as well. [ABSTRACT FROM AUTHOR]

We introduce a new hybrid control strategy, which is conceptually different from the commonly used synergistic hybrid approaches, to efficiently deal with the problem of the undesired equilibria that precludes smooth vectors fields on $SO(3)$ from achieving global stability. The key idea consists in constructing a suitable potential function on $SO(3)\times \mathbb {R}$ involving an auxiliary scalar variable, with flow and jump dynamics, which keeps the state away from the undesired critical points while, at the same time, guarantees a decrease of the potential function over the flows and jumps. Based on this new hybrid mechanism, a hybrid feedback control scheme for the attitude tracking problem on $SO(3)$ , endowed with global asymptotic stability and semiglobal exponential stability guarantees, is proposed. This control scheme is further improved through a smoothing mechanism that removes the discontinuities in the input torque. The third hybrid control scheme, proposed in this article, removes the requirement of the angular velocity measurements, while preserving the strong stability guarantees of the first hybrid control scheme. This approach has also been applied to the tracking problem on $SE(3)$ to illustrate its advantages with respect to the existing synergistic hybrid approaches. Finally, some simulation results are presented to illustrate the performance of the proposed hybrid controllers. [ABSTRACT FROM AUTHOR]

This article focuses on model predictive control (MPC) design in the context of sampled-data control systems with full-state measurements. It is shown that recent results on this area can be successfully generalized to cope with sampled-data MPC. The open-loop plant is subjected to polytopic parameter uncertainty and at sampling times, a controlled output variable satisfies a set of convex constraints. A guaranteed ${\mathcal H}_2$ performance index with infinity horizon is minimized such that the feedback control preserves asymptotic stability and feasibility. The design conditions are expressed through differential linear matrix inequalities. Continuous-time systems are treated with no kind of discrete-time modeling approximation. Comparisons with classical methods from the literature dealing with continuous-time systems are presented and discussed. Examples are included for illustration. [ABSTRACT FROM AUTHOR]

This article studies the multiagent control problem for affine, rigid, and translational formation, with the aim of developing a unified distributed control strategy for global and finite-time convergence. Global stabilization of rigid formation in arbitrary dimensional spaces still remains open and challenging. This article provides a general solution to this open problem based on the sliding mode control idea. The control law design consists of two parts: the main control force regulates the trajectories of all agents to reach a sliding surface defined by an affine formation space in finite time and remain in it thereafter; the extra control force governs certain chosen leader agents towards the desired formation in the sliding surface. The article then presents in detail two approaches for designing the extra control force, one based on distance constraints and the other based on relative position constraints. For the first time, the proposed sliding mode formation control laws solve the open problem of (almost) global and finite-time stabilization of affine, rigid, and translational formations in any dimensional space. Simulation results are provided to illustrate the effectiveness of the proposed control schemes. [ABSTRACT FROM AUTHOR]

This article addresses sampled-data control of 2-D Kuramoto–Sivashinsky equation over a rectangular domain $\Omega$. We suggest to divide the 2-D rectangular $\Omega$ into $N$ subdomains, where sensors provide spatially averaged or point state measurements to be transmitted through communication network to the controller. Note that, differently from 2-D heat equation, here, we manage with sampled-data control under point measurements. We design a regionally stabilizing sampled-data controller applied through distributed in space characteristic functions. Sufficient conditions ensuring regional stability of the closed-loop system are established in terms of linear matrix inequalities (LMIs). By solving these LMIs, we find an estimate on the set of initial conditions starting from which the state trajectories of the system are exponentially converging to zero. A numerical example demonstrates the efficiency of the results. [ABSTRACT FROM AUTHOR]

In this article, we address Bayesian persuasion between a sender and a receiver with state-dependent quadratic cost measures for general classes of distributions. The receiver seeks to make mean-square-error estimate of a state based on a signal sent by the sender while the sender signals strategically in order to control the receiver’s estimate in a certain way. Such a scheme could model, e.g., deception and privacy, problems in multiagent systems. Existing solution concepts are not viable since here the receiver has continuous action space. We show that for finite state spaces, optimal signaling strategies can be computed through an equivalent linear optimization problem over the cone of completely positive matrices. We then establish its strong duality to a copositive program. To exemplify the effectiveness of this equivalence result, we adopt sequential polyhedral approximation of completely positive cones and analyze its performance numerically. We also quantify the approximation error for a quantized version of a continuous distribution and show that a semi-definite program relaxation of the equivalent problem could be a benchmark lower bound for the sender’s cost for large state spaces. [ABSTRACT FROM AUTHOR]

A key challenge in the synthesis and subsequent analysis of supervisory controllers is the impact of state-space explosion caused by concurrency. The main bottleneck is often the memory needed to store the composition of plant and requirement automata and the resulting supervisor. Partial-order reduction (POR) is a well-established technique that alleviates this issue in the field of model checking. It does so by exploiting redundancy in the model with respect to the properties of interest. For controller synthesis, the functional properties of interest are nonblockingness, controllability, and least-restrictiveness, but also performance properties, such as throughput and latency are of interest. We propose an on-the-fly POR on the input model that preserves both functional and performance properties in the synthesized supervisory controller. This improves the scalability of the synthesis (and any subsequent performance analysis). Synthesis experiments show the effectiveness of the POR on a set of realistic manufacturing system models. [ABSTRACT FROM AUTHOR]

This article presents a small-gain theorem for networks composed of a countably infinite number of finite-dimensional subsystems. Assuming that each subsystem is exponentially input-to-state stable, we show that if the gain operator, collecting all the information about the internal Lyapunov gains, has a spectral radius less than one, the overall infinite network is exponentially input-to-state stable. The effectiveness of our result is illustrated through several examples including nonlinear spatially invariant systems with sector nonlinearities and a road traffic network. [ABSTRACT FROM AUTHOR]

This article investigates the enforcement of generalized mutual exclusion constraints (GMECs) and deadlock-freeness on a time Petri net (TPN) system with uncontrollable transitions, motivated by the fact that the existing methods enforcing GMECs may degrade the performance of a closed-loop system and lead to deadlock states. A supervisor enforcing a set of GMECs and deadlock-freeness on an underlying untimed Petri net system is assumed to be available. By exploiting timing information and mathematical programming, a control function is designed to restrict the firing intervals of transitions such that a TPN system can avoid entering forbidden states. The key idea behind the proposed approach is the online computation of a graph, called reduced modified state class graph (RMSCG), that is an extension of the partial modified state class graph recently introduced by the authors. Based on the RMSCG, an online control synthesis procedure is developed, which can enforce the originally given GMECs and deadlock-freeness in a maximally permissive way. [ABSTRACT FROM AUTHOR]

We develop a discrete-time optimal control framework for systems evolving on Lie groups. Our article generalizes the original differential dynamic programming method, by employing a coordinate-free, Lie-theoretic approach for its derivation. A key element lies, specifically, in the use of quadratic expansion schemes for cost functions and dynamics defined on Lie groups. The obtained algorithm iteratively optimizes local approximations of the control problem, until reaching a (sub)optimal solution. On the theoretical side, we also study the conditions under which convergence is attained. Details about the behavior and implementation of our method are provided through a simulated example on $TSO(3)$. [ABSTRACT FROM AUTHOR]

We focus on the proportional-integral-derivative (PID) controller design for linear time-delay systems. All the controller gains ($k_P$ , $k_I$ , and $k_D$) and the delay ($\tau$) are treated as free parameters and no particular constraints are imposed on the controlled plants. Such a problem (involving totally four free parameters) is of theoretical as well as practical importance, but, to the best of the authors’ knowledge, it has not been fully explored. First, we will develop an algebraic algorithm to solve the complete stability problem w.r.t. $\tau$. Consequently, for any given PID controller vector $(k_P, k_I, k_D)$ , the distribution of ${NU}(\tau)$ (${NU}(\tau)$ denotes the number of characteristic roots in the right-half plane, as a function of $\tau$) can be accurately obtained and the exhaustive stability range of $\tau$ may be automatically calculated. Next, a global understanding of the distribution of ${NU}(\tau)$ over the whole $(k_P, k_I, k_D)$ -space may be achieved and all structural changes regarding the ${NU}(\tau)$ distribution can be analytically determined. To achieve such a goal, a complete positive real root classification (for some appropriate auxiliary characteristic equation) will be explicitly proposed. Finally, we will give a new methodology, a new parameter-space approach, for determining the stability set in the $(k_P, k_I, k_D, \tau)$ -space. [ABSTRACT FROM AUTHOR]

The problem of maximally permissive deadlock avoidance for sequential resource allocation systems (RAS) is a well-defined problem in the current controls literature. The corresponding supervisor is known as the maximally permissive deadlock avoidance policy (DAP), and it can be perceived as a classifier effecting the dichotomy of the underlying state space into its “safe” and “unsafe” subspaces. In the deployment of the maximally permissive DAP, an important issue is the selection of an effective and computationally efficient representation of the aforementioned dichotomy. A popular such representation is the “linear classifier,” where the admissibility of any given RAS state is resolved based on its ability to satisfy a given set of linear inequalities. However, linear classifiers cannot provide effective representation of the maximally permissive DAP for all RAS instantiations. Hence, this article provides a methodology for synthesizing linear DAPs for any given RAS instance that might not be maximally permissive in the original sense of this term, but observe a more relaxed notion of “maximality” that is defined within the particular space of the DAPs that admit a linear representation. The presented developments formally define this new DAP class, and provide the necessary algorithms for the synthesis or the systematic approximation of maximal linear DAPs for any given RAS instance. [ABSTRACT FROM AUTHOR]