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Polytopic Matrix Factorization (PMF) is introduced as a flexible data decomposition tool with potential applications in unsupervised learning. PMF assumes a generative model where observations are lossless linear mixtures of some samples drawn from a particular polytope. Assuming that these samples are sufficiently scattered inside the polytope, a determinant maximization based criterion is used to obtain latent polytopic factors from the corresponding observations. This article aims to characterize all eligible polytopic sets that are suitable for the PMF framework. In particular, we show that any polytope whose set of vertices have only permutation and/or sign invariances qualifies for PMF framework. Such a rich set of possibilities enables elastic modeling of independent/dependent latent factors with combination of features such as relatively sparse/antisparse subvectors, mixture of signed/nonnegative components with optionally prescribed domains., KUIS AI Lab AI Fellowship

Design conditions for existence of linear state feed-back control for uncertain discrete-time stochastic systems with state-multiplicative noise, whose region of uncertainty is a polytope, are presented in the paper. Adapting the enhanced form of the bounded real lemma for this class of systems, an LMI based procedure is provided for computation of the gain matrix of linear state control law, giving conditions guaranteeing the resultant closed-loop systems to be asymptotically stable. The approach is illustrated by example demonstrating the validity of the proposed method.

This paper defines a security injection region (SIR) to guarantee reliable operation of water distribution systems (WDS) under extreme conditions. The model of WDSs is highly nonlinear and nonconvex. Understanding the accurate SIRs of WDSs involves the analysis of nonlinear constraints, which is computationally expensive. To reduce the computational burden, this paper first investigates the convexity of the SIR of WDSs under certain conditions. Then, an algorithm based on a monotone inner polytope sequence is proposed to effectively and accurately determine these SIRs. The proposed algorithm estimates a sequence of inner polytopes that converge to the whole convex region. Each polytope adds a new area to the SIR. The algorithm is validated on two different WDSs, and the conclusion is drawn. The computational study shows this method is applicable and fast for both systems.

A controller based on regional pole placement for the handling stability of an intelligent ground vehicle is proposed in this paper. The 2 degree-of-freedom model for the vehicle was applied to analysis the uncertainty generated by the tires and model of the vehicle. Another parameter that must be considered is the change in the vehicle’s speed. A polytope with finite vertices was applied for the uncertainty generated by the vehicle’s speed in the model. The linear matrix inequality (LMI) method was used to solve the gain of the controller’s matrix. Simulations were also used to verify the performance of the proposed controller. The simulation results prove that the controller can improve the vehicle’s handling stability against the uncertainty generated by the parameters.

A more stationary and discriminative embedding is necessary for robust classification of images. We focus our attention on the newel CapsNet model and we propose the angular margin loss function in composition with margin loss. We define a fixed classifier implemented with fixed weights vectors obtained by the vertex coordinates of a simplex polytope. The advantage of using simplex polytope is that we obtain the maximal symmetry for stationary features angularly centred. Each weight vector is to be considered as the centroid of a class in the dataset. The embedding of an image is obtained through the capsule network encoding phase, that is identified as digitcaps matrix. Based on the centroids from the simplex coordinates and the embedding from the model, we compute the angular distance between the image embedding and the centroid of the correspondent class of the image. We take this angular distance as angular margin loss. We keep the computation proposed for margin loss in the original architecture of CapsNet. We train the model to minimise the angular between the embedding and the centroid of the class and maximise the magnitude of the embedding for the predicted class. The experiments on different datasets demonstrate that the angular margin loss improves the capability of capsule networks with complex datasets.

When operating autonomous surface vessels in uncertain environments with dynamic obstacles, planning safe trajectories and evaluating collision risk is key to navigating safely. In order to perform these tasks, it is important to have a computationally efficient and adaptable obstacle representation to allow for quick and robust predictions of the obstacle trajectory. This paper presents a novel space-time obstacle representation, which is able to predict the reachable set for a dynamic obstacle under uncertainty. This is done by projecting the area occupied by the obstacle forward in time, using a set of velocities representing the possible maneuvers that the obstacle may take. Under some mild assumptions, we show how the space-time obstacle can be implemented in a computationally efficient way, using both convex polytopes and ellipsoids. Additionally, we show how the space-time obstacle representation can be used for risk assessment, collision avoidance and trajectory planning for autonomous surface vessels.

This paper addresses the problem of reachable set estimation for switched genetic regulatory networks with mixed delays and bounded disturbances. A delay-dependent sufficient condition is investigated to guarantee that the reachable set of genetic regulatory networks under consideration is contained within a Cartesian product of two polytopes or spheres. The sufficient condition involves only some simple inequalities, which can be easily verified by using the usual tool software. A numerical example is given to present that the proposed method is effective. Compared with the Lyapunov–Krasovskii functional method that has been used in many literature, the proposed method have two advantages: (i) No Lyapunov– Krasovskii functional is required; and (ii) Less computational complexity is involved.

This paper investigates an $H_{\infty}$ gain-scheduling controller designed method based on the direct yaw moment control (DY C) for the good perform of vehicle maneuverability and lateral stabilization. To make the designed controller more practical, both the network-induced delay and time-varying longitudinal velocity are considered. For reducing the conservativeness, an improved polytope structure with finite vertex is adopted to depict the velocity-dependent auxiliary parameters. Based on the designed DY C controller, the vehicle driving safety and maneuverability are enhanced, the $H_{\infty}$ performance of vehicle lateral dynamic system under different operations is satisfied. Several simulations are presented to verify the validity of the designed controller.

An approach to the analysis and design of continuous T-S fuzzy control system is proposed in this paper. The fuzzy logic controller has two consequents in each rule. They are numerator part and denominator part and are equivalent to any proper practical controller. It is shown that the overall closed-loop system behaves like an uncertain polytope of polynomials and the system stability can be checked by using some graphical robust stability criteria. The complex process on finding a common Lyapunov function to guarantee the system stability can be omitted. An illustrative example will be given to demonstrate the ability of the design procedure in the proposed approach.

In this paper, we presented a sufficient condition on the frequency domain for the absolutely stable analysis of the Takagi-Sugeno (T-S) fuzzy control system. For modeling the plant and controller we used transfer functions instead of the state description and showed that the overall T-S fuzzy plant is a polytope of transfer functions. Besides, by applying local proportional controllers, the bounds of proportional parameter of the overall fuzzy controller can be computed. Hence the robust absolutely stability criterions with graphical interpretation for closed-loop T-S fuzzy control system were derived.

We present a general approach for constructing proofs of motion planning infeasibility. Effective high-dimensional motion planners, such as sampling-based methods, are limited to probabilistic completeness, so when no plan exists, these planners either do not terminate or can only run until a timeout. We address this completeness challenge by augmenting a sampling-based planner with a method to create an infeasibility proof in conjunction with building the search tree. An infeasibility proof is a closed polytope that separates the start and goal into disconnected components of the free configuration space. We identify possible facets of the polytope via a nonlinear optimization procedure using sampled points in the non-free configuration space. We identify the set of facets forming the separating polytope via a linear constraint satisfaction problem. This proof construction is valid for general (i.e., non-Cartesian) configuration spaces. We demonstrate this approach on the low-dimensional Jaco manipulator and discuss engineering approaches to scale to higher dimensional spaces.

Legged robots have great potential to perform complex loco-manipulation tasks, yet it is challenging to keep the robot balanced while it interacts with the environment. In this paper we investigated the use of additional contact points for maximising the robustness of loco-manipulation motions. Specifically, body-ground contact was studied for its ability to enhance robustness and manipulation capabilities of quadrupedal robots. We proposed equipping the robot with prongs: small legs rigidly attached to the body which create body-ground contact at controllable point-contacts. The effect of these prongs on robustness was quantified by computing the Smallest Unrejectable Force (SUF), a measure of robustness related to Feasible Wrench Polytopes. We applied the SUF to evaluate the robustness of the system, and proposed an effective approximation of the SUF that can be computed at near-real-time speed. We developed a hierarchical quadratic programming based whole-body controller that can control stable interaction when the prongs are in contact with the ground. This novel prong concept and complementary control framework were implemented on hardware to validate their effectiveness by showing increased robustness and newly enabled loco-manipulation tasks, such as obstacle clearance and manipulation of a large object.

This paper presents and classifies different methodologies for aggregating the load shifting potentials of several plug-in electric vehicles (EVs) for consideration in energy system models. These approaches range from modeling single charging events to approximation of EV fleets. We conclude that most of the applied approaches show shortages and that their suitability depends on the number of time steps of the energy system model and the EV fleet size. For distributed systems and limited time span, modeling single EVs is suitable and can provide exact results when used in optimization models. For large EV fleets and centralized optimization approaches, approximation methods seem to be appropriate. Aggregating the flexibility of multiple EVs to polytopes exactly describes the solution space for the optimization but is an NP-hard task. Consequently, approximation methods to this so-called Minkowski sum are presented, which may improve the consideration of EVs’ flexibility in energy system models.

The term of Ehrhart tensor polynomials is a natural popularization of the Ehrhart polynomial of lattice polytopes which is a much more used in different applications. As our fundamental instrument, we present $\mathrm{h}^{\mathrm{r}}$-tensor polynomials which expand the idea of the Ehrhart $\mathrm{h}*$-polynomial for a polytope, and networks. It also examines their coefficients for positive semi definiteness rather than the standard $\mathrm{h}*-$ polynomial. This paper gives the proof of two theorems that compute the coefficients of the Ehrhart Tensor polynomials (h3 and $\mathrm{h}^{4}$). The positivity for these coefficients is also proved. Some applications on the Ehrhart tensor polynomials are given.

We introduce R3T, a reachability-based variant of the rapidly-exploring random tree (RRT) algorithm that is suitable for (optimal) kinodynamic planning in nonlinear and hybrid systems. We developed tools to approximate reachable sets using polytopes and perform sampling-based planning with them. This method has a unique advantage in hybrid systems: different dynamic modes in the reachable set can be explicitly represented using multiple polytopes. We prove that under mild assumptions, R3T is probabilistically complete in kinodynamic systems, and asymptotically optimal through rewiring. Moreover, R3T provides a formal verification method for reachability analysis of nonlinear systems. The advantages of R3T are demonstrated with case studies on nonlinear, hybrid, and contact-rich robotic systems.

In this paper a new algorithm for multi-class classification is presented. The algorithm, that is named Vertex Feature Classification (VFC), maps input data sets into an ad-hoc built space, called "simplex space" in order to perform geometric classification. More precisely, each class is first associated to a specific vertex of the polytope computed in the feature space. Successively, pattern classification is performed according to the geometric arrangement of patterns in a higher dimensional feature space. The experimental results, carried out on datasets of the UCI Machine Learning Repository, demonstrate the accuracy of the new algorithm is comparable with KNN, VDA and SVM, without or with a little training phase. An important aspect of this algorithm is its training time, which takes often a few milliseconds. Furthermore, the algorithm is robust and computationally efficient.

We propose a model-based approach to design feedback policies for dexterous robotic manipulation. The manipulation problem is formulated as reaching the target region from an initial state for some non-smooth nonlinear system. First, we use trajectory optimization to find a feasible trajectory. Next, we characterize the local multi-contact dynamics around the trajectory as a piecewise affine system, and build a funnel around the linearization of the nominal trajectory using polytopes. We prove that the feedback controller at the vicinity of the linearization is guaranteed to drive the nonlinear system to the target region. During online execution, we solve linear programs to track the system trajectory. We validate the algorithm on hardware, showing that even under large external disturbances, the controller is able to accomplish the task., Comment: ICRA 2020 (Final Version)

The paper is dedicated to deriving a mathematical model of a water level control system for a boiler unit with interval parameters. It also describes a synthesis method of a robust controller providing a desired degree of an aperiodic stability for a system with interval parameters. The method is based on a robust extension of a D-partition method. The method proposed is used to synthesize a P-controller of the control system considered; step responses of the system synthesized are plotted to prove stability of its extreme operating modes. In order to analyze robust control quality of the system synthesized, a multiparametric interval root locus is plotted.

The polytope containment problem is deciding whether a polytope is a contained within another polytope. This problem is rooted in computational convexity, and arises in applications such as verification and control of dynamical systems. The complexity heavily depends on how the polytopes are represented. Describing polytopes by their hyperplanes (H-polytopes) is a popular representation. In many applications we use affine transformations of H-polytopes, which we refer to as AH-polytopes. Zonotopes, orthogonal projections of H-polytopes, and convex hulls/Minkowski sums of multiple H-polytopes can be efficiently represented as AH-polytopes. While there exists efficient necessary and sufficient conditions for AH-polytope in H-polytope containment, the case of AH-polytope in AH-polytope is known to be NP-complete. In this paper, we provide a sufficient condition for this problem that is cast as a linear program with size that grows linearly with the number of hyperplanes of each polytope. Special cases on zonotopes, Minkowski sums, convex hulls, and disjunctions of H-polytopes are studied. These efficient encodings enable us to designate certain components of polytopes as decision variables, and incorporate them into a convex optimization problem. We present examples on the zonotope containment problem, polytopic Hausdorff distances, zonotope order reduction, inner approximations of orthogonal projections, and demonstrate the usefulness of our results on formal controller verification and synthesis for hybrid systems., Comment: Shorter version submitted to CDC 2019

We introduce a moment-based framework to design robust energy-maximising optimal controllers for Wave Energy Converters (WECs). The technique explicitly allows model uncertainty in the computation of the optimal control input, by defining a suitable uncertainty polytope. The resulting robust optimisation is formulated as a minimax problem which has to be solved only at a small number of points of this uncertainty set. The objective function under the proposed strategy is shown to be of quadratic-type and the optimal solution is proven to be unique, providing a computationally efficient robust optimal control framework for WECs. The performance of the proposed controller is demonstrated by means of an application case, which considers a heaving point absorber WEC with imprecisely known model parameters.

This paper studies the problem of "isotropically rounding" a polytope K ⊆ R^n, that is, computing a linear transformation which makes the uniform distribution on the polytope have roughly identity covariance matrix. It is assumed that K ⊆ R^n is defined by m linear inequalities. We introduce a new variant of the ball walk Markov chain and show that, roughly, the expected number of arithmetic operations per-step of this Markov chain is O(m) that is sub-linear in the input size mn -- the per-step time of all prior Markov chains. Subsequently, we apply this new variant of the ball walk to obtain a rounding algorithm that gives a factor of √n improvement on the number of arithmetic operations over the previous bound which uses the hit-and-run algorithm. Since the cost of the rounding pre-processing step is in many cases the bottleneck in improving sampling or volume computation running time bounds, our results imply improved bounds for these tasks. Our algorithm achieves this improvement by a novel method of computing polytope membership, where one avoids checking inequalities which are estimated to have a very low probability of being violated. We believe that this method is likely to be of independent interest for constrained sampling and optimization problems.

The paper is dedicated to a problem of parametric synthesis of a controller for a control system having interval parameters and affine uncertainty of characteristic polynomial coefficients. Such polynomial can be represented as a sum of productions of interval parameters and polynomials including zero and real poles. The paper proposes a method of synthesizing a linear robust controller placing system poles allocation areas according to desired control quality. The method is based on dividing parameters of a controller in two groups: restricted ones and unrestricted ones. Restricted parameters of a controller provide desired allocation of a dominant pole; unrestricted parameters determine allocation of all other poles of a system. In this case, the right border of an allocation area of a dominant pole defines minimal stability degree of a system synthesized. Values of controller parameters are proposed to be found via D-partition in vertices of a interval parameters’ polytope. An example of the proposed method application is provided.

While recent studies have shed light on the expressivity, complexity and compositionality of convolutional networks, the real inductive bias of the family of functions reachable by gradient descent on natural data is still unknown. By exploiting symmetries in the preactivation space of convolutional layers, we present preliminary empirical evidence of regularities in the preimage of trained rectifier networks, in terms of arrangements of polytopes, and relate it to the nonlinear transformations applied by the network to its input.

The aim of this paper is development of a distributed robust controller for distributed generators (DGs) interfaced to a DC microgrid through a boost converter. The suggested controller is robust static state feedback local controller based on $\mathbf{H}_{\mathbf{2}}/\mathbf{H}_{\infty}/\mathbf{pole}$ -placement/polytope technique. The proposed technique considers the system robustness against uncertainty, the system speed, and the system damping simultaneously, through construction a set of linear matrix inequalities (LMI) and polytopic representation of the system. The control problem is solved using LMI optimization approach, Using MATLAB LMI toolbox, because of LMI ability to handle multi-objective design problems. To test the effectiveness of the proposed technique, different simulation cases are carried out under diverse disturbances such as generation outage and variation of constant power loads (CPL). Eventually, the results of the suggested technique are compared with three other tuning techniques known as $\mathbf{H2}/\mathbf{H}\infty, \mathbf{H}_{\mathbf{2}}/\mathbf{H}_{\infty}/\mathbf{pole}$ -placement, and H 2 techniques. The comparisons prove the superiority of the proposed technique.

In this paper, we address target positioning in scenarios where the reference nodes, denoted as anchors, are not distributed at the perimeter of the area where the target is, but are concentrated in a very small region and target is outside this region. This scenario may be meaningful in smart building applications, where anchor nodes cannot be distributed and cabled in the monitored area. On the other hand, anchors may be installed on a single hotspot to be placed at the center of the environment of interest. In this case, the target has to be localized outside the polytope identified by the anchors. To this end, we investigate Ultra WideBand (UWB)-based target positioning with Time Difference of Arrival (TDoA) processing at the anchors. A comparative analysis between geometric and Particle Swarm Optimization (PSO) algorithms is carried out. Our results show accurate angle of arrival estimation accuracy. Moreover, while PSO guarantees a better performance, in terms of average position estimation error, the “dispersion” of position estimation (i.e., the standard deviation of the position error) is higher than in the case of geometric algorithms.

This paper addresses the problem of determining the peak of the impulse response of polytopic linear time-varying (LTV) systems. In particular, linear systems affected linearly by time-varying structured uncertainty constrained into a polytope are considered. A novel condition for establishing upper bounds of the sought peak is proposed in terms of feasibility of a system of linear matrix inequalities (LMIs). The proposed condition is sufficient for any degree of a polynomial introduced to describe the impulse response, moreover, its conservatism can be decreased by increasing this degree. Some examples illustrate the use of the proposed condition and show the application to a DC motor with time-varying uncertain parameters.

The paper is dedicated to the problem of placing roots of interval system characteristic polynomial the coefficients of which are included in a parametric polytope, according to desired control quality. To provide applicable values of stability degree and oscillability degree, a pair of dominant poles is placed in desired areas of a complex plane. Other poles of the system, which are unrestricted poles, are placed in a desired allocation area, the border of which is defined to observe the poles dominance principle. In order to place poles according to desired control quality, a set of parametric polytope's vertices, which determine poles allocation, is found on the basis of angular properties of interval root locus. The parameters of a controller providing desired allocation of dominant and unrestricted poles are found with the help of D-partition basing on the previously obtained vertex polynomials.

For continued reliability and stability of the modern power system, the new controls that are designed for it, must be functionally flexible and computationally advanced. In accordance with this realization, this paper presents the design of a coordinated wide-area damping controller (CWADC) that employs a partial state feedback control mechanism for damping inter-area oscillations. For developing the control, an enhanced version of selective modal analysis (SMA) is proposed, which is then combined with a linear matrix inequality (LMI)-based polytope. To ensure robustness of the control, the option of selecting PMU inputs from alternate locations is also explored. A 16 machine, 68 bus test system is used to illustrate the proposed technique, while a 29 machine, 127 bus equivalent model of the Western Electricity Coordinating Council (WECC) system is used to demonstrate its applicability to larger systems.

This paper considers finding an optimal path for a robot from a given initial position to a final position. The dynamics of the robot is assumed to be a discrete LTI system with Gaussian process noise (to account for the uncertainties). The obstacles in the robot's configuration space are taken to be convex polytopes. The inputs that can be applied to the robot at each instant belongs to a known finite set. As the dynamics of the robot includes a Gaussian process noise, its motion is a Gaussian process with a mean dependent on the input applied at each instant. The probability of collision with the obstacles corresponding to each input is then computed for its feasibility. An algorithm resembling the dynamic programming is presented for growing a tree of state distributions. It is also shown that this tree contains the optimal path. Numerical simulations and hardware experiments (with a Kobuki robot) are performed for validating the algorithm.

Formation control problem has been studied in this paper, where the additive communication noises are considered in the position and velocity vectors. First, the proposed problem is transformed into the convergence problem of infinite products of general stochastic matrices. Based on the convex analysis theory, several convex polytopes have been constructed in order to analyze this problem. Then, a sufficient condition is derived to guarantee that the position with communication noises of agents can approach the desired position as well as the velocity with communication noises of agents. Finally, an example is presented to show the effectiveness of the result.

We consider the domain of non-empty convex and compact subsets of a finite dimensional Euclidean space to represent partial or imprecise points in computational geometry. The convex hull map on such imprecise points is given domain-theoretically by an inner and an outer convex hull. We provide a practical algorithm to compute the inner convex hull when there are a finite number of convex polytopes as partial points. A notion of pre-inner support function is introduced, whose convex hull gives the support function of the inner convex hull in a general setting. We then show that the convex hull map is Scott continuous and can be extended to finitely generable subsets, represented by the Plotkin power domain of the underlying domain. This in particular allows us to compute, for the first time, the convex hull of attractors of iterated function systems in fractal geometry. Finally, we derive a program logic for the convex hull map in the sense of the weakest pre-condition for a given post-condition and show that the convex hull predicate transformer is computable.

We show how to exploit the piecewise affine structure of the optimal control law in networked linear-quadratic MPC. The complete explicit solution is not required in this approach, but the current piece, i.e., the feedback law and its polytope of validity, is computed on demand instead. Assuming such a piece has been determined for the current state, a lean local computational node suffices to check whether future states are still in the current polytope and to evaluate the optimal affine feedback law if that is the case. Whenever the system leaves the current polytope, the local node requests a central, powerful node to determine a new control law and polytope by solving a MPC problem, and to send the resulting law and polytope over the network. In this contribution we extend the approach by not sending the information for a single control law, but for multiple control laws for the predicted system behavior. We show that the number of requests and the amount of transmitted data can be reduced by using active set updates on the local node.

A robust gain-scheduled H ∞ controller is proposed to improve vehicle lateral stability and handling performance of in-wheel-motor-driven electric vehicles by managing the external yaw moment. To address uncertainties in vehicle lateral dynamics, uncertain factors such as tire cornering stiffness and vehicle mass are represented via the norm-bounded uncertainty, and then linear parameter-varying vehicle model is built, which depends affinely on the time-varying longitudinal velocity described via a polytope with finite vertices. Meanwhile, a hyper-trapezoidal polytope is applied to reduce conservative. In addition, the quadratic D-stability is also considered to enhance the transient response of the closed-loop system. In term of a set of linear matrix inequalities, the gain-scheduled H ∞ controller is finally completed. Simulations with a high-fidelity, CarSim®, full-vehicle model verify the effectiveness of the proposed controller.

Collision detection has attracted attention of researchers for decades in the field of computer graphics, robot motion planning, computer aided design, etc. A large number of successful algorithms have been proposed and applied, which make use of convex polytopes and bounding volumes as primitives. However, algorithms for those shapes rely significantly on the complexity of the meshes. This paper deals with collision detection for shapes with simple and exact mathematical descriptions, such as ellipsoids and superquadrics. These primitives have a wide range of applications in representing complex objects and have much fewer parameters than meshes. The foundation of the proposed collision detection scheme relies on the closed-form Minkowski sums between ellipsoids and superquadrics in n-dimensional Euclidean space. The basic idea here is to shrink the ellipsoid into a point and expand each superquadric into a new offset surface with closed-form parametric expression. The solutions for detecting relative positions between a point and a general convex differentiable parametric surface in both 2D and 3D are derived, leading to an algorithm for exact collision detection. To compare between exact and inexact algorithms, an accuracy metric is introduced based on the Principal Kinematic Formula (PKF). The proposed algorithm is then compared with existing wellknown algorithms: Gilbert-Johnson-Keerthi (GJK) and Algebraic Separation Conditions (ASC). The results show that the proposed algorithm performs competitively with these efficient checkers.

Three mathematical models of a placement problem in VLSI design is offered, where modules are squares, octagons, and rhombuses, and 450-rotation is permitted. For that, the generalized placement problem of signed permutation polytopes within a signed permutation polytope is constructed based on a concept of a circular-like object. The way to formalize 450-rotation is presented, which applies the theory of continuous functional representations of discrete sets.

This paper analyzes the transient stability of power systems with uncertain parameters. More precisely, we suppose that the inertia constants and damping coefficients of generators are uncertain, and consider the problem of checking the stability of an equilibrium state for all possible values of the uncertain parameters, and estimating the intersection of the region of attraction for those parameter values. By using a polytope representation for the uncertain parameters, we present a method to solve the problem. In the presented method, we solve a sum of squares (SOS) programming problem based on the polytope representation. It is proven that if this SOS programming problem is feasible, then our method provides a solution to the analysis problem of the transient stability.

Model predictive control (MPC) under chance constraints has been a promising solution to complicated control problems subject to uncertain disturbance. However, traditional approaches either require exact knowledge of probabilistic distributions, or rely on massive multi-scenarios that are generated to represent uncertainties. In this paper, a novel approach is proposed based on actively learning a compact high-density region from available data in form of a polytope. This is achieved by adopting the support vector clustering, which has been recently utilized in data-driven robust optimization. A new strategy is developed to calibrate the size of the polytope, which provides appropriate probabilistic guarantee. Finally the optimal control problem is cast as a robust optimization problem, which can be efficiently handled by existing numerical solvers. The proposed method commonly requires less data samples than traditional approaches, and can help reducing the conservatism, thereby enhancing the practicability of model predictive control. The efficacy of the proposed method is verified based on a simulated example.

In this paper, we present a new low-complexity algorithm for computing the projection onto the parity polytope in the context of ADMM-based LP decoding. This projection is the heart of the ADMM algorithm, that usually involves a sorting operation, which is the main effort of the projection. Our proposed algorithm relies on new findings in the recursive structure of the parity polytope and works by fixing selected components in an iterative manner. The absence of the sorting operation makes it preferable for efficient hard- and software implementations. In addition, we can show that this new projection algorithm requires up to 37% less arithmetic operations compared to state-of-the-art projections.

We investigate a correct-by-construction synthesis of piecewise affine feedback controllers designed to satisfy the strict safety specifications set forth by the adaptive cruise control (ACC) problem. Our design methodology is based on the formulation of the ACC problem as a reach control problem on a polytope in a 2D state space. The boundaries of this polytope, expressed as linear constraints on the states, arise from the headway and velocity safety requirements imposed by the ACC problem statement. We propose a model for the ACC problem, develop a controller that satisfies the ACC requirements, and produce simulations for the closed-loop system.