Feedback equivalence of constant linear systems

In this paper, we present a sampling-based verification algorithm for embedded robotic systems with continuous dynamics and uncertainty due to adversaries, unmodeled disturbance inputs, unknown parameters, or initial conditions. The algorithm attempts to find inputs (and resulting trajectories) that falsify the specifications of the system thus providing examples of bad inputs to the system. The system is said to be verified if the algorithm cannot find falsifying inputs. The main contribution of the paper is the analysis of the effects of discretization of the state and input spaces that are inherent to sampling-based techniques. We derive conditions that guarantee resolution completeness. These provide sufficient, although conservative, conditions for verifying Lipschitz continuous (but possibly non-smooth) dynamic systems without known analytical solutions. We analyze the effects of transformations of the input and state space on these conditions. The main results of this paper are illustrated with several simple examples.

In electrical grids all generators work and provide the power to the loads. When the load on the generator increases, the speed of generator rotor decreases resulting in a reduction in the frequency of the grid. During restoration, it is needed to keep frequency response in the allowable range. For this purpose, this work is dedicated to present a method to keep frequency response of a hydro-power system at a required value. In this paper, firstly, the state space model for Hydro-power system unit was presented, and the controllability and observability of the system was discussed. Next, full state feedback controller and tracking gain were designed to optimize the performance of the system and decrease the effect of the frequency load. Then, an optimal closed-loop controller using algebraic Riccati equation was developed. We studied the frequency load effect before and after these designs and trying to decrease this effect. In simulation results, state space feedback controller decreases the average over shoot from 10.73% to 6.5% and the average error from 47.8 to zero. For comparison between the two controllers, LQR optimal controller decreases the settling time to 8 sec instead of 11.6 sec that is obtained from state space feedback controller. In general, the designed LQR optimal controller provides best performance in terms of overshoot, error, and time.

<p>This paper demonstrates the pontential of verification based on state spaces reduced by equivalence relations. The basic observation is that quite often, some states of a system are similar, i.e., they induce similar behaviours. Similarity can be formally expressed by defining equivalence relations on the set of states and the set of actions of a system under consideration. A state space can be constructed, in which the nodes correspond to equivalence classes of states and the arcs correspond to equivalence classes of actions. Such a state space is often much smaller than the ordinary full state space, but does allow derivation of many verification results.</p><p>Other researches have taken advantage of the symmetries of systems, which induce a certain kind of equivalence. The contribution fo this paper is to show that a more general notion of equivalence is useful. As example, a communication protocol modelled in the formalism of Coloured Petri Nets is verified. Aided by a computer tool supporting state spaces with equivalence classes, significant reduction of state spaces are exhibited.</p><p><strong>Topics</strong>. State space reduction methods, equivalence vs. symmetry, High-level Petri Nets, communication protocols.</p>

Dynamic programming techniques have proven much more flexible than calculus of variations and other techniques in performing redundancy resolution through global optimization of performance indices. When the state and input spaces are discrete, and the time horizon is finite, they can easily accommodate generic constraints and objective functions and find Pareto-optimal sets. Several implementations have been proposed in previous works, but either they do not ensure the achievement of the globally optimal solution, or they have not been demonstrated on robots of practical relevance. In this communication, recent advances in dynamic programming redundancy resolution, so far only demonstrated on simple planar robots, are extended to be used with generic kinematic structures. This is done by expanding the Robot Operating System (ROS) and proposing a novel architecture meeting the requirements of maintainability, re-usability, modularity and flexibility that are usually required to robotic software libraries. The proposed ROS extension integrates seamlessly with the other software components of the ROS ecosystem, so as to encourage the reuse of the available visualization and analysis tools. The new architecture is demonstrated on a 7-DOF robot with a six-dimensional task, and topological analyses are carried out on both its state space and resulting joint-space solution.

Determination of control and state space trajectory which realizes given output time history of nonlinear dynamic system is considered in the paper. Necessary and sufficient conditions for existence of solution are formulated for a class of systems described by ordinary differential equations X=F(t,X,U) with right hand side as a vector function of class Cn, where n is dimension of state space. Existence and uniqueness of solution are studied as a local system properties, i.e in the viciniti of considered point in state and input space. A procedure which leads to a solution by means of system of nonlinear algebraic equations and classical initial-value problem solving is presented.

A fundamental challenge in automated reasoning about programming assignments at scale is clustering student submissions based on their underlying algorithms. State-of-the-art clustering techniques are sensitive to control structure variations, cannot cluster buggy solutions with similar correct solutions, and either require expensive pair-wise program analyses or training efforts. We propose a novel technique that can cluster small imperative programs based on their algorithmic essence: (A) how the input space is partitioned into equivalence classes and (B) how the problem is uniquely addressed within individual equivalence classes. We capture these algorithmic aspects as two quantitative semantic program features that are merged into a program's vector representation. Programs are then clustered using their vector representations. The computation of our first semantic feature leverages model counting to identify the number of inputs belonging to an input equivalence class. The computation of our second semantic feature abstracts the program's data flow by tracking the number of occurrences of a unique pair of consecutive values of a variable during its lifetime. The comprehensive evaluation of our tool SemCluster on benchmarks drawn from solutions to small programming assignments shows that SemCluster (1) generates far fewer clusters than other clustering techniques, (2) precisely identifies distinct solution strategies, and (3) boosts the performance of clustering-based program repair, all within a reasonable amount of time.

The operator Carleson measure criterion for admissibility of control operators for diagonal semigroups on [formula omitted

This paper discusses nonnegativity and positivity concepts and related properties for the state and output trajectory solutions of dynamic linear time-invariant systems described by functional differential equations subject to point time delays. The various nonnegativities and positivities are introduced hierarchically from the weakest one to the strongest one while separating the corresponding properties when applied to the state space or to the output space as well as for the zero-initial state or zero-input responses. The formulation is first developed by defining cones for the input, state and output spaces of the dynamic system, and then extended, in particular, to cones being the three first orthants each being of the corresponding dimension of the input, state, and output spaces.

Recursive least-squares estimates for processes that can be generated from finite-dimensional linear systems are usually obtained via an n \times n matrix Riccati differential equation, where n is the dimension of the state space. In general, this requires the solution of n(n + 1)/2 simultaneous nonlinear differential equations. For constant parameter systems, we present some new algorithms that in several cases require only the solution of less than 2np or n(m + p) simultaneous nonlinear differential equations, where m and p are the dimensions of the input and observation processes, respectively. These differential equations are said to be of Chandrasekhar type, because they are similar to certain equations introduced in 1948 by the astrophysicist S. Chandrasekhar, to solve finite-interval Wiener-Hopf equations arising in radiative transfer. Our algorithms yield the gain matrix for the Kalman filter directly without having to solve separately for the error-covariance matrix and potentially have other computational benefits. The simple method used to derive them also suggests various extensions, for example, to the solution of nonsymmetric Riccati equations.

Patient similarity measurement is an essential step in discovering clinically meaningful subgroups and building case retrieval systems. Most existing studies implement this procedure using similarity measurement algorithms on the multivariate clinical time-series (input space) or the low-dimensional patient representation (representation space) learned by a representation learning model. However, they either suffer from the adverse effects of irrelevant variables in the data or fail to assess the fine-grained similarity underneath the disease progress. In this paper, we propose a method to measure more fine-grained patient similarity in the state space, where each patient is represented by a series of state representations that reveal the dynamic health status. We discuss three desiderata, including stability, personality, and interpretability, for the state representations, and on this basis, develop a supervised predictive model that learns good state representations for identifying similar patients and predicting patient outcomes. Experimental results on the publicly available dataset MIMIC-III show that our method offers a promising direction for precisely identifying similar patients at the state trajectory level, as well as accurately predicting outcomes.

Computing the reliability kernel of a time-variant system: Application to a corroded beam

In this paper a new method of modelling and controller design for continuous systems is introduced. Petri Net is a useful tool for modelling, analysis and controller synthesis in discrete event systems. Continuous-Time Delay-Petri Nets (CTDPN) are presented to challenge with other types of continuous dynamic system modelling tools. This article focuses on an approach to design a controller using Petri Net for continuous dynamic systems. It is shown here that this method simplifies system analysis and controller design by first, converting the continuous state space to a discrete state space and then applying CTDPN, to model and analyse it. Finally, a state feedback control algorithm is adapted to be used for models which are derived by CTDPN approach. Simulation results show the efficiency of the proposed approach comparing to state space feedback controller in terms of the simplicity of the system analysis and the time it takes to simulate the close-loop system. DOI: http://dx.doi.org/10.5755/j01.itc.45.4.13665

In this paper, we show that a minimal state space realization in Jordan canonical form for linear multivariable continuous‐time systems described by rational transfer function matrices could be obtained in a natural and basic way by using the concept of Nerode equivalence. While the minimal state space realization is known, the contribution of this note is to provide an alternative realization procedure which is directly introduced in a simple and self‐contained manner. Both scalar and multivariable cases in the continuous‐time setting are discussed. The presentation also provides a concrete construction of rational vector function with preassigned poles to interpolate any 2‐norm bounded analytical vector function in the open left‐half of the complex plane. The basic idea of Nerode equivalence is that the state can be identified with a corresponding equivalence class of inputs. For a linear finite dimensional time‐invariant continuous‐time system, the zero state is identified with the kernel of certain Hankel operator. This characterization via Nerode equivalence class sheds light on the construction of state for general nonlinear input–output systems.

In recent work we have introduced performance evaluation programming - a programming paradigm supporting performance analysis and formal verification of concurrent programs. For software verification the ranges of variables must be bounded, and programs are translated into finite-state continuous-time Markov chains verified formally by using probabilistic model checking techniques. Our present aim is to increase the size of the state space of programs that can be verified formally. The solution investigated in this paper is inspired by equivalence partitioning - a well-known software testing technique. We provide experimental results showing that, by partitioning the state space of concurrent programs into behavioral equivalence classes, the performance evaluation programming paradigm can be used to design and verify formally concurrent programs with large state spaces.

Topological states in photonics offer novel prospects for guiding and manipulating photons and facilitate the development of modern optical components for a variety of applications. Over the past few years, photonic topology physics has evolved and unveiled various unconventional optical properties in these topological materials, such as silicon photonic crystals. However, the design of such topological states still poses a significant challenge. Conventional optimization schemes often fail to capture their complex high dimensional design space. In this manuscript, we develop a deep learning framework to map the design space of topological states in the photonic crystals. This framework overcomes the limitations of existing deep learning implementations. Specifically, it reconciles the dimension mismatch between the input (topological properties) and output (design parameters) vector spaces and the non-uniqueness that arises from one-to-many function mappings. We use a fully connected deep neural network (DNN) architecture for the forward model and a cyclic convolutional neural network (cCNN) for the inverse model. The inverse architecture contains the pre-trained forward model in tandem, thereby reducing the prediction error significantly.