State Space Of System Research Articles

On the computation of physically realizable generalized flat inputs for nonlinear single‐output systems from a geometric point of view

AbstractPhysical realizability is a structural property of (classical) flat inputs for state space systems which originate from higher order differential equations. These state space systems contain a derivative array, i.e., an integrator chain. Flat input injections into this array, however, structurally prevents from the reversal of the order reduction. Using a generalized flat input definition, we demonstrate how physical realizability can be achieved.

Scheduling and Power Control for Wireless Multicast Systems via Deep Reinforcement Learning.

Multicasting in wireless systems is a natural way to exploit the redundancy in user requests in a content centric network. Power control and optimal scheduling can significantly improve the wireless multicast network’s performance under fading. However, the model-based approaches for power control and scheduling studied earlier are not scalable to large state spaces or changing system dynamics. In this paper, we use deep reinforcement learning, where we use function approximation of the Q-function via a deep neural network to obtain a power control policy that matches the optimal policy for a small network. We show that power control policy can be learned for reasonably large systems via this approach. Further, we use multi-timescale stochastic optimization to maintain the average power constraint. We demonstrate that a slight modification of the learning algorithm allows tracking of time varying system statistics. Finally, we extend the multi-time scale approach to simultaneously learn the optimal queuing strategy along with power control. We demonstrate the scalability, tracking and cross-layer optimization capabilities of our algorithms via simulations. The proposed multi-time scale approach can be used in general large state-space dynamical systems with multiple objectives and constraints, and may be of independent interest.

Sliding Mode Control of Electro-Hydraulic Servo System Based on Optimization of Quantum Particle Swarm Algorithm

This paper investigates a sliding mode controller based on quantum particle swarm optimization algorithm (QPSO) to solve the nonlinearity of electro-hydraulic servo systems, external disturbance problems, and jitter of sliding mode controller. The electro-hydraulic servo system state space equations are established, constructing the sliding surface according to the tracking error and obtaining the output of the sliding mode controller. The ITAE metric is used as an adaptation function of the QPSO algorithm to evaluate the parameters in the sliding mode controller, which has good engineering utility and parameter selectivity. The QPSO algorithm is used to increase the randomicity of the search and to expand the search space, which can effectively prevent falling into a local optimum solution. Finally, a comparative simulation is presented to illustrate global search performance of QPSO algorithm and the effectiveness and applicability of the proposed control method.

Data-driven dynamic interpolation and approximation

The behavioral system theory and in particular a result that became known as the “fundamental lemma” give the theoretical foundation for nonparametric representations of linear time-invariant systems based on Hankel matrices constructed from data. These “data-driven” representations led in turn to new system identification, signal processing, and control methods. This paper shows how the approach can be used further on for solving interpolation, extrapolation, and smoothing problems. The solution proposed and the resulting method are general – can deal simultaneously with missing, exact, and noisy data of multivariable systems – and simple—require only the solution of a linear system of equations. In the case of exact data, we provide conditions for existence and uniqueness of solution. In the case of noisy data, we propose an approximation procedure based on ℓ1-norm regularization and validate its performance on real-life datasets. The results have application in missing data estimation and trajectory planning. They open a practical computational way of doing system theory and signal processing directly from data without identification of a transfer function or state-space system representation.

Assessment of State-Space Building Energy System Models in Terms of Stability and Controllability

Building energy management system involves the development of control strategies for the heating, ventilation, and air-conditioning (HVAC), as well as lighting, systems. Building energy modeling is a significant part of designing such strategies. In order to analyze the feasibility of a building energy system model for any desired control strategy, a mathematical assessment tool is developed in this paper. A multi-input multi-output (MIMO) building energy system model, consisting of an outdoor wall, an external wall, two partition walls, one roof, and a ceiling, has been considered as the virtual test setup. A methodology for conducting stability and controllability assessment tests on the building energy model is proposed using inverse dynamics input theory (IDIT). IDIT enables the decoupling of control variables so as to enable the conversion of an MIMO system to a number of independent single-input single-output systems. The controllability is assessed based on the design properties for continuous systems: asymptotes and transmission zeros. The results show that the relative humidity and air temperature of the building space were controllable for all operating points; however, in unconditioned situations, where the humidity levels of the building space were greater than that of the outdoor levels, the models were unstable.

Contribution of the generalized nonunique inverses to the minimum-energy control theory: The inverse model control investigation

The innovative analytical approach to the minimum-energy design problem of the inverse model control (IMC) state-space structures is presented in this work. Following the recent papers, it should be concluded that the optimal behavior of the IMC strategy cannot longer be associated with the application of the well-known Moore–Penrose minimum-norm inverse. However, the minimum-energy IMC-oriented scheme has only be obtained through heuristic methods. Nevertheless, in the recent authors’ work, it has been proven for the first time that such an issue can be considered in an analytical manner. Yet, the obtained results have only been valid for the second-order state-space systems. Therefore, the motivation instance proposed in the manuscript, confirming the possibility of extending such paradigm to higher-order plants, will certainly contribute to the introduction of the new unified minimum-energy IMC theory canon. Since the nonunique σ and H inverses can successfully be employed in the robustification of the discussed control strategy, they can also be helpful in the case of our essential considerations. Thus, from now on the yet unexplored research area can now be investigated in the analytical manner, what has never been seen before in the modern IMC-originated control theory and practice. The predefined methodology clearly fills the gap in the analytical control design procedures and opens a new chapter in the knowledge related to the well-known and broadly accepted multivariable control canons.

Strong and weak-star stabilisation of bilinear systems on nonreflexive Banach spaces

This paper deals with bilinear systems evolving in a Banach state space (which is not necessarily reflexive). To the best of our knowledge, strong and weak-star stabilisation of control systems in nonreflexive state spaces has not been studied yet. This work is a first attempt towards providing sufficient conditions that guarantee the weak-star stabilisation of the system at hand. The strong stabilisation is stated as well. The approach uses results relying on convex analysis and some properties of the duality mapping and linear semigroup theory. Applications to reaction–diffusion and transport equations are further presented.

Dynamic Expectation Maximization Algorithm for Estimation of Linear Systems with Colored Noise

The free energy principle from neuroscience has recently gained traction as one of the most prominent brain theories that can emulate the brain’s perception and action in a bio-inspired manner. This renders the theory with the potential to hold the key for general artificial intelligence. Leveraging this potential, this paper aims to bridge the gap between neuroscience and robotics by reformulating an FEP-based inference scheme—Dynamic Expectation Maximization—into an algorithm that can perform simultaneous state, input, parameter, and noise hyperparameter estimation of any stable linear state space system subjected to colored noises. The resulting estimator was proved to be of the form of an augmented coupled linear estimator. Using this mathematical formulation, we proved that the estimation steps have theoretical guarantees of convergence. The algorithm was rigorously tested in simulation on a wide variety of linear systems with colored noises. The paper concludes by demonstrating the superior performance of DEM for parameter estimation under colored noise in simulation, when compared to the state-of-the-art estimators like Sub Space method, Prediction Error Minimization (PEM), and Expectation Maximization (EM) algorithm. These results contribute to the applicability of DEM as a robust learning algorithm for safe robotic applications.

Moment equations and quasi-moment neglect closure approximations for non-linear dynamic systems under Erlang renewal impulse process excitations

In the present paper the moment equations technique together with two closure approximations is developed for a non-linear dynamic system subjected to an impulse process excitation driven by an Erlang renewal counting process. The original non-Markov problem is converted into a Markov one at the expense of augmentation of the state space of the dynamic system by the Markov states of the introduced auxiliary jump process. Differential equations for moments are derived from the explicit integro-differential equations governing the joint probability density — discrete distribution function (Iwankiewicz,2008), which is a different approach from that of a generating equation for moments (Iwankiewicz,2014). For non-linear dynamic systems with polynomial-type non-linearities two closure approximation techniques, based on neglecting of quasi-moments above certain order, are developed. The first closure technique is the result of application of quasi-moment closure directly to the usual, unconditional moments. The second one, a modified closure approximation technique, is based on the representation of the joint probability density — discrete distribution function in the form consisting of a spike and of the continuous part (cf. (Iwankiewicz,1990). As an example of a non-linear system the oscillator with cubic restoring force term is considered. The equations for moments up to the fourth-order are derived where the usual and the modified quasi-moment neglect closure approximations are used for redundant fifth- and sixth order moments. Mean value and variance of the response have been determined with the aid of the developed approximate techniques and verified against Monte Carlo simulations.

Negative imaginariness and positive realness analysis of state-space symmetric systems with interval uncertainty

In this article, the state-space symmetric systems with symmetrical interval uncertainty that have positive real and negative imaginary properties are studied. First, a necessary and sufficient test in view of a state matrix is derived for a state-space symmetric system to be negative imaginary, which allows having poles at the origin. Second, bounds on symmetrical interval uncertainty that guarantee the positive realness and negative imaginariness of state-space symmetric systems are provided. Finally, the main results are illustrated by a resistor–capacitor network and a numerical design example.

Impact of Quadrature Booster on Power-System State Estimation in Polar Coordinates

The paper concerns the estimation of the state of a power system in which there is a phase shifter called a quadrature booster. The aim of the paper is a comparative analysis of two different cases including the quadrature booster in the state estimation. In the first case, the quadrature booster is represented by a model consisting of two real voltage sources, one in series with a power line and the other in a shunt branch. In the second case, in the power system model, the real branch with the quadrature booster is represented as off at the end where the considered quadrature booster is actually installed. The state estimation is assumed to be carried out in the polar coordinate system. The properties of the state estimation are characterized by: the number of iterations in the calculation process, the index of conditioning of the matrix of coefficients in the equations to be solved (cond(G)), and ratio Je/Jm, which is a measure of the accuracy of the estimation. Using IEEE 14-bus test system, investigations are carried out in such a way as to cover the entire state space of the power system as possible. In the investigations, Monte Carlo experiments are carried out for each of the considered cases of the state estimation. Each of these cases is also analyzed from the point of view of the assumed definition of the state estimation. Investigations show that in the first of the previously described cases, the state estimation is more accurate, but there are more iterations in the calculations and worse conditioning of the estimation process. The comparative analysis also shows that, the accuracy of the results obtained in each of the considered cases is practically independent of the coordinate system in which the estimation calculations are performed. Taking into account the number of iterations in the estimation process and index cond(G), it can be concluded that the implementation of each of the above-mentioned estimation cases in the rectangular coordinate system is more reasonable.

From lakes and glades to viability algorithms: automatic classification of system states according to the topology of sustainable management

The framework Topology of Sustainable Management by Heitzig et al. (Earth Syst Dyn 7:21. https://doi.org/10.5194/esd-7-21-2016, 2016) distinguishes qualitatively different regions in state space of dynamical models representing manageable systems with default dynamics. In this paper, we connect the framework to viability theory by defining its main components based on viability kernels and capture basins. This enables us to use the Saint-Pierre algorithm to visualize the shape and calculate the volume of the main partition of the Topology of Sustainable Management. We present an extension of the algorithm to compute implicitly defined capture basins. To demonstrate the applicability of our approach, we introduce a low-complexity model coupling environmental and socioeconomic dynamics. With this example, we also address two common estimation problems: an unbounded state space and highly varying time scales. We show that appropriate coordinate transformations can solve these problems. It is thus demonstrated how algorithmic approaches from viability theory can be used to get a better understanding of the state space of manageable dynamical systems.

An Algebraic Approach to Identifiability

This paper addresses the problem of identifiability of nonlinear polynomial state-space systems. Such systems have already been studied via the input-output equations, a description that, in general, requires differential algebra. The authors use a different algebraic approach, which is based on distinguishability and observability. Employing techniques from algebraic geometry such as polynomial ideals and Gröbner bases, local as well as global results are derived. The methods are illustrated on some example systems.

Nested Gaussian filters for recursive Bayesian inference and nonlinear tracking in state space models

We introduce a new sequential methodology to calibrate the fixed parameters and track the stochastic dynamical variables of a state-space system. The proposed method is based on the nested hybrid filtering (NHF) framework of [1], that combines two layers of filters, one inside the other, to compute the joint posterior probability distribution of the static parameters and the state variables. In particular, we explore the use of deterministic sampling techniques for Gaussian approximation in the first layer of the algorithm, instead of the Monte Carlo methods employed in the original procedure. The resulting scheme reduces the computational cost and so makes the algorithms potentially better-suited for high-dimensional state and parameter spaces. We describe a specific instance of the new method and then study its performance and efficiency of the resulting algorithms for a stochastic Lorenz 63 model and for a stochastic volatility model with real data.

A model-based reinforcement learning approach for maintenance optimization of degrading systems in a large state space

Scheduling maintenance tasks based on the deteriorating process has often been established on degradation models. However, the formulas of the degradation processes are usually unknown and hard to be determined for a system working in practices. In this study, we develop a model-based reinforcement learning approach for maintenance optimization. The developed approach determines maintenance actions for each degradation state at each inspection time over a finite planning horizon, supposing that the degradation formula is known or unknown. At each inspection time, the developed approach attempts to learn an optimal assessment value for each maintenance action to be performed at each degradation state. The assessment value quantifies the goodness of each state-action pair in terms of minimizing the accumulated maintenance costs over the planning horizon. To optimize the assessment values when a well-defined degradation formula is known, we customize a Q-learning method with model-based acceleration. When the degradation formula is unknown or hard to be determined, we develop a Dyna-Q method with maintenance-oriented improvements, in which an environment model capturing the degradation pattern under different maintenance actions is learned at first; Then, the assessment values are optimized while considering the stochastic behavior of the system degradation. The final maintenance policy is acquired by performing the maintenance actions associated with the highest assessment values. Experimental studies are presented to illustrate the applications.

Optimal sparse control of interval networks: Voltage source converter based DC grid study

Decentralized optimal control of interval state space systems involves significant complications due to the required sparsity, robustness, and convergence rate. By adding an l1 term to the standard linear quadratic optimization, the obtained composite index, accommodating the sparsity, contains both differentiable and non-differentiable terms. Knowing that the choice of optimization techniques are problem-dependent, in this paper, several smart provisions such as “Active set”, “Coordinate descent”, and separately managing the differentiable part of the cost function are put together to form a relatively fast algorithm. In addition, a rule is introduced to adjust automatically the degree of sparsity and a constraint to confine the interval closed-loop system eigenvalues to the desired s plane region. As a result, a robust sparse decentralized control (RSDC) approach for interval networks is established. The RSDC method is utilized for the control of a multi-terminal VSC (Voltage Source Converter ) -based DC (Direct Current) grid with parametric uncertainty under various external random perturbations. The test results show that the proposed method with just 20 feedback links (out of available 100) performs superiorly to the alternative ones in terms of sparsity, convergence rate, and time-domain performances, verified through extensive simulations.

Fault Tolerant Control of Sensor Faults in Microgrid Inverter Control System

Aiming at sensor drift fault, the design problem of active fault tolerant controller for island-AC microgrid inverter system is studied. The controller has good ability to suppress external finite energy disturbance. Firstly, a closed-loop system state space model is constructed for an isolated AC microgrid inverter system with external finite energy disturbance. Then, when the sensor fault occurs in the control system, the fault information is taken as an auxiliary state vector, and the estimated value of the sensor fault and the system state variable is obtained simultaneously through the proposed extended observer. A fault tolerant controller is designed based on the idea of fault compensation. Finally, the correctness and effectiveness of this method are verified by programming simulation in MATLAB.

Group Theory: Mathematical Expression of Symmetry in Physics

The present article reviews the multiple applications of group theory to the symmetry problems in physics. In classical physics, this concerns primarily relativity: Euclidean, Galilean, and Einsteinian (special). Going over to quantum mechanics, we first note that the basic principles imply that the state space of a quantum system has an intrinsic structure of pre-Hilbert space that one completes into a genuine Hilbert space. In this framework, the description of the invariance under a group G is based on a unitary representation of G. Next, we survey the various domains of application: atomic and molecular physics, quantum optics, signal and image processing, wavelets, internal symmetries, and approximate symmetries. Next, we discuss the extension to gauge theories, in particular, to the Standard Model of fundamental interactions. We conclude with some remarks about recent developments, including the application to braid groups.

Simultaneous estimation and modeling of nonlinear, non-Gaussian state-space systems

This paper presents a framework for simultaneous estimation and modeling of nonlinear, non-Gaussian state-space systems. In the proposed approach, uncertainty in motion model parameters is incorporated to avoid overconfidence in state prediction and better account for modeling inaccuracies. The additional original contribution of a model correction stage improves nonlinear model parameter estimates in order to enhance the accuracy of state estimation. The presented nonlinear/non-Gaussian Simultaneous Estimation And Modeling (SEAM) approach was compared with contemporary estimation techniques using a Monte-Carlo simulation study. This study showed that the proposed method successfully reduces estimation error relative to existing approaches even when substantial model parameter uncertainty and multi-modal sensor noise are present. The framework has potential use in a wide range of applications where state-space estimation is employed, including robotics, signal processing, and controls.

On σ-entropy Analysis of Linear Stochastic Systems in State Space

In this paper the problem of random disturbance attenuation capabilities in linear continuous systems is studied. It is supposed that the system operates under random disturbances with bounded σ-entropy level. σ-entropy norm indicates a performance index of the continuous system on the set of the random signals with bounded σ-entropy. This paper presents a time-domain solution to the calculation of σ-entropy norm of the continuous linear time-invariant system. σ-entropy norm is defined after solving coupled matrix equations: one algebraic Riccati equation, one nonlinear equation over log determinant function, and two Lyapunov equations.