Set In State Space Research Articles

A Dynamic Harmonic Regression Approach to Power System Modal Identification and Prediction

—Time series models provide a powerful tool to extract nonstationary features from measured data. In this article, a statistical framework based upon a dynamic harmonic regression model for examining modal behavior is provided. In this model, temporal patterns in measured data are modeled within a stochastic state space setting. Estimates of the states or time-varying parameters are then obtained using an optimal estimation method based on the Kalman filter. Techniques to estimate future values of the unobserved signal are also analyzed. The widely applicable technique is illustrated on both simulated and measured data. Factors that affect the performance of the method are discussed, including the effects of non-linear trends, data quality, and sampling design. Connections with other modal identification methods are also investigated.

Linear matrix inequalities (LMIs) observer and controller design synthesis for parabolic PDE

In this work, an observer based controller synthesis is proposed based on the linear matrix inequality (LMI) framework to achieve stabilization of the parabolic PDE in the presence of input constraints. A novel feature of the proposed synthesis is to construct the LMI formulation within the modal space by converting the original parabolic PDE into an infinite-dimensional abstract state space setting. The state feedback controller and the Luenberger observer are developed by accounting for the entire infinite number of modal states, thereby stabilizing the system and reconstructing the state rigorously. The input constraints naturally existing in realistic applications are considered in the design framework. Finally, since initial modal states are hardly known in advance, the LMI formulation, based on the augmented modal state space, including unstable modal states, its estimation error and fast modal output, is developed to maximize the region of attraction (ROA) for the real process state, such that the controller and the observer are robust enough to the error in the initial state guess. By considering a numerical example of an unstable parabolic PDE, we demonstrate that if feasible, the state feedback controller and the observer, generated by the LMI formulation, have capability to drive the process state to the equilibrium steady state.

An adaptive learning damage estimation method for structural health monitoring

Structural health monitoring is an important problem of interest in many civil infrastructure and aerospace applications. In the last few decades, many techniques have been investigated to address the detection, estimation, and classification of damage in structural components. One of the key challenges in the development of real-world damage identification systems, however, is variability due to changing environmental and operational conditions. Conventional statistical methods based on static modeling frameworks can prove to be inadequate in a dynamic and fast changing environment, especially when a sufficient amount of data is not available. In this paper, a novel adaptive learning structural damage estimation method is proposed in which the stochastic models are allowed to perpetually change with the time-varying conditions. The adaptive learning framework is based on the use of Dirichlet process (DP) mixture models, which provide the capability of automatically adjusting to structure within the data. Specifically, time–frequency features are extracted from periodically collected structural data (measured sensor signals), that are responses to ultrasonic excitation of the material. These are then modeled using a DP mixture model that allows for a growing, possibly infinite, number of mixture components or latent clusters. Combined with input from physically based damage growth models, the adaptively identified clusters are used in a state-space setting to effectively estimate damage states within the structure under varying external conditions. Additionally, a data selection methodology is implemented to enable judicious selection of informative measurements for maximum performance. The utility of the proposed algorithm is demonstrated by application to the estimation of fatigue-induced damage in an aluminum compact tension sample subjected to variable-amplitude cyclic loading.

Necessary and Sufficient Razumikhin-Type Conditions for Stability of Delay Difference Equations

This technical note considers stability analysis of time-delay systems described by delay difference equations (DDEs). All existing analysis methods for DDEs that rely on the Razumikhin approach provide sufficient, but not necessary conditions for asymptotic stability. Nevertheless, Lyapunov-Razumikhin functions are of interest because they induce invariant sets in the underlying state space of the dynamics. Therefore, we propose a relaxation of the Razumikhin conditions and prove that the relaxed conditions are necessary and sufficient for asymptotic stability of DDEs. For linear DDEs, it is shown that the developed conditions can be verified by solving a linear matrix inequality. Moreover, it is indicated that the proposed relaxation of Lyapunov-Razumikhin functions has an important implication for the construction of invariant sets for linear DDEs.

A stochastic games framework for verification and control of discrete time stochastic hybrid systems

We describe a framework for analyzing probabilistic reachability and safety problems for discrete time stochastic hybrid systems within a dynamic games setting. In particular, we consider finite horizon zero-sum stochastic games in which a control has the objective of reaching a target set while avoiding an unsafe set in the hybrid state space, and a rational adversary has the opposing objective. We derive an algorithm for computing the maximal probability of achieving the control objective, subject to the worst-case adversary behavior. From this algorithm, sufficient conditions of optimality are also derived for the synthesis of optimal control policies and worst-case disturbance strategies. These results are then specialized to the safety problem, in which the control objective is to remain within a safe set. We illustrate our modeling framework and computational approach using both a tutorial example with jump Markov dynamics and a practical application in the domain of air traffic management.

Non-parametric estimation of forecast distributions in non-Gaussian, non-linear state space models

Abstract The object of this paper is to produce non-parametric maximum likelihood estimates of forecast distributions in a general non-Gaussian, non-linear state space setting. The transition densities that define the evolution of the dynamic state process are represented in parametric form, but the conditional distribution of the non-Gaussian variable is estimated non-parametrically. The filtered and prediction distributions are estimated via a computationally efficient algorithm that exploits the functional relationship between the observed variable, the state variable and a measurement error with an invariant distribution. Simulation experiments are used to document the accuracy of the non-parametric method relative to both correctly and incorrectly specified parametric alternatives. In an empirical illustration, the method is used to produce sequential estimates of the forecast distribution of realized volatility on the S&P500 stock index during the recent financial crisis. A resampling technique for measuring sampling variation in the estimated forecast distributions is also demonstrated.

Model Reduction of Descriptor Systems by Interpolatory Projection Methods

In this paper, we investigate interpolatory projection framework for model reduction of descriptor systems. With a simple numerical example, we first illustrate that employing subspace conditions from the standard state space settings to descriptor systems generically leads to unbounded H2 or H-infinity errors due to the mismatch of the polynomial parts of the full and reduced-order transfer functions. We then develop modified interpolatory subspace conditions based on the deflating subspaces that guarantee a bounded error. For the special cases of index-1 and index-2 descriptor systems, we also show how to avoid computing these deflating subspaces explicitly while still enforcing interpolation. The question of how to choose interpolation points optimally naturally arises as in the standard state space setting. We answer this question in the framework of the H2-norm by extending the Iterative Rational Krylov Algorithm (IRKA) to descriptor systems. Several numerical examples are used to illustrate the theoretical discussion.

Spike oscillations

According to Belinski\ifmmode \check{i}\else \v{i}\fi{}, Khalatnikov and Lifshitz (BKL), a generic spacelike singularity is characterized by asymptotic locality: Asymptotically, toward the singularity, each spatial point evolves independently from its neighbors, in an oscillatory manner that is represented by a sequence of Bianchi type I and II vacuum models. Recent investigations support this conjecture but with a modification: Apart from local BKL behavior there also exists formation of spatial structures (``spikes'') at, and in the neighborhood of, certain spatial surfaces that break asymptotic locality; the complete description of a generic spacelike singularity involves spike oscillations, which are described by sequences of Bianchi type I and certain inhomogeneous vacuum models. In this paper we describe how BKL and spike oscillations arise from concatenations of exact solutions in a Hubble-normalized state space setting, suggesting the existence of hidden symmetries and showing that the results of BKL are part of a greater picture.

Controllability of a class of Volterra equations in Hilbert spaces with completely monotone kernel

We consider a parabolic Volterra integro-differential equation in Hilbert space with a completely monotone convolution kernel and a sectorial operator. Utilizing a state space setting, we show that for a large class of kernels the state cannot be controlled exactly to zero. On the other hand, equations of our type are always approximately controllable, provided the control operator has dense range.

Stability Analysis of 2D Discrete Linear System Described by the Fornasini-Marchesini Second Model with Actuator Saturation

This paper proposes a novel antiwindup controller for 2D discrete linear systems with saturating controls in Fornasini-Marchesini second local state space (FMSLSS) setting. A Lyapunov-based method to design an antiwindup gain of 2D discrete systems with saturating controls is established. Stability conditions allowing the design of antiwindup loops, in both local and global contexts have been derived. Numerical examples are provided to illustrate the applicability of the proposed method.

LQ-boundary control of a diffusion-convection-reaction system

In this work, the boundary control of a distributed parameter system modelled by linear parabolic partial differential equations (PDEs) with spatially varying coefficients is studied. An infinite-dimensional state space setting is considered and an exact transformation of the boundary actuation is realised to obtain an evolutionary model. The evolutionary model which incorporates the spatially varying coefficients of the underlying set of the PDEs is used for subsequent linear quadratic regulator synthesis. The formulated linear quadratic-state feedback controller is applied to a nonlinear model of the reactor and its performance is studied.

Optimal Control for Stochastic Volterra Equations with Completely Monotone Kernels

In this paper, we study a class of optimal control problems for stochastic Volterra equations in infinite dimensions. We are concerned with a class of stochastic Volterra integro-differential problem with completely monotone kernels, where we assume that the noise enters the system when we introduce a control. We provide a semigroup setting for the problem, by the state space setting; the applications to optimal control provide other interesting results and require a precise description of the properties of the generated semigroup. In our stochastic optimal control problems, the drift term of the equation has a linear growth in the control variable, the cost functional has a quadratic growth, and the control process belongs to the class of square integrable, adapted processes with no bound assumed on it. Our main results are the existence for the optimal feedback control, the identification of the optimal cost with the value $Y_0$ of the maximal solution $(Y,Z)$ of the backward stochastic differential equation, the existence of a weak solution to the so-called closed loop equation and, finally, the construction of an optimal feedback in terms of the process $Z$.

Discounted Approximations for Risk-Sensitive Average Criteria in Markov Decision Chains with Finite State Space

This work concerns Markov decision processes with finite state space and compact action set. The performance of a control policy is measured by a risk-sensitive average cost criterion and, under standard continuity-compactness conditions, it is shown that the discounted approximations converge to the optimal value function, and that the superior and inferior limit average criteria have the same optimal value function. These conclusions are obtained for every nonnull risk-sensitivity coefficient, and regardless of the communication structure induced by the transition law.

Linear quadratic optimal boundary control of a diffusion-convection-reaction system

AbstractIn this work, the boundary control of a distributed parameter system (DPS) modeled by parabolic partial differential equations with spatially varying coefficients is studied. An infinite dimensional state space setting is formulated and an exact transformation of the boundary actuation is realized to obtain an evolutionary model. The evolutionary model is used for subsequent linear quadratic regulator synthesis which incorporates the spatially varying coefficients of the underlying set of the PDEs. The formulated LQR controller is applied to the nonlinear model of the system and its performance is studied.

Dilated LMI characterisations for linear time-invariant singular systems

This article explores, based on projection lemma, dilated linear matrix inequality (LMI) characterisations for linear time-invariant singular systems. The deduced formulations cover not only the existing results reported in the literature, but also complete some missing LMI conditions. This article also lightens the mutual relations of these characterisations and clarifies the relation between the dilated LMIs for the conventional state-space systems and those for singular systems. The well-known results for the state-space setting reported in the literature can be reviewed as special cases.

On the computation of convex robust control invariant sets for nonlinear systems

In this paper we provide a method to compute robust control invariant sets for nonlinear discrete-time systems. A simple criterion to evaluate if a convex set in state space is a robust control invariant set for a nonlinear uncertain system is presented. The criterion is employed to design an algorithm for computing a polytopic robust control invariant set. The method is based on the properties of DC functions, i.e. functions which can be expressed as the difference of two convex functions. Since the elements of a wide class of nonlinear functions have DC representation or, at least, admit an arbitrarily close approximation, the method is quite general. The algorithm requires relatively low computational resources.

A Hamiltonian State Space Approach for 3D Analysis of Circular Cantilevers

A 3D exact analysis of extension, torsion and bending of a cantilever of a circular cross section is studied with emphasis on the fixed-end effect. Through Hamiltonian variational formulation, the basic equations of elasticity in cylindrical coordinates and the boundary conditions of the problem are formulated into the state space setting in which the state vector comprises the displacement vector and the conjugate stress vector as the dual variables. Upon delineating the Hamiltonian characteristics of the system, 3D solutions for transversely isotropic circular cantilevers subjected to an axial force, a torque, terminal couples and transverse forces are determined, thereby, the fixed-end effects and applicability of the solutions of generalized plane strains and the elementary theory of bending of beams are evaluated.

Optimal control of transport-reaction system with time varying spatial domain

This paper considers the multiscale optimal control of crystal growth. The optimal control is realized for the crystal pulling arm modeled by the standard rigid body dynamics, while the underlying dynamics of the diffusion of heat in the crystal growth region is given by parabolic partial differential equations (PDEs) with time varying spatial domain. The underlying transport-reaction system is developed from the first principles and the associated dynamics is analyzed in appropriate functional state space setting. The complete description of the evolutionary parabolic domain time varying PDE is provided and explored within the coupled master-slave control setting. Numerical simulations demonstrate an optimal pulling evolution rate and its effects on the temperature profile in the crystal with the time varying domain.

Discrete time, finite state space mean field games

In this paper we study a mean field model for discrete time, finite number of states, dynamic games. These models arise in situations that involve a very large number of agents moving from state to state according to certain optimality criteria. The mean field approach for optimal control and differential games was introduced by Lasry and Lions (2006, 2007) [3–5]. The discrete time, finite state space setting is motivated both by its independent interest as well as by numerical analysis questions which appear in the discretization of the problems introduced by Lasry and Lions. The main contribution of this paper is the exponential convergence to equilibrium of the initial-terminal value problem.

A new look at state-space models for neural data.

State space methods have proven indispensable in neural data analysis. However, common methods for performing inference in state-space models with non-Gaussian observations rely on certain approximations which are not always accurate. Here we review direct optimization methods that avoid these approximations, but that nonetheless retain the computational efficiency of the approximate methods. We discuss a variety of examples, applying these direct optimization techniques to problems in spike train smoothing, stimulus decoding, parameter estimation, and inference of synaptic properties. Along the way, we point out connections to some related standard statistical methods, including spline smoothing and isotonic regression. Finally, we note that the computational methods reviewed here do not in fact depend on the state-space setting at all; instead, the key property we are exploiting involves the bandedness of certain matrices. We close by discussing some applications of this more general point of view, including Markov chain Monte Carlo methods for neural decoding and efficient estimation of spatially-varying firing rates.