Form Of State-space Model Research Articles

Design of a linear quadratic regulator for nonlinear systems modeled via norm-bounded linear differential inclusions

AbstractThis paper proposes the design of a linear quadratic regulator (LQR) for a class of nonlinear systems modeled via norm-bounded linear differential inclusions (NLDIs). The system to be controlled is originally described by a nonlinear dynamical model in state-space form. For synthesis purposes, however, this nonlinear system is represented in the form of an NLDI. Using an approach based on the mean-value theorem, this NLDI considers the nonlinear terms in the Taylor series expansion of the nonlinear equations as uncertainties in the model. The main contribution of this paper is the construction of a procedure to design an LQR controller for the NLDI representation of the nonlinear system, in such a way that the properties guaranteed to the NLDI model by the controller will also be valid for the underlying nonlinear system. The control design problem is formulated as an optimization problem in the form of bilinear matrix inequalities (BMIs) and solved via an iterative process known as V-K iterations. A numerical example is presented at the end of this paper to demonstrate the effectiveness of the LQR controller to this class of systems.

Identification of speculative bubbles using state-space models with Markov-switching

In this paper we use a state-space model with Markov-switching to detect speculative bubbles in stock-price data. To this end we express a present-value stock-price model in state-space form which we estimate using the Kalman filter. This procedure enables us to estimate a two-regime Markov-switching specification of the unobservable bubble process. The respective Markov-regimes represent two distinct phases in the bubble process, namely one in which the bubble survives and one in which it collapses. We ultimately identify bursting stock-price bubbles by statistically separating both Markov-regimes from each other. In an empirical analysis we apply our methodology to a plethora of artificial and real-world data sets. Our study has two major findings. First, we find significant Markov-switching structures in real-world stock-price bubbles. Second, in the stock markets considered our identification procedure correctly detects most speculative periods which have been classified as such by economic historians.

Estimating exponential affine models with correlated measurement errors: Applications to fixed income and commodities

Exponential affine models (EAMs) are factor models popular in financial asset pricing requiring a dynamic term structure, such as for interest rates and commodity futures. When implementing EAMs it is usual to first specify the model in state-space form (SSF) and then to estimate it using the Kalman filter. To specify the SSF, a structure of the measurement error must be provided which is not specified in the EAM itself. Different specifications of the measurement errors will result in different SSFs, leading to different parameter estimates. In this paper we investigate the influence of the measurement error specification on the parameter estimates. Using market data for both fixed income and commodities we provide evidence that measurement errors are cross-sectionally and serially correlated, which is not consistent with the independent identically distributed (iid) assumptions commonly adopted in the literature. Using simulated data we show that measurement error assumptions affect parameter estimates, especially in the presence of serial correlation. We provide a new specification, the augmented state-space form (ASSF), as a solution to these biases and show that the ASSF gives much better estimates than the basic SSF.

State-space approach to nonlinear predictive generalised minimum variance control

A nonlinear predictive generalised minimum variance control algorithm is introduced for the control of nonlinear discrete-time multivariable systems. The plant model is represented by the combination of a very general nonlinear operator and also a linear subsystem which can be open-loop unstable and is represented in state-space model form. The multi-step predictive control cost index to be minimised involves both weighted error and control signal costing terms. The solution for the control law is derived in the time domain using a general operator representation of the process. The controller includes an internal model of the nonlinear process, but because of the assumed structure of the system, the state observer is only required to be linear. In the asymptotic case, where the plant is linear, the controller reduces to a state-space version of the well-known GPC controller.

Control of proton exchange membrane fuel cells using data driven state space models

Proton exchange membrane fuel cells (PEMFCs) are increasingly being researched upon due to their potential toward sustainable energy generation. Toward improved productivity of PEMFCs, it is important to develop systematic approaches for optimization and control of their operations. PEMFCs pose interesting challenges toward these tasks due to their complex behavior such as nonlinearity and spatial variations. While first principles model based approaches could be used, a more mathematically attractive and cost-effective alternative is to use empirical modeling approaches for representing the system dynamics toward optimization and control. In this paper, we propose to use a novel, innovation form of state space models that facilitate the development of advanced control algorithms such as linear quadratic Gaussian (LQG) and model predictive control (MPC), and provide improved disturbance rejection necessary for these applications. We demonstrate the applications of such model based algorithms via simulations involving a distributed along-the-channel model of the PEMFC, and also present experimental validation on a PEMFC setup.

Oversampled phase tracking in digital communications with large excess bandwidth

This paper deals with the on-line carrier phase estimation in a digital receiver. We consider a Brownian phase evolution in a data aided scenario. The proposed study uses an oversampled signal model after matched filtering, leading to a coloured reception noise and a non-stationary power signal. The contribution of this paper is twofold. First, we derive the Bayesian Cramér–Rao bound for this estimation problem. Then, based on a state-space model formulation of the problem, we propose an extended Kalman filter to approach this lower bound for a BOC shaping pulse. Our numerical results illustrate the gain resulting from the use of an oversampled version of the received signal to estimate the phase offset, obtaining better performances than using a classical synchronizer.

Estimating Exponential Affine Models with Correlated Measurement Errors: Applications to Fixed Income and Commodities

Exponential affine models (EAMs) are factor models popular in financial asset pricing requiring a dynamic term structure, such as for interest rates and commodity futures. When implementing EAMs it is usual to first specify the model in state space form (SSF) and then to estimate it using the Kalman filter. To specify the SSF, a structure of the measurement error must be provided which is not specified in the EAM itself. Different specifications of the measurement errors will result in different SSFs, leading to different parameter estimates. In this paper we investigate the influence of the measurement error specification on the parameter estimates. Using market data for both fixed income and commodities we provide evidence that measurement errors are cross-sectionally and serially correlated, which is not consistent with the independent identically distributed (iid) assumptions commonly adopted in the literature. Using simulated data we show that measurement error assumptions affect parameter estimates, especially in the presence of serial correlation. We provide a new specification, the augmented state space form (ASSF), as a solution to these biases and show that the ASSF gives much better estimates than the basic SSF.

Methods for robust control

Robust control allows policymakers to formulate policies that guard against model misspecification. The principal tools used to solve robust control problems are state-space methods [see Hansen, L.P., Sargent T.J., 2008. Robustness. Princeton University Press; Giordani, P., Söderlind, P., 2004. Solution of macromodels with Hansen–Sargent robust policies: some extensions. Journal of Economic Dynamics and Control 28 (12), 2367–2397]. In this paper we show that the structural-form methods developed by Dennis [2007. Optimal policy rules in rational-expectations models: new solution algorithms. Macroeconomic Dynamics 11 (1), 31–55] to solve control problems with rational expectations can also be applied to robust control problems, with the advantage that they bypass the task, often onerous, of having to express the reference model in state-space form. In addition, we show how to implement two different timing assumptions with distinct implications for the robust policy and the economy. We apply our methods to a New Keynesian Dynamic Stochastic General Equilibrium model and find that robustness has important effects on policy and the economy.

Methods for Robust Control

Robust control allows policymakers to formulate policies that guard against model misspecification. The principal tools used to solve robust control problems are state-space methods (see Hansen and Sargent, 2006, and Giordani and Soderlind, 2004). In this paper we show that the structural-form methods developed by Dennis (2006) to solve control problems with rational expectations can also be applied to robust control problems, with the advantage that they bypass the task, often onerous, of having to express the reference model in state-space form. Interestingly, because state-space forms and structural forms are not unique the two approaches do not necessarily return the same equilibria for robust control problems. We apply both state-space and structural solution methods to an empirical New Keynesian business cycle model and find that the differences between the methods are both qualitatively and quantitatively important. In particular, with the structural-form solution methods the specification errors generally involve changes to the conditional variances in addition to the conditional means of the shock processes.

MODELING AND CONTROL OF THE ACTIVE SUSPENSION SYSTEM USING PROPORTIONAL INTEGRAL SLIDING MODE APPROACH

The purposes of this paper are to present a new method in modeling an active suspension system for half-car model in state space form and to develop a robust strategy in controlling the active suspension system. Proportional integral sliding mode control strategy is proposed for the system. A simulation study is performed to prove the effectiveness and robustness of the control approach and performance of the controller is compared to the linear quadratic regulator and the existing passive suspension system.

Unknown input modeling and robust fault diagnosis using black box observers

A key issue that needs to be addressed while performing fault diagnosis using black box models is that of robustness against abrupt changes in unknown inputs. A fundamental difficulty with the robust FDI design approaches available in the literature is that they require some a priori knowledge of the model for unmeasured disturbances or modeling uncertainty. In this work, we propose a novel approach for modeling abrupt changes in unmeasured disturbances when innovation form of state space model (i.e. black box observer) is used for fault diagnosis. A disturbance coupling matrix is developed using singular value decomposition of the extended observability matrix and further used to formulate a robust fault diagnosis scheme based on generalized likelihood ratio test. The proposed modeling approach does not require any a priori knowledge of how these faults affect the system dynamics. To isolate sensor and actuator biases from step jumps in unmeasured disturbances, a statistically rigorous method is developed for distinguishing between faults modeled using different number of parameters. Simulation studies on a heavy oil fractionator example show that the proposed FDI methodology based on identified models can be used to effectively distinguish between sensor biases, actuator biases and other soft faults caused by changes in unmeasured disturbance variables. The fault tolerant control scheme, which makes use of the proposed robust FDI methodology, gives significantly better control performance than conventional controllers when soft faults occur. The experimental evaluation of the proposed FDI methodology on a laboratory scale stirred tank temperature control set-up corroborates these conclusions.

Reduction of Power System Model Using Balanced Realization, Routh and Pade Approximation Methods

A time domain model reduction technique based on balanced realization has been applied to a single machine power system model i.e., a synchronous machine connected to an infinite bus through transmission line. The controllability and observability grammians have been computed by solving the Lyapunov equations. The high order system is transformed into balanced form by using the similarity transformation matrix to bring the grammian matrices to the balanced format, which is used to formulate the reduced order model in state space form. The results have also been compared with the Routh and Pade approximation methods, the two well-known frequency domain techniques. The comparison of time and frequency responses shows that, the reduced order models satisfactorily approximate the original system.

Earnings per Share: Stylized Facts and New Paradigms

The paradigm that inspired the equity price bubble of the 1990s is inconsistent with the long-term earnings expectations that a rational agent could draw from the stylized facts available in real time. The paper identifies the stylized facts that characterize the behavior of accounting earnings per share (EPS) that Standard & Poor's has reported for the S&P 500 index since 1935. Estimation of an unobserved components model in state-space form shows that the long-run component of earnings is predictable, and that its growth rate has obeyed a deterministic process for three-quarters of a century. It is impossible to reconcile a deterministic slope in log EPS with the new paradigms and other delusions regarding earnings that periodically generate equity bubbles. The evidence is inconsistent with market rationality, and buttresses a behavioral theory of finance in which folly occasionally occupies center stage.

RECURRENT NEURAL NETWORKS ARE UNIVERSAL APPROXIMATORS

Recurrent Neural Networks (RNN) have been developed for a better understanding and analysis of open dynamical systems. Still the question often arises if RNN are able to map every open dynamical system, which would be desirable for a broad spectrum of applications. In this article we give a proof for the universal approximation ability of RNN in state space model form and even extend it to Error Correction and Normalized Recurrent Neural Networks.

Continuous time system identification using subspace methods

System identification is a well known technique for developing mathematical models based on plant input and output data sequences. Models that describe the systems may be in various forms and one of the possibilities is a state space model formulation. The state space mathematical modelling involves vectors and matrices in a unique geometrical framework. It offers the key advantages on providing low parameter sensitivity with respect to perturbation for high order systems and also has shown its ability to present multi-input and multi-output systems with minimal state dimensions. We use a time domain subspace approach in conjunction with Laguerre filters and instrumental variables to develop a mathematical formulation of the state space model for identification of a continuous time system. The method aims at searching for accurate matrices of the state space model to ensure that the constructed model closely mimics the actual system as well as provide information for the purpose of control system design. The subspace identification algorithm provides state space models with better conditioning, improved quality and easily maintainable parametrisation. The algorithm is validated with identification of two systems: a simulated plant, and a magnetic bearing system. For both systems, the computer simulation results demonstrate that the obtained model describes the system closely.

Finding module-based gene networks with state-space models - Mining high-dimensional and short time-course gene expression data

This study explores some problems to analyze time-course gene expression data by state-space models (SSMs). One problem is regarding the methods of parameter estimation and determination of the dimension of the internal state variable. Although several methods have been applied, there are few literature studies which with to compare them. Thus, this paper gives a brief review of the existing literature that use the SSM to analyze the gene expression time-course data. Another problem is the identifiability of the model. If the parameters of SSMs are simply estimated without any constraints for parameter space, they lack identifiability. To identify a system uniquely, it requires a specific algorithm to estimate the parameters with some constraints. For that purpose, an identifiable form of SSMs and an algorithm for estimating parameters are derived. The last problem is the extraction of biological information by interpreting the estimated parameters, such as mechanism of gene regulations at the module level. For that one, this paper explores methods to extract further information using the estimated parameters, that is, reconstruction of a module network from time-course gene expression data

H∞/µ - SYNTHESIS for CUK CONVERTOR CIRCUIT CONTROLLER

The present work deals with robust stability analysis and synthesis of Cuk converter circuit .Mathematical model in state space form is developed for the circuit .A general system matrix is derived for the system with perturbations in a chosen elements values in the system. H∞ - design technique is applied to design a feedback controller for the system with its nominal values. A µ - synthesis technique is used to design a controller for the system to guaranty its robust stability .The system with designed controller of the two above techniques, is investigated. It is observed that the controller obtained from µ - synthesis technique guaranties the robustness of the Cuk converter circuit with 25% simultaneous uncertainties in all perturbed elements. Key wards: Robust Stability, H∞ - Design Technique, µ - Synthesis Technique

A ´BIG LEAF, BIG FRUIT, BIG SUBSTRATE´ MODEL FOR EXPERIMENTS ON RECEDING HORIZON OPTIMAL CONTROL OF NUTRIENT SUPPLY TO GREENHOUSE TOMATO

A dynamic model was set up to describe the mineral content, fruit dry matter content, and biomass of greenhouse tomato, for use in an experiment aiming at controlling the fertigation so as to reach the best compromise between disinfection costs of the recycled water and income from fruit fresh weight production, while ensuring that the dry matter of the fruits is above a given quality standard. The model describes the effect of mineral shortage on growth, and has a simple mechanism to relate dry matter content to the total ion concentration in the substrate. Optimal control requires a quantitative model in state-space form. In order to reduce on-line computation time, the number of states was kept within reasonable limits by lumping leaves, fruits and substrate into three compartments. Therefore the model is labelled `big leaf, big fruit, big substrate? or 3Bigs model. Using parameters from literature and some calibration on previous experiments, quite acceptable fits were obtained for biomass, and dry matter and K, N, and Ca contents of the fruits. The model was subsequently used in a receding horizon optimal control experiment. Ion-concentrations in the drain as measured by ion-specific electrodes are fed back to correct the state of the model before the next values of the required drain flow and the fertilizer fluid aliquots are computed. Although it was not possible to operate the controller for more than six weeks, it could be observed that the controller behaviour was consistent with expectations in view of the given model and goal function. The fresh weight yield and the dry matter constraint dominate the behaviour. The results also suggest that currently used ion concentrations are higher than necessary for uninhibited growth.

On The Study Of The Dynamics Of The Zvs Three-phase Dc/dc Converter

This paper presents the theoretical analysis concerning the operating stages of the three-phase dc/dc converter associated with the hybridge rectifier, in order to obtain its dynamic model. The nonlinear instant model in state space form is obtained by the definition of a switching function q(t). From this model, an average nonlinear model is proposed, as well as a linearized model. The dynamic model, the controller design, and also experimental results obtained from a prototype of 6kW are presented.

Identification of dynamic systems under closed-loop control

This paper introduces a novel modelling framework for identifying dynamic models of systems that are under feedback control. These models are identified under closed-loop conditions and produce a joint representation that includes both the plant and controller models in state space form. The joint plant/controller model is identified using subspace model identification (SMI), which is followed by the separation of the plant model from the identified one. Compared to previous research, this work (i) proposes a new modelling framework for identifying closed-loop systems, (ii) introduces a generic structure to represent the controller and (iii) explains how that the new framework gives rise to a simplified determination of the plant models. In contrast, the use of the conventional modelling approach renders the separation of the plant model a difficult task. The benefits of using the new model method are demonstrated using a number of application studies.