Linear State Dynamics Research Articles

LINEAR STATE FEEDBACK DESIGN FOR SURGE CONTROL OF AXIAL FLOW COMPRESSOR DYNAMICS

Based on Moore and Greitzer's model [15], the surge behavior of axial flow compression system was found to be attributed to the appearance of Andronov-Hopf bifurcationin [12]. In this paper, we focus on the design of control law to prevent and/or delay the occurrence of the bifurcations, which will then suppress the surge behavior. Both types of linear dynamic state feedback and linear washout filter controllers are proposed. The proposed washout design does not require the explicit knowledge of the operating point and keep the original equilibrium characteristic while the dynamic feedback design will change the equilibrium operating points for pre-surge and post-surge regime. In addition, the washout type control scheme can relax the implementation of the control law in the practical applications. Numerical results demonstrate the effectiveness of the proposed designs and the superiority of the washout filter scheme.

On a Nonrenewable Resource Extraction Game Played by Asymmetric Firms

A differential game of extraction of a nonrenewable resource is taken into account, where two firms compete over time and their two terminal times of extraction are two different random variables. The winning firm will be the only one remaining in the game after the first one retires. We explicitly compute the Hamilton---Jacobi---Bellman equations of the model and solve them in an asymmetric game with logarithmic payoff structure and linear state dynamics.

On observation distributions for state space models of population survey data

1. State space models are starting to replace more simple time series models in analyses of temporal dynamics of populations that are not perfectly censused. By simultaneously modelling both the dynamics and the observations, consistent estimates of population dynamical parameters may be obtained. For many data sets, the distribution of observation errors is unknown and error models typically chosen in an ad-hoc manner. 2. To investigate the influence of the choice of observation error on inferences, we analyse the dynamics of a replicated time series of red kangaroo surveys using a state space model with linear state dynamics. Surveys were performed through aerial counts and Poisson, overdispersed Poisson, normal and log-normal distributions may all be adequate for modelling observation errors for the data. We fit each of these to the data and compare them using AIC. 3. The state space models were fitted with maximum likelihood methods using a recent importance sampling technique that relies on the Kalman filter. The method relaxes the assumption of Gaussian observation errors required by the basic Kalman filter. Matlab code for fitting linear state space models with Poisson observations is provided. 4. The ability of AIC to identify the correct observation model was investigated in a small simulation study. For the parameter values used in the study, without replicated observations, the correct observation distribution could sometimes be identified but model selection was prone to misclassification. On the other hand, when observations were replicated, the correct distribution could typically be identified. 5. Our results illustrate that inferences may differ markedly depending on the observation distributions used, suggesting that choosing an adequate observation model can be critical. Model selection and simulations show that for the models and parameter values in this study, a suitable observation model can typically be identified if observations are replicated. Model selection and replication of observations, therefore, provide a potential solution when the observation distribution is unknown.

Modeling hourly ozone concentration fields

This paper compares two methods built on a hierarchical Bayesian foundation and designed for modeling hourly ozone concentrations over the eastern United States. One, a dynamic linear state space model (DLM) that has been proposed earlier, lies in a very contemporary setting where two historical paths to temporal process models, the Kalman filter and the dynamic system with random perturbations, converge. The other, which we call the Bayesian spatial predictor (BSP), is a Bayesian alternative to the purely spatial method of kriging. The DLM as a dynamic system model has parameters that are states of the process which generate the ozone and change with time. More specifically, the model includes a time-varying site invariant mean field as well as time-varying coefficients for 24 and 12 hour diurnal cyclic components. The resulting model’s great flexibility comes at the cost of complexity, forcing the use of an MCMC approach and very time-consuming computations. Thus, the size of the DLM’s spatial domain of applicability has to be restricted and the number of monitoring sites that can be treated limited. The paper’s assessment of the DLM reveals other difficulties that point to the need to consider a less flexible competitor, a Bayesian spatial predictor (BSP). The two methods are compared in a variety of ways and overall conclusions given. In particular, the conclusions point to the BSP as the more practical alternative for spatial prediction.

Rheological Properties of Polypropylene/Attapulgite Nanocomposite

The rheological properties of nanocomposites consisting of organic modified attapulgite and polypropylene are investigated. The morphology of the nanocomposites was studied with scanning electron microscopy and X-ray diffraction. Linear dynamic viscoelasticity and steady state flow step of the nanocomposites are presented. For the nanocomposites, the deviation from linear behavior occurs earlier than pure polypropylene at the strain. The storage moduli, loss moduli and dynamic viscosities of the nanocomposites increase monotonically with organoclay concentration. However, the nanocomposites show greater shear thinning tendency than pure polypropylene because of the orientation of the organoclay fibers. Therefore, the nanocomposites have higher moduli but better processibility compared with pure polypropylene.

Dynamic Modeling with Structural Equations and Stochastic Differential Equations: Applications with the German Socio-economic Panel

The paper discusses an application of linear dynamic models to multi-wave longitudinal data. Starting from three-wave and four-wave simplex models using standard structural equations, linear dynamic state space models with stochastic differential equations are presented. The main differences between longitudinal structural equations (static view) and stochastic differential equations (dynamic view) are emphasized. Substantively, the models prove the relation, stability and change of two concepts in a period of 10 years: National Identity and Intention to stay in Germany. Data from a sample of migrant workers in Germany included in the German Socio-economic Panel (GSOEP) are used for the analyses. Results and further developments of dynamic models are discussed in the final section.

On‐line estimation and path planning for multiple vehicles in an uncertain environment

AbstractA unified approach to cooperative target tracking and path planning for multiple vehicles is presented. All vehicles, friendly and adversarial, are assumed to be aircraft. Unlike the typical target tracking problem that uses the linear state and nonlinear output dynamics, a set of aircraft nonlinear dynamics is used in this work. Target state information is estimated in order to integrate into a path planning framework. The objective is to fly from a start point to a goal in a highly dynamic, uncertain environment with multiple friendly and adversarial vehicles, without collision. The estimation architecture proposed is consistent with most path planning methods. Here, the path planning approach is based on evolutionary computation technique which is then combined with a nonlinear extended set membership filter in order to demonstrate a unified approach. A cooperative estimation approach among friendly vehicles is shown to improve speed and routing of the path. Copyright © 2004 John Wiley & Sons, Ltd.

Approximation of trajectories of non-linear systems by iterates of systems with linear state dynamics

A modification of an iterative Picard process is proposed to approximate the output of a non-linear system by a concatenation of systems with linear state dynamics and non-linear outputs. A local uniform convergence result is given. The motivating example is a non-linear system that arises in surface plasmon resonance experiments to determine protein–protein interaction constants. We show with simulations that for this example the approximants converge not only locally but over the full time interval of interest in the application.

An approach to discrete-time incentive feedback Stackelberg games

A substantial effort has been devoted to various incentive Stackelberg solution concepts. Most of these concepts work well in the sense that the leader can get his desired solution in the end. Yet, most incentive strategies developed thus far include either the follower's control, which may not be realistic in practice, or delays in the state, which makes stabilization more difficult to achieve. In this paper, we obtain the team-optimal state feedback Stackelberg strategy (with no delays) of an important class of discrete-time two-person nonzero-sum dynamic games characterized by linear state dynamics and quadratic cost functionals.

Real-Time Parameter Estimation in the Frequency Domain

A method for real-time estimation of parameters in a linear dynamic state space model was developed and studied. The application is aircraft dynamic model parameter estimation from measured data in flight for indirect adaptive or reconfigurable control. Equation error in the frequency domain was used with a recursive Fourier transform for the real-time data analysis. Linear and nonlinear simulation examples and flight test data from the F-18 High Alpha Research Vehicle HARV) were used to demonstrate that the technique produces accurate model parameter estimates with appropriate error bounds. Parameter estimates converged in less than 1 cycle of the dominant dynamic mode natural frequencies, using control surface inputs measured in flight during ordinary piloted maneuvers. The real-time parameter estimation method has low computational requirements, and could be implemented aboard an aircraft in real time.

Robust state feedback control of LTV systems: nonlinear is better than linear

Examples of linear control systems with fast time-varying uncertain coefficients are given, which can be stabilized by a nonlinear memoryless state feedback, but cannot be stabilized by a linear time-invariant dynamic state feedback. By means of one of these examples the authors show that the closed loop quadratic stability margin may be infinitely smaller than the actual stability margin.

Improving stability margins via dynamic-state feedback for systems with constant uncertainty

It is well known that if a linear system with time-varying uncertainty in the system matrix and/or the input connection matrix is quadratically stabilizable by linear dynamic state feedback, then it is also quadratically stabilizable by linear static state feedback. In this paper, we provide an example of a system with unknown constant real uncertainty which is stabilizable by a linear, dynamic-state feedback controller but not by a static-state feedback controller.

Robust H∞ state feedback control for linear systems with state delay and parameter uncertainty

This paper addresses the problem of robust H ∞ state feedback control design for linear systems with state delay and parameter uncertainty. Based on a positive-definite solution of a modified algebraic Riccati equation, the proposed robust H ∞ static state feedback controller stabilizes uncertain time-delay systems independently of delays, and guarantees an H ∞ norm bound constraint on disturbance attenuation for all admissible uncertainties. Furthermore, it is shown that no advantage can be obtained from the use of a proper linear dynamic state feedback compared with the use of static state feedback. Illustrative examples are given to demonstrate the applicability of the proposed approach.

Combined Deterministic and Stochastic System Identification and Realization: DSR - A Subspace Approach Based on Observations

A numerically stable and general algorithm for identification and realization of a complete dynamic linear state space model, including the system order, for combined deterministic and stochastic systems from time series is presented. A special property of this algorithm is that the innovations covariance matrix and the Markov parameters for the stochastic sub-system are determined directly from a projection of known data matrices, without e.g. recursions of non-linear matrix Riccatti equations. A realization of the Kalman filter gain matrix is determined from the estimated extended observability matrix and the Markov parameters. Monte Carlo simulations are used to analyze the statistical properties of the algorithm as well as comparing with existing algorithms.

On positively invariant sets for discrete-time linear systems with disturbance: an application of maximal disturbance sets

This note concerns the investigation of maximal sets, with respect to which a set is positively invariant, and, given a set a, some properties of /spl Delta/-p.i. sets are analyzed. The approach uses one step of the backward recursion introduced in d'Alessandro and De Santis (1991), where feasibility and optimality conditions for linear dynamical input and/or state constrained systems were analyzed. In Section II, the author gives the main definitions and some basic results. In the subsequent Section III some properties of /spl Delta/-p.i. sets are investigated. The particular case of polyhedral sets is analyzed in Section IV. In Section V, the author shows an application to linear systems with parametric uncertainties in the dynamic matrix A. The author develops a technique to determine a set in the parameters space such that, for any value of the parameter in the set, a given set in the state space is positively invariant. In the last section a numerical example is developed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nonlinear state feedback for ℓ 1 optimal control

This paper considers ℓ 1 optimal control problems with full state feedback. In contrast to H ∞ optimal control, previous work has shown that linear ℓ 1 optimal controllers can be dynamic and arbitrarily high order. However, this paper shows that continuous memoryless nonlinear state feedback perform as well as dynamic linear state feedback. The derivation, which is nonconstructive, relies on concepts from viability theory.

On the Hankel-norm approximation of upper-triangular operators and matrices

A matrix T = Tij ∞=-∞ , which consists of a doubly indexed collection Tij of operators, is said to be upper when Tij = 0 for i > j. We consider the case where the Tij are finite matrices and the operator T is bounded, and such that the Tij are generated by a strictly stable, non-stationary but linear dynamical state space model or colligation. For such a model, we consider model reduction, which is a procedure to obtain optimal approximating models of lower system order. Our approximation theory uses a norm which generalizes the Hankel norm of classical stationary linear dynamical systems. We obtain a parametrization of all solutions of the model order reduction problem in terms of a fractional representation based on a non-stationary J-unitary operator constructed from the data. In addition, we derive a state space model for the so-called maximum entropy approximant. In the stationary case, the problem was solved by Adamyan, Arov and Krein in their paper on Schur-Takagi interpolation. Our approach extends that theory to cover general, non-Toeplitz upper operators.

New approximation for the partially observed jump linear quadratic problem

We consider the Jump Linear Quadratic Problem where linear state dynamics are made contingent upon the markovian transition of a regime variable. It is desired to regulate the state while minimizing a quadratic performance index. In the case of partial observations the exact solution has proved to be elusive and we present a new approximation based on the optimal solution of an average version of the original problem.

Numerical solution for a class of open-loop differential games via the shifted Chebyshev polynomial approximation

Shifted Chebyshev polynomial functions are used to solve a class of open-loop differential games described by linear state dynamics and quadratic objective functions. Using linear transformation of the variables, the necessary conditions that the open-loop differential games must satisfy are reduced to functional differential equations. The solutions of the functional equations are then obtained in a series of shifted Chebyshev polynomial approximations. The computational procedure is simple and available for computer implementation. It is also shown that the computational results provide satisfactory accuracy compared with those of the exact solutions.

Closed-loop strategies in continuous dynamic game with multi-level of hierarchy

The derivation is given of the closed-loop Stackelberg strategy for a class of continuous three-player non-zero-sum differential games using the idea of the team optimal Stackelberg strategy. The game systems are described by linear state dynamics and quadratic objective functions. First, a definition of the three-player hierarchical equilibrium is given, then a general theorem which was developed by Basar (1981) is examined and applied to the game under consideration to deduce some sufficient conditions for the solution of the game to exist. A simple example is given.