Linear State Equations Research Articles

Application of the LMI Approach in the Robust Force Control of Servo-Hydraulic Actuator with Parametric Uncertainties

Linear matrix inequalities (LMI) method is proposed to design the robust controller for the servo-hydraulic actuator with parametric uncertainties. The pretreatments are adopted to convert the nonlinear dynamic models into linear state equations using the linear fractional transformation (LFT) approach, which facilitates conveniently utilizing the LMI method to calculate the state feedback controller. The supervising parameters, including the system output and special derivative output generated from the uncertain items, are proposed to model a state matrix equation for representing the dynamic system with the parameter variations and disturbances. LMI control base on the H∞ control schematic is finally employed to carry out the state feedback controller for the servo-hydraulic actuator with parametric uncertainties. The results demonstrate the efficiency of dynamical performance with small settling time and overshoot compared with the quantitative feedback theory (QFT) approach.

Stochastic Controllability of Systems with Multiple Delays in Control

Stochastic Controllability of Systems with Multiple Delays in ControlFinite-dimensional stationary dynamic control systems described by linear stochastic ordinary differential state equations with multiple point delays in control are considered. Using the notation, theorems and methods used for deterministic controllability problems for linear dynamic systems with delays in control as well as necessary and sufficient conditions for various kinds of stochastic relative controllability in a given time interval are formulated and proved. It will be proved that, under suitable assumptions, relative controllability of an associated deterministic linear dynamic system is equivalent to stochastic relative exact controllability and stochastic relative approximate controllability of the original linear stochastic dynamic system. As a special case, relative stochastic controllability of dynamic systems with a single point delay is also considered. Some remarks and comments on the existing results for stochastic controllability of linear dynamic systems are also presented.

A convenient approach for estimating time-dependent structural reliability in the load space

A procedure, formulated in the space of the load processes, is described for estimating the reliability of structures subject to multi-parameter time-varying loading. For most realistic reliability problems the load process space is of low order. As a result, the required multidimensional integration is significantly simplified. The proposed approach also has well defined steps. As a result, there is increased transparency and reduced problems of integration instability and non-convergence. Both loads and resistances are described in terms of random variable parameters and time dependent structural resistances can be considered. Numerical examples are given to illustrate the proposed method. Example applications are given for a fixed base rigid-plastic portal frame subjected to time dependent loads and resistances. Linear and non-linear limit state equations and Normal and non-Normal distribution of the random variables are considered and compared, in some cases, to the results evaluated using Monte Carlo simulation.

Infinite Horizon and Ergodic Optimal Quadratic Control for an Affine Equation with Stochastic Coefficients

We study stochastic optimal control problems with a quadratic cost and linear state equation that involves stochastic coefficients and control dependent noise and, moreover, is perturbed by an affine term. Both the infinite horizon case and the ergodic case are treated. To this purpose we introduce a backward stochastic Riccati equation and a dual backward stochastic equation, both considered in the whole time line. Besides some stabilizability conditions we prove the existence of a solution for the two previous equations defined as limit of suitable finite horizon approximating problems. This allows us to perform the synthesis of the optimal control.

Optimal Filtering for Incompletely Measured Polynomial Systems with Multiplicative Noise

In this paper, the optimal filtering problem for polynomial system states with polynomial multiplicative noise over linear observations with an arbitrary, not necessarily invertible, observation matrix is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. Thus, the Ito differentials for the optimal estimate and error variance corresponding to the stated filtering problem are first derived. A transformation of the observation equation is introduced to reduce the original problem to the previously solved one with an invertible observation matrix. The procedure for obtaining a closed system of the filtering equations for any polynomial state with polynomial multiplicative noise over linear observations is then established, which yields the explicit closed form of the filtering equations in the particular cases of linear and bilinear state equations. In an example, the performance of the designed optimal filter is verified against those of the optimal filter for a quadratic state with a state-independent noise and a conventional extended Kalman---Bucy filter.

Stochastic controllability and minimum energy control of systems with multiple delays in control

In the paper finite-dimensional stationary dynamical control systems described by linear stochastic ordinary differential state equations with multiple point delays in the control are considered. Using notations, theorems and methods taken directly from deterministic controllability problems for linear dynamical systems with delays in control, necessary and sufficient conditions for different kinds of stochastic relative controllability in a given time interval are formulated and proved. It will be proved that under suitable assumptions relative controllability of a deterministic linear associated dynamical system is equivalent to stochastic relative exact controllability and stochastic relative approximate controllability of the original linear stochastic dynamical system. As a special case relative stochastic controllability of dynamical systems with single point delay is also considered. Some remarks and comments on the existing results for stochastic controllability of linear dynamical systems are also presented. In the second part of the paper minimum energy control problem is considered. Under the assumption that system is stochastically relatively exactly controllable analytical formula for minimum energy control is given.

Optimal design under the one-dimensional wave equation

An optimal design problem governed by the wave equation is examined in detail. Specifically, we seek the time-dependent optimal layout of two isotropic materials on a 1-d domain by minimizing a functional depending quadratically on the gradient of the state with coefficients that may depend on space, time and design. As it is typical in this kind of problems, they are ill-posed in the sense that there is not an optimal design. We therefore examine relaxation by using the representation of two-dimensional ($(x,t)\\in\\bkR^{2}$) divergence free vector fields as rotated gradients. By means of gradient Young measures, we transform the original optimal design problem into a non-convex vector variational problem, for which we can compute an explicit form of the \`\`\\emph{constrained quasiconvexification }" of the cost density. Moreover, this quasiconvexification is recovered by first or second-order laminates which give us the optimal microstructure at every point. Finally, we analyze the relaxed problem and some numerical experiments are performed. The perspective is similar to the one developed in previous papers for linear elliptic state equations. The novelty here lies in the state equation (the wave equation), and our contribution consists in understanding the differences with respect to elliptic cases.

A generalized linear equation for non-linear diffusion in external fields and non-ideal systems

Based on macroscopic thermodynamic analysis, we have established a generalized linear theory for non-linear diffusion in external fields and non-ideal systems, which was classically described by the Fokker–Planck equation (or the Smoluchowski equation) and the non-linear Fickian equation, respectively. The new theory includes three basic equations expressed in ‘apparent variables’ as defined in this paper: (i) a generalized linear flux equation for non-linear diffusion; (ii) an apparent mass conservation equation and (iii) a generalized linear non-steady state equation for non-linear diffusion. Our analysis shows that (i) all of the existing linear and non-linear equations are the special cases of the new non-steady state general diffusion equation. It was also demonstrated that the general equation of the non-steady state is equivalent to the Fokker–Planck equation; (ii) coupling diffusion with multiple driving forces can be unified to a single force: the apparent concentration gradient; (iii) the exact relationship between diffusion coefficient and concentration in the non-linear Fickian equation under non-ideal conditions could be established and (iv) the potential energy is conservative in a diffusion process. An application of the generalized linear equation showed that the solution is simple. For the first time, an analytic solution of the Smoluchowski equation with a time-dependent potential in the algebraic form was obtained.

Stochastic Controllability of Linear Systems With State Delays

Stochastic Controllability of Linear Systems With State DelaysA class of finite-dimensional stationary dynamic control systems described by linear stochastic ordinary differential state equations with a single point delay in the state variables is considered. Using a theorem and methods adopted directly from deterministic controllability problems, necessary and sufficient conditions for various kinds of stochastic relative controllability are formulated and proved. It will be demonstrated that under suitable assumptions the relative controllability of an associated deterministic linear dynamic system is equivalent to the stochastic relative exact controllability and the stochastic relative approximate controllability of the original linear stochastic dynamic system. Some remarks and comments on the existing results for the controllability of linear dynamic systems with delays are also presented. Finally, a minimum energy control problem for a stochastic dynamic system is formulated and solved.

GMV control of non-linear continuous-time systems including common delays and state-space models

A non-linear generalized minimum variance control law is proposed for the control of non-linear continuous-time multivariable systems with common delays on input and output channels. The quadratic cost index involves both error and control signal costing terms. The solution for the control law is obtained using a non-linear operator representation of the plant and a linear state-equation model for the disturbance and reference models. The reference and disturbance models are represented by linear subsystems. However, the plant model can be in a very general non-linear operator form, which could involve state-space, transfer operators or non-linear function look up tables. The structure of the system and criterion is chosen so that a simple controller structure and solution is obtained. The controller obtained is simple to implement, particularly in one form, which might be considered to be a state-space version of a non-linear Smith predictor. The results are related to those for discrete-time systems but the presence of the transport delay terms complicates the solution rather more in the continuous-time case.

Optimal filtering for polynomial system states with polynomial multiplicative noise

AbstractIn this paper, the optimal filtering problem for polynomial system states with polynomial multiplicative noise over linear observations is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. As a result, the Ito differentials for the optimal estimate and error variance corresponding to the stated filtering problem are first derived. The procedure for obtaining a closed system of the filtering equations for any polynomial state with polynomial multiplicative noise over linear observations is then established, which yields the explicit closed form of the filtering equations in the particular cases of a linear state equation with linear multiplicative noise and a bilinear state equation with bilinear multiplicative noise. In the example, performance of the designed optimal filter is verified for a quadratic state with a quadratic multiplicative noise over linear observations against the optimal filter for a quadratic state with a state‐independent noise and a conventional extended Kalman–Bucy filter. Copyright © 2006 John Wiley & Sons, Ltd.

GENETIC APPROACH FOR A LOCALISATION PROBLEM BASED UPON PARTICLE FILTERS

Localisation, i.e., estimating a robot pose relative to a map of an environment, is one of the most important problems in mobile robotics. The literature offers several possible approaches to deal with such task, and optimal algorithms can be devised relying on particular constraints (linear state equations, Gaussian posterior density, etc.). Here, we propose a preliminary study for an algorithm able to approximate a large number of probability distributions when a map of the environment is available. This work improves the particle filters strategies presented in literature, reducing the number of particles needed to solve the localisation problem. The algorithm relies upon a suitable clustering of the particle set and a genetic approach for the resampling step.

Relationship between (2 + 1) and (3 + 1)-Friedmann–Robertson–Walker cosmologies; linear, non-linear, and polytropic state equations

It is shown that Friedmann–Robertson–Walker (FRW) cosmological models coupled to a single scalar field and to a perfect fluid fitting a wide class of matter perfect fluid state equations, determined in (3+1) dimensional gravity can be related to their (2+1) cosmological counterparts, and vice-versa, by using simple algebraic rules relating gravitational constants, state parameters, perfect fluid and scalar field characteristics. It should be pointed out that the demonstration of these relations for the scalar fields and potentials does not require the fulfilment of any state equation for the scalar field energy density and pressure. As far as to the perfect fluid is concerned, one has to demand the fulfilment of state equations of the form p+ρ = γ f(ρ). If the considered cosmologies contain the inflation field alone, then any (3+1) scalar field cosmology possesses a (2+1) counterpart, and vice-versa. Various families of solutions are derived, and we exhibited their correspondence; for instance, solutions for pure matter perfect fluids and single scalar field fulfilling linear state equations, solutions for scalar fields coupled to matter perfect fluids, a general class of solutions for scalar fields subjected to a state equation of the form p φ+ρφ = γρφ β are reported, in particular Barrow–Saich, and Barrow–Burd–Lancaster–Madsen solutions are exhibited explicitly, and finally perfect fluid solutions for polytropic state equations are given.

Bond graph models of structured parameter uncertainties

Taking into account uncertainties in model parameter values is a crucial point for studying robustness in modeling and in control. This paper proposes to construct in a systematic and graphical procedure two forms of the linear state equation usually found in the literature for dealing with the robustness problem in the case of structured uncertainties on parameters. It is shown how to model uncertainties in a bond graph approach, in the case of additive or multiplicative parametric variations.

Transfer Alignment Error Compensator Design Based on Robust State Estimation

This paper examines the transfer alignment problem of the StrapDown Inertial Navigation System (SDINS), which is subject to the ship’s roll and pitch. Major error sources for velocity and attitude matching are lever arm effect, measurement time delay and ship-body flexure. To reduce these alignment errors, an error compensation method based on state augmentation and robust state estimation is devised. A linearized error model for the velocity and attitude matching transfer alignment system is derived first by linearizing the nonlinear measurement equation with respect to its time delay and dominant Y-axis flexure, and by augmenting the delay state and flexure state into conventional linear state equations. Then an H∞ filter is introduced to account for modeling uncertainties of time delay and the ship-body flexure. The simulation results show that this method considerably decreases azimuth alignment errors considerably.

Boundary control problems with convex cost and dynamic programming in infinite dimension part II: Existence for HJB

This is the second of two papers on boundary optimal control problems with linear state equation and convex cost arising from boundary control of PDEs and the the associated Hamilton--Jacobi--Bellman equation. In the first paper we studied necessary and sufficient conditions of optimality (Pontryagin Maximum Principle). In this second paper we will apply Dynamic Programming to show that the value function of the problem is a solution of an integral version of the HJB equation, and moreover that it is the pointwise limit of classical solutions of approximating equations.

Dynamic analysis of neural encoding by point process adaptive filtering.

Neural receptive fields are dynamic in that with experience, neurons change their spiking responses to relevant stimuli. To understand how neural systems adapt their representations of biological information, analyses of receptive field plasticity from experimental measurements are crucial. Adaptive signal processing, the well-established engineering discipline for characterizing the temporal evolution of system parameters, suggests a framework for studying the plasticity of receptive fields. We use the Bayes' rule Chapman-Kolmogorov paradigm with a linear state equation and point process observation models to derive adaptive filters appropriate for estimation from neural spike trains. We derive point process filter analogues of the Kalman filter, recursive least squares, and steepest-descent algorithms and describe the properties of these new filters. We illustrate our algorithms in two simulated data examples. The first is a study of slow and rapid evolution of spatial receptive fields in hippocampal neurons. The second is an adaptive decoding study in which a signal is decoded from ensemble neural spiking activity as the receptive fields of the neurons in the ensemble evolve. Our results provide a paradigm for adaptive estimation for point process observations and suggest a practical approach for constructing filtering algorithms to track neural receptive field dynamics on a millisecond timescale.

Boundary-control problems with convex cost and dynamic programming in infinite dimension. I. The maximum principle

This is the first of two papers on boundary optimal control problems with linear state equation and convex cost arising from boundary control of PDEs and the associated Hamilton--Jacobi--Bellman equation. In this paper we study necessary and sufficient conditions of optimality (Pontryagin maximum principle), and study the properties of a family of approximating problems that will be useful both in this paper and in the sequel. In the second paper we will apply dynamic programming to show that the value function of the problem is a solution of an integral version of the HJB equation, and moreover that it is the pointwise limit of classical solutions of approximating equations.

State Modeling and Estimation of Controllable Noisy Markov Chains

A filtration algorithm of current states of a discrete sequence, optimal by the criterion of the minimum root-mean-square estimation error (MRMSEE), based on a model of linear state and observation equations, introduced for the Markov discrete sequence (Markov chain) was synthesized.

Modelling growth of drug resistant cancer populations as the system with positive feedback

We are concerned with dynamical properties of models of emergence of resistance of cancer cells to chemotherapy, based on recent progress in molecular biology. The model of the process has a form of the infinite system of linear and bilinear state equations. We show that, it can be decomposed into two parts: the first one is finite dimensional bilinear, and the second one is infinite dimensional linear, coupled by the positive feedback. Then, we use some results of the theory of feedback systems to study the asymptotic properties of the process.