Linear Parameter Systems Research Articles

Smooth representation of a parametric polyhedral convex set with application to sensitivity in optimization

We show in this paper that if a polyhedral convex set is defined by a parametric linear system with smooth entries, then it possesses local smooth representation almost everywhere. This result is then applied to study the differentiability of the solutions and the marginal functions of several classes of parametric optimization problems.

FIRST HARMONIC ANALYSIS OF PLANAR LINEAR SYSTEMS WITH SINGLE SATURATED FEEDBACK

A first harmonic approach (describing function method) is used here to analyze the dependence of periodic orbits on the control parameters of planar linear systems with single saturated feedback. It is shown that, if the open-loop system has at least one unstable eigenvalue, periodic orbits converge monotonically to an unstable equilibrium point as the control gains go to infinity.

Turbulence and separation induced pressure fluctuations on a finite circular cylinder — application of a linear unsteady strip theory

Abstract In a turbulent stream, the fluctuating pressures on bluff bodies are mainly caused by the oncoming turbulence. Further unsteadiness is engendered by the wake turbulence and by the vortex shedding. Although the gross properties of the flow around a body and in particular the separated flow are affected by the intensity and the scale of the incident turbulence, these sources of pressure fluctuations are thought to be statistically independent. On this basis, the physical system of a turbulent flow around a body is approximated by an artificial black box model. This model corresponds to a constant parameter linear system with multiple stationary random inputs, i.e. turbulent velocity fluctuations far upstream the body. The model output is equal to an instantaneous surface pressure, which is subdivided into a turbulence induced contribution and an extraneous noise term to account for the residual fluctuations. By means of velocity/pressure cross spectral densities (i) significant velocity fluctuations, (ii) the associated admittance functions and (iii) the conditioned and residual pressure spectra are identified experimentally for the flow around a surface mounted circular cylinder in an atmospheric boundary layer flow. In a first approximation, the longitudinal u′(t) and lateral velocity fluctuation ν′(t) of a single stream filament are thought to be the only decisive inputs to the multivariate system (linear unsteady strip theory). The latter induces maximum spectral amplitudes in the mid-frequency range, whereas the longitudinal component achieves maximum effectiveness at the lowest frequencies. However, significant deviations between the (u, ν)-conditioned and the factual pressure spectrum remain. Depending on the circumferential angle and in a certain frequency range, vortex shedding contributes up to 45% to the total pressure RMS-values.

Eigenstructure Perturbation for Robust Stability Analysis of Linear State Space Models

This paper addresses the nature of eigenavalue and eigenvector perutrbations in the robust stability analysis of linear interval parameter systems. The paper uses the fact that the robust stability problem can be converted to a robust non-singularity problem involving the original matrix and the associated bialterrate sum matrix (which we label as the 'tilde' matrix). The special nature of the 'tilde' matrix is exploited to arrive at the main result of the paper which involves the special nature of the 'eigenstructure behavior' in the 'tilde' matrix space. This result is deemed to pave the way for a possible extreme point result for the problem of checking the robust stability of interval matrices.

New stability/performance results for singularly perturbed systems

The guardian map theory of generalized stability of parametrized linear time-invariant systems is used to prove new results on stability/performance of linear time-invariant singularly perturbed systems with an exogenous parameter, i.e. systems that contain, in addition to a singular parameter ϵ, another uncertain parameter μ. The results give necessary and sufficient conditions for generalized stability for all sufficiently small values of ϵ and all values of μ in a given interval [ μ 1, μ 2]. In addition, explicit expressions for the largest upper bound ϵ ∗ on ϵ for guaranteed stability and performance are given and the cases leading to finite or infinite ϵ ∗ are clearly delineated. Thus the results represent a significant addition to the classical Klimushev-Krasovskii theorem, while at the same time providing closed-form expressions for the maximum parameter range for stability and performance.

Kalman filter with hypothesis testing: A tool for estimating uncertain parameters

In this paper we present a simple and practical algorithm for the estimation of uncertain parameters of linear systems. The uncertainty is twofold, involving random observation noise, and possible jumps in the parameter values. The jumps may occur at unknown points in time, and are of unknown magnitudes and directions. The algorithm is based on the Kalman filter, with a single-sample hypothesis test, which is used to employ a three-state decision rule (yes, no, maybe). The "maybe" choice invokes a fading memory Kalman filter. The overall algorithm contains the constant parameter filter, fading memory filter, and the set of tests and rules that enable it to switch back and forth between the two filters. Application examples are presented.

System identification via singular value decomposition

The problem of identifying linear time invariant system parameters from their truncated impulse response data is studied. The authors analyse the algorithm originally proposed by Kung, and later modified by others, which is based on the singular value decomposition of the Hankel matrix. They propose a modified algorithm which generates more accurate results.

Linear systems identification from random threshold binary data

A new identification problem of estimating parameters of linear dynamic systems from random threshold binary observations of its output and input is stated. The only available data are collected as a result of checking whether a signal reached a randomly specified threshold at a randomly chosen instant of time. The proposed estimation algorithm is based on the celebrated von Neumann theorem, which was earlier used mainly for generating random numbers. Strong consistency of parameters estimate from low-cost output binary observations is proved, assuming deterministic input signal of a finite duration. Possibilities of relaxing the assumption used in the theoretical part of the paper are considered by means of simulations.

Circulatory control and the supercontrollers

NEUROHUMORAL SYSTEMS AS SUPERCONTROLLERS: The brain and closely linked hormone systems play a crucial part in short- and long-term cardiovascular control and have many features of adaptive control systems. The cardiovascular control system is a multivariate system, while changes in environmental conditions often result in alterations in system parameters and other non-linearities, in contrast to the fixed parameters of linear control systems. In blood pressure control these features are exemplified by diurnal circadian fluctuations, alterations in lifestyle and psychosocial stress. Because the neurohumoral controllers are involved in virtually all aspects of homeostasis, they can be regarded as supercontrollers. THE CIRCULATORY SYSTEM AND THE BRAIN: Analysis in conscious animals of the effects of circulatory disturbances suggests that the central nervous system integrates information from multiple sources of afferents. Integration of the information associated with most reflex and behavioural disturbances is mediated by many neuron groups at different levels of the neuraxis, including suprapontine brain regions. The disturbances considered include baroreflexes in intact animals, some central actions of alpha-methyldopa and reflex responses to hypoxia and haemorrhage. The operations involve the brain in comparisons of the relative magnitude of different inputs, while the occurrences of non-linear changes in baroreflex properties signify alterations in the parameters of the controller. NEUROHUMORAL MECHANISMS AND CARDIOVASCULAR DEVELOPMENT: Neurohumoral mechanisms also play a key role in cardiovascular development. Increased sympathetic activity early in life causes hypertension in spontaneously hypertensive rats (SHR) and accounts for the differences in blood pressure and structural variables from corresponding values in Wistar-Kyoto (WKY) rats. In contrast, the renin-angiotensin system affects early cardiovascular development in the same way in each strain, so that it is unlikely to be a cause of hypertension in SHR. However, after drug withdrawal following treatment of young rats with the angiotensin converting enzyme inhibitor enalapril, there were between-strain differences in late cardiovascular development. Late development is relatively small in SHR compared to WKY rats, which contributes to the long-term attenuation of hypertension in SHR and to the normalization of blood pressure in WKY rats.

On the Adaptive Control of Jump Parameter Systems via Nonlinear Filtering

In this paper we first present an error analysis for the process of estimates generated by the Wonham filter when it is used for the estimation of the (finite set-valued) jump-Markov parameters of a random parameter linear stochastic system and further give bounds on certain functions of these estimates. We then consider a certainty equivalence adaptive linear-quadratic Gaussian feedback control law using the estimates generated by the nonlinear filter and demonstrate the global existence of solutions to the resulting closed-loop system. A stochastic Lyapunov analysis establishes that the certainty equivalence law stabilizes the Markov jump parameter linear system in the mean square average sense. The conditions for this result are that certain products of (i) the parameter process jump rate and (ii) the solution of the control Riccati equation and its second derivatives should be less than certain given bounds. An example is given where the controlled linear system has state dimension 2. Finally, the stabilizing properties of certainty equivalence laws which depend on (i) the maximum likelihood estimate of the parameter value and (ii) a modified version of this estimate are established under certain conditions.

Parallel computation of a Krylov matrix for a sparse and structured input

As in our previous work, we reduce parallel computation of a Krylov matrix to solving a parametrized linear system of equations. This time we show that such a method is effective in the cases of banded matrices, sparse and structured matrices and triangular matrices.

Parameter estimation of continuous linear systems using functional approximation

A novel method to estimate unknown system parameters for SISO linear time invariant systems using functional approximation is proposed. It is shown that the differential equation of a linear time invariant system can be approximated by a finite Taylor series and the unknown system parameters are then estimated by using the information from the Laplace transforms of the input and the output.

Estimation of complex Fourier coefficients

Estimating complex Fourier coefficients of periodic functions from additively error disturbed observations is discussed. This is the first step in most methods for estimating the parameters of linear dynamic systems from noise disturbed responses to periodic perturbation signals. In this type of estimation problem the period is known. Estimating complex Fourier coefficients is also the first step in procedures for reconstructing periodic images from noisy observations. In these problems the period is usually unknown and the Fourier coefficients and the period have to be estimated jointly. Complex valued estimators are proposed for both types of problem. The real and imaginary parts of the unknown complex valued parameters are not treated as separate real parameters. Statistical and numerical properties of the proposed estimators are described.

Calculation of Markov parameters from the transfer function matrix of MIMO systems

This paper presents a simple computational algorithm for determining the Markov parameters of a multivariable linear system specified either by a transfer function matrix or a right matrix fraction description. In addition, the procedure gives the observability/controllability index and the order of the minimal realization. The dual version of the algorithm can be used when the transfer function matrix is given by a left matrix fraction description. Computational examples are given to show the feasibility of the suggested procedure

An algorithm for calculating Markov parameters from transfer function matrices

A simple computational algorithm is presented for determining the Markov parameters of a multivariable linear system specified by a transfer function matrix or right matrix fraction description. In addition, the procedure gives the observability/controllability index and the order of the minimal realization. The dual version of the algorithm can be used when the transfer function matrix is given by a left matrix fraction description. Computational examples are given to show the feasibility of the suggested procedure.

Combined instrumental variable and subspace fitting approach to parameter estimation of noisy input-output systems

The paper considers the problem of estimating the parameters of linear discrete-time systems from noise-corrupted input-output measurements, under fairly general conditions: the output and input noises may be auto-correlated and they may be cross-correlated as well. By using the instrumental-variable (IV) principle a covariance matrix is obtained, the singular vectors of which bear complete information on the parameters of the system under study. A weighted subspace fitting (WSF) procedure is then employed on the sample singular vectors to derive estimates of the parameters of the system. The combined IV-WSF method proposed in the present paper is noniterative and simple to use. Its large-sample statistical performance is analyzed in detail and the theoretical results so obtained are used to predict the behavior of the method in samples with practical lengths. Several numerical examples are included to show the agreement between the theoretically predicted and the empirically observed performances.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stability of Recursive Stochastic Tracking Algorithms

First, the paper gives a stability study for the random linear equation $X_{n+1}=(I-A_{n})x_n$. It is shown that for a quite general class of random matrices $\{A_n\}$ of interest, the stability of such a vector equation can be guaranteed by that of a corresponding scalar linear equation, for which various results are given without requiring stationary or mixing conditions. Then, these results are applied to the main topic of the paper, i.e., to the estimation of time varying parameters in linear stochastic systems, giving a unified stability condition for various tracking algorithms including the standard Kalman filter, least mean squares, and least squares with forgetting factor.

Recursive methods for parameter estimation of linear lumped systems using Hermite polynomials

A recursive method for estimation of the parameters of a linear lumped parameter system is developed. An example is given to illustrate the algorithm, and results of the estimation in the presence of white noise are also given.

A decoupled approach for optimal estimation of transfer function parameters from input-output data

A theoretical and algorithmic framework is proposed for optimal identification of rational transfer function parameters of discrete-time linear systems from input-output (IO) data. The nonlinear criterion is theoretically decoupled into a purely linear problem for estimating the optimal numerator and a nonlinear problem for the optimal denominator. The proposed decoupled approach has reduced computational requirements when compared with existing algorithms that estimate the parameters simultaneously.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Identification of linear systems with noisy input using input-output cumulants

Abstract An identification method is presented for estimating the parameters of a discrete-time linear dynamic system excited by non-gaussian input signals using the fourth-order cumulants of the input and output signals, both of which are contaminated by additive gaussian noise. Two types of estimators of the fourth-order cumulants of the input and output signals are proposed for this method. The first is conventional. The second, which is new allows us only to have a recursive algorithm for computing the parameter estimators. The parameter estimators obtained by this algorithm are shown to be strongly consistent under certain weak conditions. Simulation examples are included to demonstrate the effectiveness of the proposed method.