Linear Parameter Systems Research Articles

Simultaneous control of linear systems by state feedback

In this paper, a new method for finding a state feedback matrix in order to control simultaneously a collection of linear systems (of the same size) by using similarity operations is presented. For stabilization of all the systems, it is necessary that the eigenvalues of the closed-loop systems lie inside a specified region in the left hand side of the complex plane. This aim is achieved by solving linear and nonlinear parametric systems of equations using nonlinear programming. The presented method is implemented in two examples and the results are verified in view of the norm of the state feedback matrix and stabilizability.

Identification of dynamical systems with fractional derivative damping models using inverse sensitivity analysis

The problem of identifying parameters of time invariant linear dynamical systems with fractional derivative damping models, based on a spatially incomplete set of measured frequency response functions and experimentally determined eigensolutions, is considered. Methods based on inverse sensitivity analysis of damped eigensolutions and frequency response functions are developed. It is shown that the eigensensitivity method requires the development of derivatives of solutions of an asymmetric generalized eigenvalue problem. Both the first and second order inverse sensitivity analyses are considered. The study demonstrates the successful performance of the identification algorithms developed based on synthetic data on one, two and a 33 degrees of freedom vibrating systems with fractional dampers. Limited studies have also been conducted by combining finite element modeling with experimental data on accelerances measured in laboratory conditions on a system consisting of two steel beams rigidly joined together by a rubber hose. The method based on sensitivity of frequency response functions is shown to be more efficient than the eigensensitivity based method in identifying system parameters, especially for large scale systems.

A global optimisation method for computing interval hull solution for parametric linear systems

An algorithm for computing the interval solution for a parametric interval linear system is presented. The basic idea behind the research is to combine an interval global optimisation method with the Direct Method for Checking the Monotonicity (MCM) of the parametric solution. The MCM is used to perform the monotonicity test to speed up the convergence of the global optimisation. Other acceleration techniques such as subdivision direction selection rules, multisection and the midpoint test are involved as well and checked for usefulness. By using the proposed algorithm, several examples of parametric linear systems are solved. The research proves that the proposed monotonicity test is crucial for the convergence of the interval global optimisation used for computing the interval hull for parametric solution sets, whereas other accelerating techniques are not relevant. The presented algorithm can be useful for solving real-life problems concerning structure mechanics.

Stability analysis for point delay fractional description models via linear matrix inequalities

This paper is devoted to linear parameter systems under linear fractional representations (LFR) of parameter-dependent nonlinear systems with real–rational nonlinearities and point-delayed dynamics. The robust global asymptotic stability of the system either independent of or dependent on the delay sizes is investigated. The associate matrix inequalities are related to the time-derivatives of appropriate Lyapunov functions at all the vertices of the polytope which contains the parameterized uncertainties.

A two-step method to identify parameters of piecewise linear systems

Based on the direct parameter estimation method and the Legendre series approximation, a two-step method is presented to estimate by measured data the physical parameters, i.e., mass, damping, stiffness and knot, of nonlinear systems with symmetrical piecewise linear restoring force. At first, the piecewise linear restoring force of the system is approximated by Legendre series. A least squares direct parameter estimation method is adopted to identify the mass and damping parameters of the system and the corresponding coefficients of the Legendre series. Secondly, the physical parameters of the piecewise linear restoring force, i.e., the primary and the secondary stiffness and the knots, are extracted from the estimated Legendre coefficients. Neither iteration nor search process is needed in this method. The validity of the method is demonstrated by numerical simulation on a single-degree-of-freedom and a three-degree-of-freedom system with both noise-free data and noise-polluted data.

Impulsive synchronization of chaotic Lur'e systems via partial states

In this Letter, the problem of impulsive synchronization of two identical Lur'e systems via partial states is studied. The problem arises in the situation when only partial states of the driven systems are available. By using the method of the variation of parameters for linear impulsive systems and some analysis technique, a sufficient condition for the existence of the impulsive control law for synchronization via partial states is derived. The sufficient condition is given in terms of linear matrix inequalities concerning the interconnection matrices. By using this result, we propose a new impulsive synchronization scheme for a class of Lur'e systems. The new impulsive synchronization scheme only exerts the impulsive input on partial states of the driven system and is characterized by a set of conditions related to the impulsive interval bound, the impulsive magnitude and a coupling condition between them. The proposed impulsive synchronization method is illustrated through Chua's circuit.

Mathematical modeling for nonlinear control: a Hamiltonian approach

Modern model-based nonlinear control requires a good mathematical description of the system we want to control, for both, the system analysis and the controller design. Obviously, the term nonlinear system is too broad, and one is interested in subclasses of nonlinear systems with at least two properties. These classes should cover real world problems, and there should exist controller design methods, powerful enough to admit a systematic design of the closed loop with certain properties. Now, classical Hamiltonian systems have a rich mathematical structure, which has been extended such that dissipative effects and inputs, outputs, or better ports, are included in this class. This contribution starts with a Hamiltonian description of linear time invariant lumped parameter systems to motivate the introduction of certain mathematical ideas, which will be exploited in the nonlinear case afterwards. After that the approach will be extended to the distributed parameter case. Finally, the applicability of the presented methods is shown with the help of a piezoelectric elastic structure.

Visualizing parametric solution sets

We characterize the boundary ∂Σp of the solution set Σp of a parametric linear system A(p)x=b(p) where the elements of the n×n matrix and the right-hand side vector depend on a number of parameters p varying within interval bounds. The characterization of ∂Σp is by means of pieces of parametric hypersurfaces, the latter represented by their coordinate functions depending on corresponding subsets of n-1 parameters. The presented approach has a direct application for efficient visualization of parametric solution sets by utilizing some plotting functions supported by Mathematica and Maple.

On sequential parameter estimation of a linear regression process

This paper presents a sequential estimation procedure for unknown parameters of a stochastic linear regression. As examples the sequential estimation problem of two dynamic parameters in stochastic linear systems with memory and in autoregressive processes is solved. The estimation procedure is based on the least squares method with weights and yields estimators with guaranteed accuracy in the sense of the Lq–norm (q ≥ 2). The proposed procedure works in the mentioned examples for all possible values of the unknown dynamic parameters on the plane R2 with the exception of some lines. The asymptotic behavior of the duration of observations is investigated.

On strong consistency of a class of recursive stochastic Newton-Raphson type algorithms with application to robust linear dynamic system identification

The recursive stochastic algorithms for estimating the parameters of linear discrete-time dynamic systems in the presence of disturbance uncertainty has been considered in the paper. Problems related to the construction of min-max optimal recursive algorithms are demonstrated. In addition, the robustness of the proposed algorithms has been addressed. Since the min-max optimal solution cannot be achieved in practice, an approximate optimal solution based on a recursive stochastic Newton-Raphson type procedure is suggested. The convergence of the proposed practically applicable robustified recursive algorithm is established theoretically using the martingale theory. Both theoretical and experimental analysis related to the practical robustness of the proposed algorithm are also included. .

Seismic Vibration Control of Nonlinear Structures Using the Liquid Column Damper

A design methodology for a variation of the conventional liquid column damper (LCD) (by attaching a spring connection to the LCD) as a seismic vibration control device for structures with nonlinear behavior has been proposed in this paper. The proposed method with the nonconventional damper has the advantage of being applicable to nonlinear structures having high initial stiffness (short natural period) in the linear range below yield with subsequent period lengthening in the postyield phase, where the conventional LCD would be ineffective. The reason for the lack of effectiveness is not only due to the damper parameters designed for the linear structure becoming inapplicable when the structures move to the inelastic regime but also because of the restriction imposed by the high natural period of the conventional damper on its applicability to comparatively stiff structures (for tuning). The methodology in this paper, thereby incorporating the use of a modified model of the LCD system, namely a spring-connected one, removes the requirement on the natural period of the liquid in the LCD and further, the design is based on the parameters of an equivalent linear system for the nonlinear structure. The latter has been represented by a single-degree-of-freedom system with bilinear hysteresis. A procedure for obtaining the equivalent linear system for the nonlinear structure by adopting a temporally averaged linearization technique has been outlined. A few different response parameters known to have damaging effects on structures, such as number/extent of nonlinear excursions of yield level, permanent set, decay of vibration (duration of response), in addition to the reduction in peak response has been the focus of this paper. The efficacy of the proposed method has been examined through simulation studies using recorded accelerograms.

A Nonlinear Observer for Rigid Body Attitude Estimation using Vector Observations

This work proposes a nonlinear observer for attitude estimation on SO(3), exploiting the information of vector observations and biased angular rate measurements. It is shown that the attitude and bias estimation errors converge exponentially fast to the origin, for arbitrary angular velocity trajectories. The proposed attitude feedback law is an explicit function of the vector measurements and observer estimates, and convergence rate bounds are obtained using recent results for parametrized linear time-varying systems. The stability and convergence properties of the estimation errors are evidenced in simulation for time-varying angular velocities.

Detection and identification of nonlinearities by amplitude and frequency modulation analysis

This paper presents an amplitude and frequency modulation method (AFMM) for extracting characteristics of nonlinear systems and intermittent transient responses by processing stationary/transient responses using the empirical mode decomposition, Hilbert–Huang transform (HHT), and nonlinear dynamic characteristics derived from perturbation analysis. A sliding-window fitting (SWF) method is derived and used to show the physical implications of the proposed method and other methods for data processing and transformation. Similar to the wavelet transform, the SWF uses windowed regular harmonics and function orthogonality to extract time-localized regular and/or distorted harmonics, and then the amplitude and frequency modulations of the harmonics are used to identify system nonlinearities. On the other hand, the HHT uses the apparent time scales revealed by the signal's local maxima and minima and cubic splines of the extrema to sequentially sift components of different time scales, starting from high-frequency to low-frequency ones. Because HHT does not use predetermined basis functions and function orthogonality for component extraction, it provides more accurate instant amplitudes and frequencies of extracted components for accurate estimation of system characteristics and nonlinearities. Moreover, because the first component extracted by HHT contains all original discontinuities, its time-varying amplitude and frequency are excellent indicators for pinpointing times and locations of impulsive external loads. However, the discontinuity-induced Gibbs’ phenomenon makes HHT analysis inaccurate around the two data ends. On the other hand, the SWF analysis is not affected by Gibbs’ phenomenon, but it cannot extract accurate time-varying frequencies and amplitudes because of the use of predetermined basis functions, function orthogonality, and windowed curve fitting for component extraction. Numerical results show that the proposed AFMM can provide accurate estimation of softening and hardening effects, different orders of nonlinearity, linear and nonlinear system parameters, and time instants of intermittent transient responses.

An importance sampling procedure for estimating failure probabilities of non-linear dynamic systems subjected to random noise

The first passage problem for linear and non-linear oscillators excited by white and coloured noise are considered. An iterative variance reduction scheme is used in a framework of a measure change in the space of sample functions according to the Girsanov transformation, which is based on introducing a Markov control process. It is proved that a good approximation to the optimal stochastic control process can be obtained from an equivalent white noise excited linear oscillator. It is shown that this leads to very accurate estimates of the failure probability of the original system. The advantage of this procedure is that expressions for the parameters of the equivalent linear system and the design point oscillations, which are needed to find the control process, are available analytically. The number of samples, the variance of the failure probability estimates and the computational time are reduced significantly compared with direct Monte Carlo simulations.

Tekrarlamalı Gauss-Seidel Yardımcı Değişkenler Algoritması İle Transfer Fonksiyonu Parametrelerinin Yansız Tahmini

In this paper, a recursive algorithm based on the use of Gauss-Seidel iterations and instrumental variables together is introduced for unbiased estimation of transfer function parameters of linear time invariant discrete-time systems. Furthermore, a stochastic convergence analysis of the proposed algorithm is performed and it is shown that the proposed algorithm is an unbiased parameter estimator that gives optimum solution of normal equations even if the measurement noise is colored. The proposed algorithm is used for estimation of transfer function parameters of a sample second order system and compared with similar algorithms by a simulation study. According to the results obtained, it is shown that the proposed algorithm is a good alternative to the others by viewpoints of computational complexity and convergence rate.

Orthoregressive estimates for the parameters of systems of linear difference equations

The problem of estimating the parameters of linear difference systems of equations from the observations of the segments of solutions with additive stochastic perturbations is considered. The methods of multivariate orthogonal regression are applied to obtain the estimators. The results of comparative study of the methods are exposed from the standpoint of information on linear constraints in observations. The properties of the estimators in the limit cases of a gross sample are studied. For small perturbations, a scheme for comparison of estimators by linear approximations is proposed.

A robust semi-definite optimization based solution to the robust order reduction problem for parametric uncertain dissipative linear systems

In this paper, we address the problem of reducing the order of a linear system affected by uncertainties from the robust dissipative perspective introduced in Barb et al. (2003, Robust dissipativity of interval uncertain linear systems. SIAM J. Control Optim., 41, 1661–1695.). First, we show that all major balanced truncation techniques developed and reported in the literature of the last two decades can be treated in a uniform fashion within the framework of dissipative systems. Accordingly, we shall generalize these results to uncertain dissipative systems. The key role is played by balancing two positive definite robust solutions to the uncertain dissipativity linear matrix inequalities (LMIs) associated with the linear system in question and its dual. Determining the maximal level of uncertainty for which such two solutions exist and computing them efficiently is well known to be non-poly NP-hard. Our method is based on determining robust tractable approximations of these NP-hard entities by following the novel method known as Matrix-Cube theory. The proven results are accompanied by a numerical example.

Solving parametric fuzzy systems of linear equations by a nonlinear programming method

Linear systems of equations, with uncertainty on the parameters, play a major role in various problems in economics and finance. In this paper parametric fuzzy linear systems of the general form A 1 x + b 1 = A 2 x + b 2, with A 1, A 2, b 1 and b 2 matrices with fuzzy elements, are solved by means of a nonlinear programming method. The relation between this methodology and the algorithm proposed in Muzzioli and Reynaerts [(2006) Fuzzy Sets and Systems, in press] is highlighted. The methodology is finally applied to an economic and a financial problem.

Condensed Model Identification and Recovery for Structural Damage Assessment

This study aims to develop a system identification methodology for determining structural parameters of linear dynamic system, taking into consideration of practical constraints such as insufficient sensors. A new methodology called the condensed model identification and recovery method is presented for identification of full stiffness matrices for damage assessment based on incomplete measurement. With the proposed methodology, it is possible to obtain several condensed stiffness matrices, so as to identify individual stiffness coefficients for the structure. Three different model condensation methods, namely static condensation, dynamic condensation, and system equivalent reduction expansion process, are adopted. Having identified the condensed model, the stiffness parameters in the entire system are recovered by extracting sufficient information with either fixed sensor or repositioned sensor approach. The efficiency of the proposed technique is shown by numerical simulation study for multistory shear buildings subjected to random forces, accounting for effects of signal noise. In addition, laboratory experiments are carried out to illustrate the performance of the proposed method. It is shown both numerically and experimentally that the proposed methodology gives reasonably accurate identification in terms of locating and quantifying structural damage.

Best performing parameters of linear and non-linear seismic base-isolator systems obtained by the power flow analysis

In this paper the problem of finding the best performing base isolator parameters of a seismic excited structure is treated. The structural and earthquake parameters and the isolator type are considered assigned. The approach here proposed is based on the minimization of the power flow transmitted by the isolator to the structure. The power flow is a statistical measure of the power transmitted to the structure during an earthquake. The earthquake is idealized as a filtered stochastic process. This approach considers the structure–isolator–excitation system as a super-structure in which the structure, the isolator and the excitation filter are the sub-structures. It guarantees a very good global behaviour of the structure, both in terms of relative displacements and in terms of accelerations. The isolator types considered in this work are the linear, the yielding and the friction ones.